==== TABLE OF CONTENTS ====
Sharp-nodal-length-on-smooth-surfaces//bibliography.tex
Sharp-nodal-length-on-smooth-surfaces//descent.tex
Sharp-nodal-length-on-smooth-surfaces//introduction.tex
Sharp-nodal-length-on-smooth-surfaces//length.tex
Sharp-nodal-length-on-smooth-surfaces//paper.tex
Sharp-nodal-length-on-smooth-surfaces//persistence.tex
Sharp-nodal-length-on-smooth-surfaces//preamble.tex
Sharp-nodal-length-on-smooth-surfaces//profiles.tex
Sharp-nodal-length-on-smooth-surfaces//subdivision.tex
===========================

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//bibliography.tex ====
\newpage
\begin{thebibliography}{99}

\bibitem{Bruning1978}
J.~Br\"uning,
\emph{\"Uber Knoten von Eigenfunktionen des Laplace--Beltrami-Operators},
Math. Z. \textbf{158} (1978), no.~1, 15--21.
\href{https://doi.org/10.1007/BF01214561}{doi:10.1007/BF01214561}.

\bibitem{DeTurckKazdan1981}
D.~M.~DeTurck and J.~L.~Kazdan,
\emph{Some regularity theorems in Riemannian geometry},
Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{14} (1981), no.~3, 249--260.
\href{https://doi.org/10.24033/asens.1405}{doi:10.24033/asens.1405}.

\bibitem{DonnellyFefferman1988}
H.~Donnelly and C.~Fefferman,
\emph{Nodal sets of eigenfunctions on Riemannian manifolds},
Invent. Math. \textbf{93} (1988), no.~1, 161--183.
\href{https://doi.org/10.1007/BF01393691}{doi:10.1007/BF01393691}.

\bibitem{DonnellyFefferman1990}
H.~Donnelly and C.~Fefferman,
\emph{Nodal sets for eigenfunctions of the Laplacian on surfaces},
J. Amer. Math. Soc. \textbf{3} (1990), no.~2, 333--353.
\href{https://doi.org/10.1090/S0894-0347-1990-1035413-2}{doi:10.1090/S0894-0347-1990-1035413-2}.

\bibitem{Dong1992}
R.-T.~Dong,
\emph{Nodal sets of eigenfunctions on Riemann surfaces},
J. Differential Geom. \textbf{36} (1992), no.~2, 493--506.
\href{https://doi.org/10.4310/JDG/1214448750}{doi:10.4310/JDG/1214448750}.

\bibitem{ImbertSilvestre2013}
C.~Imbert and L.~Silvestre,
\emph{$C^{1,\alpha}$ regularity of solutions of some degenerate fully
non-linear elliptic equations},
Adv. Math. \textbf{233} (2013), 196--206.
\href{https://doi.org/10.1016/j.aim.2012.07.033}{doi:10.1016/j.aim.2012.07.033}.
The numbering cited here is that of
\href{https://arxiv.org/abs/1201.3739v1}{arXiv:1201.3739v1}.

\bibitem{Logunov2018Lower}
A.~Logunov,
\emph{Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's
conjecture and of the lower bound in Yau's conjecture},
Ann. of Math. (2) \textbf{187} (2018), no.~1, 241--262.
\href{https://doi.org/10.4007/annals.2018.187.1.5}{doi:10.4007/annals.2018.187.1.5}.

\bibitem{LogunovMalinnikova2018}
A.~Logunov and E.~Malinnikova,
\emph{Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff
measure in dimensions two and three},
in \emph{50 Years with Hardy Spaces},
A.~Baranov, S.~Kisliakov, and N.~Nikolski (eds.),
Operator Theory: Advances and Applications, vol.~261,
Birkh\"auser, Cham, 2018, pp.~333--344.
\href{https://doi.org/10.1007/978-3-319-59078-3_17}{doi:10.1007/978-3-319-59078-3\_17}.

\bibitem{NazarovPolterovichSodin2005}
F.~Nazarov, L.~Polterovich, and M.~Sodin,
\emph{Sign and area in nodal geometry of Laplace eigenfunctions},
Amer. J. Math. \textbf{127} (2005), no.~4, 879--910.
\href{https://doi.org/10.1353/ajm.2005.0030}{doi:10.1353/ajm.2005.0030}.

\bibitem{Ransford1995}
T.~Ransford,
\emph{Potential Theory in the Complex Plane},
London Mathematical Society Student Texts, vol.~28,
Cambridge University Press, Cambridge, 1995.

\bibitem{RoyFortin2015}
G.~Roy-Fortin,
\emph{Nodal sets and growth exponents of Laplace eigenfunctions on surfaces},
Anal. PDE \textbf{8} (2015), no.~1, 223--255.
\href{https://doi.org/10.2140/apde.2015.8.223}{doi:10.2140/apde.2015.8.223}.

\bibitem{RoyFortinLq}
G.~Roy-Fortin,
\emph{Average $L^q$ growth and nodal sets of eigenfunctions of the Laplacian
on surfaces},
preprint, 2015.
\href{https://arxiv.org/abs/1510.02376}{arXiv:1510.02376}.

\bibitem{Yau1982}
S.-T.~Yau,
\emph{Problem section},
in \emph{Seminar on Differential Geometry}, S.-T.~Yau (ed.),
Annals of Mathematics Studies, vol.~102,
Princeton University Press, Princeton, NJ, 1982, pp.~669--706.

\end{thebibliography}
==== END Sharp-nodal-length-on-smooth-surfaces//bibliography.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//descent.tex ====
\section{Descent and stability}\label{sec:descent}

We iterate the subdivision estimate while the untilted growth or the
tilt is large. The first goal is a bound, independent of the number of
levels, for the total residual growth and the accumulated changes of
tilt. This bound will show that a sufficiently large initial tilt with
small residual growth forces high untilted growth to persist on a
positive fraction of paths.

Fix the integer \(A\) and constants \(C_0,C_1\) from
Proposition~\ref{prop:subdivision}.  We may increase \(C_0\) so that
\(C_0\ge 1\).
Consider a finite subdivision tree with all terminal squares at the same
level, and suppose that
\[
K_Q=k s(Q)\ge a_0>0
\]
on every square of the tree.  The square at the start of a subtree will
be denoted by \(I\), and its side satisfies \(s(I)\le 1\).
All disks used below are assumed to lie in the coordinate region of
Section~\ref{sec:subdivision}.  The arguments also apply to a subtree
in an extension of the same grid whenever its disks have this property.
Throughout this section, constants may depend on the fixed number
\(a_0\), as well as on the coordinate data and the constants already
chosen.

Choosing a child uniformly among the \(A^2\) children at each step
defines a random path through the tree.  Equivalently, one may choose a
point uniformly in \(I\) and follow its containing squares, ignoring
grid boundaries of area zero.

\begin{lemma}[One step of descent]\label{lem:descent}
There is a fixed \(L>0\) such that, at any nonterminal square \(Q\),
if \(N(Q)>L\) or \(\abs{b}>L\), one can assign a tilt \(b_q\) to each
child \(q\) so that, writing
\(m=m(Q,b)\), \(m'=m(q,b_q)\), and \(b'=b_q\), one has
\begin{equation}\label{eq:descent}
 \begin{gathered}
 \E\bigl(\abs{b'-b}\mid Q,b\bigr)\le C_1(m+f_Q),
 \qquad
 \E\bigl(m'\mid Q,b\bigr)\le \tfrac12(m+f_Q),\\
 f_Q=\frac{C_0}{K_Q}+s(Q).
 \end{gathered}
\end{equation}
Moreover, if \(Q_0=I,Q_1,\ldots,Q_d\) is any path from \(I\) to
the terminal level, then
\begin{equation}\label{eq:floors}
 \sum_{i=0}^{d}f_{Q_i}
 \le \frac{A}{A-1}
       \left(\frac{C_0}{a_0}+s(I)\right).
\end{equation}
\end{lemma}

\begin{proof}
Write \(K=K_Q\), \(s=s(Q)\), and set
\[
 S=E(Q,b)+C_0+Ks,\qquad B=Kb.
\]
Thus \(S\ge C_0\), \(E(Q,b)\le S\), and \(S\ge Ks\).
Equation~\eqref{eq:untilted-comparison} gives
\[
 N(Q)\le m(Q,b)+(R+1)\abs{b}
       \le (R+1)\bigl(m(Q,b)+\abs{b}\bigr).
\]
Consequently \(S+\abs{B}\ge KN(Q)/(R+1)\); also
\(S+\abs{B}\ge K\abs{b}\).
Choose, for example, \(L=2C_0(R+1)\).
Either of the alternatives in the lemma then implies
\[
 \frac{K}{S+\abs{B}}\le \frac1{C_0}.
\]
Using \(S\ge Ks\), we obtain the other required smallness condition,
\[
 \frac{K^2s}{S(S+\abs{B})}
 \le \frac{K}{S+\abs{B}}\le \frac1{C_0}.
\]
All the hypotheses of Proposition~\ref{prop:subdivision} hold.
Its conclusions, with \(S/K=m(Q,b)+f_Q\), are precisely
\eqref{eq:descent}.

Along a path, \(s(Q_i)=A^{-i}s(I)\) and
\(K_{Q_i}=A^{d-i}K_{Q_d}\).  Since \(K_{Q_d}\ge a_0\),
\[
 \sum_{i=0}^d \frac{C_0}{K_{Q_i}}
 \le \frac{C_0}{a_0}\sum_{\ell=0}^d A^{-\ell},
 \qquad
 \sum_{i=0}^d s(Q_i)
 =s(I)\sum_{i=0}^d A^{-i}.
\]
The geometric-series bound gives \eqref{eq:floors}.
\end{proof}

We next allow descent to stop according to the values encountered
along a path.  A stopping rule is understood to use only the squares
and tilts already visited.  In particular, a violation first detected
on arrival at a child stops the procedure at that child.

\begin{lemma}[Stopped descent]\label{lem:stopped-descent}
Start at \(I\) with a prescribed tilt \(b_I\).  Whenever a step is
made, choose the child tilts by Lemma~\ref{lem:descent}, and suppose
that its applicability condition holds at every square from which
the procedure continues.  Stop at a time \(T\), no later than the
terminal level.  Write \(m_i=m(Q_i,b_i)\) for the visited values,
including the last value \(m_T\).  There is a constant
\(C_{\mathrm{st}}\), independent of the depth and of the stopping
rule, such that
\begin{equation}\label{eq:stopped-descent}
 \E\left[
      \sum_{i=0}^{T}m_i
      +\sum_{i=0}^{T-1}\abs{b_{i+1}-b_i}
    \right]
 \le C_{\mathrm{st}}\bigl(m(I,b_I)+1\bigr).
\end{equation}
The second sum includes the step producing the last square, even if
that square causes the procedure to stop.
\end{lemma}

\begin{proof}
There are only finitely many nodes.  At each node from which descent
continues, choose one of the assignments furnished by
Lemma~\ref{lem:descent}.  These choices and the stopping rule determine
all random variables on the finite probability space of terminal
paths.  Thus only finite conditional-expectation identities are
needed.  In expressions with a fixed index, set \(m_i=0\) after
stopping; those auxiliary values do not enter the stopped sums.

Let \(d\) be the depth below \(I\), put \(m_0=m(I,b_I)\), and set
\[
 F_I=\frac{A}{A-1}\left(\frac{C_0}{a_0}+s(I)\right),
 \qquad
 M=\E\sum_{i=0}^{T}m_i.
\]
The event \(\{i<T\}\) is determined before the next child is selected.
Applying \eqref{eq:descent} on this event and summing over
\(0\le i<d\) gives
\[
 M-m_0
 =\sum_{i=0}^{d-1}\E\bigl[\mathbf 1_{\{i<T\}}m_{i+1}\bigr]
 \le \frac12\E\sum_{i=0}^{T-1}(m_i+f_{Q_i})
 \le \frac12 M+\frac12 F_I.
\]
All the \(m_i\) are nonnegative by Definition~\ref{def:growth}.
It follows that \(M\le 2m_0+F_I\).
The tilt estimate in \eqref{eq:descent} similarly yields
\[
 \E\sum_{i=0}^{T-1}\abs{b_{i+1}-b_i}
 \le C_1\E\sum_{i=0}^{T-1}(m_i+f_{Q_i})
 \le 2C_1m_0+2C_1F_I.
\]
Their sum is at most
\[
 2(1+C_1)m_0+(1+2C_1)F_I.
\]
Since \(s(I)\le1\), this is bounded by the right-hand side of
\eqref{eq:stopped-descent} with a constant depending only on the fixed
data.  This calculation includes \(T=0\), and it includes both \(m_T\)
and the incoming increment when the last square violates a stopping
condition.
\end{proof}

A large tilt whose remaining excess is small forces the original,
untilted growth to be large.  We give a separate size threshold and
ratio threshold for this statement.

\begin{lemma}[Large slope forces large growth]\label{lem:large-slope}
For the fixed \(L\) in Lemma~\ref{lem:descent}, there are constants
\(B_*\ge1\) and \(\eta_*>0\) such that every admissible square with
\(s(Q)\le1\) and \(K_Q\ge a_0\) satisfies
\begin{equation}\label{eq:large-slope}
 \abs{b}\ge B_*,
 \qquad m(Q,b)\le\eta_*\abs{b}
 \quad\Longrightarrow\quad N(Q)>L.
\end{equation}
\end{lemma}

\begin{proof}
If no such pair of thresholds existed, then for each positive integer
\(j\) there would be a square \(Q_j\) and a tilt \(b_j\) with
\[
 \abs{b_j}\ge j,\qquad
 \frac{m(Q_j,b_j)}{\abs{b_j}}\le\frac1j,\qquad
 N(Q_j)\le L.
\]
Rescale about the center of \(Q_j\) by its side \(s_j=s(Q_j)\).
In the rescaled coordinates, \(U_j\) solves
\(\Delta U_j+K_j^2p_jU_j=0\) on \(\D_R\), where
\(K_j=K_{Q_j}\), \(\norm{p_j}_\infty\le C\), and
\(\norm{\nabla p_j}_\infty\le Cs_j\).
Set
\[
 S_j=K_j\abs{b_j},\qquad B_j=K_jb_j,\qquad
 D_j=S_j+\abs{B_j}=2S_j.
\]
The hypotheses of Theorem~\ref{thm:subharmonic-profile} follow from
\[
 S_j\ge a_0j\longrightarrow\infty,\qquad
 \frac{K_j}{D_j}=\frac1{2\abs{b_j}}\longrightarrow0,\qquad
 \frac{K_j^2\norm{\nabla p_j}_\infty}{S_jD_j}
 \le\frac{Cs_j}{2\abs{b_j}^2}\longrightarrow0.
\]
Normalize the tilted measures
\[
 d\mu_j(x)=c_j e^{-2B_j\cdot x}\abs{U_j(x)}^2\,dx
\]
to have mass one on \(\D_R\).  By
Lemma~\ref{lem:profile-extraction} and
Theorem~\ref{thm:subharmonic-profile}, a subsequence has a
subharmonic profile \(V\le0\), with the identically minus infinity
case initially allowed.  The definitions give
\[
 \frac{\log\mu_j(\D_1)}{2S_j}
 =-\frac{E(Q_j,b_j)}{S_j}
 =-\frac{m(Q_j,b_j)}{\abs{b_j}}\longrightarrow0.
\]
The upper bound in \eqref{eq:profile-bounds} on the compact set
\(\overline{\D_1}\) therefore implies
\(\sup_{\overline{\D_1}}V=0\).
In particular the profile is nontrivial and attains its maximum at
an interior point of \(\D_R\).  The strong maximum principle yields
\(V=0\) on \(\D_R\).

Pass to a further subsequence so that
\(e_j=b_j/\abs{b_j}\) converges to a unit vector \(e\).
Undoing the tilt expresses the untilted growth as
\[
 \frac{E(Q_j,0)}{S_j}
 =
 \frac{1}{2S_j}\log
       \int_{\D_R}e^{2S_je_j\cdot x}\,d\mu_j(x)
 -
 \frac{1}{2S_j}\log
       \int_{\D_1}e^{2S_je_j\cdot x}\,d\mu_j(x).
\]
For any fixed \(1<r<R\), bound the first integral from below by its
restriction to \(\D_r\).  The weighted form of
Lemma~\ref{lem:profile-extraction}, applied from below on the open
disk \(\D_r\) and from above on the compact disk
\(\overline{\D_1}\), gives
\[
 \liminf_{j\to\infty}\frac{E(Q_j,0)}{S_j}
 \ge
 \sup_{\D_r} e\cdot x
 -\sup_{\overline{\D_1}} e\cdot x
 =r-1.
\]
Letting \(r\uparrow R\) gives a lower bound of \(R-1\), without any
claim about mass at the boundary of \(\D_R\).  On the other hand,
\[
 0\le\frac{E(Q_j,0)}{S_j}
 =\frac{N(Q_j)}{\abs{b_j}}
 \le\frac{L}{j}\longrightarrow0,
\]
a contradiction.
\end{proof}

\begin{proposition}[Stability of large growth]\label{prop:stability}
There are constants \(P\ge1\) and \(\sigma>0\) with the following
property.  Suppose a square \(I\) and a tilt \(b_I\) satisfy
\begin{equation}\label{eq:stability-hypotheses}
 \abs{b_I}\ge P,\qquad m(I,b_I)\le\sigma\abs{b_I}.
\end{equation}
Then at least half the paths from \(I\) to the terminal level satisfy
\(N(Q)>L\) at every visited square \(Q\), including \(I\) and the
terminal square.  Equivalently, these paths occupy a measurable set
of area at least \(\abs{I}/2\).
\end{proposition}

\begin{proof}
Let \(B_*,\eta_*\) be the thresholds from
Lemma~\ref{lem:large-slope}.  Choose \(\gamma\) so that
\[
 0<\gamma<\frac12,\qquad
 \frac{\gamma}{1-\gamma}\le\eta_*.
\]
Let \(C_{\mathrm{st}}\) be the constant from
Lemma~\ref{lem:stopped-descent}.  Choose
\[
 0<\sigma\le\min\left\{\frac{\gamma}{2},
                         \frac{\gamma}{8C_{\mathrm{st}}}\right\},
 \qquad
 P\ge\max\left\{1,\frac{B_*}{1-\gamma},
                  \frac{2L}{1-\gamma},
                  \frac{8C_{\mathrm{st}}}{\gamma}\right\}.
\]
Write \(B_0=\abs{b_I}\).  Starting at \(I\), descend using
Lemma~\ref{lem:descent} while
\[
 \abs{b_i-b_I}<\gamma B_0,\qquad m_i<\gamma B_0,
\]
and stop at the first failure of either inequality or at the terminal
level, whichever comes first.  These inequalities hold initially by
\eqref{eq:stability-hypotheses}.  Whenever they hold, we have
\[
 \abs{b_i}>(1-\gamma)B_0\ge B_*,
 \qquad \abs{b_i}>L,
 \qquad
 \frac{m_i}{\abs{b_i}}
 <\frac{\gamma}{1-\gamma}\le\eta_*.
\]
Thus every required step is available and
Lemma~\ref{lem:large-slope} gives \(N(Q_i)>L\) at each such square.

Let \(T\) be this stopping time.  Include in the failure event a
violation first produced at the terminal level.
If a failure occurs, then either \(m_T\ge\gamma B_0\), or the
triangle inequality gives
\(\sum_{i=0}^{T-1}\abs{b_{i+1}-b_i}\ge\gamma B_0\).
Consequently Lemma~\ref{lem:stopped-descent} and Markov's inequality
give
\[
 \begin{split}
 \mathbb P(\text{failure})
 &\le
 \frac{1}{\gamma B_0}
 \E\left[\sum_{i=0}^{T}m_i+
         \sum_{i=0}^{T-1}\abs{b_{i+1}-b_i}\right]\\
 &\le\frac{C_{\mathrm{st}}}{\gamma}
                \left(\sigma+\frac1P\right)
 \le\frac14.
 \end{split}
\]
On the complementary event, descent reaches the terminal level and
the two strict inequalities hold at every visited square.
All of those squares have \(N>L\).  This event has probability at
least \(3/4\), which proves the asserted lower bound of \(1/2\).
The same reasoning includes a subtree of depth zero.
\end{proof}
==== END Sharp-nodal-length-on-smooth-surfaces//descent.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//introduction.tex ====
\section{Introduction and main result}\label{sec:introduction}

For a Riemannian metric $g$, let $\Delta_g=\operatorname{div}_g\nabla_g$,
and write $Z_u=\{x:u(x)=0\}$ for the nodal set of a real function $u$.
We prove the following bound for one arbitrary fixed smooth surface.

\begin{theorem}\label{thm:main}
Let $(M,g)$ be a closed connected $C^\infty$ Riemannian surface.
There is a finite constant $C=C(M,g)$ such that every nonzero real
Laplace eigenfunction satisfying
\[
 -\Delta_g u=\lambda u,\qquad \lambda>0,
\]
obeys
\[
 \mathcal H_g^1(Z_u)\le C\sqrt\lambda.
\]
Here $\mathcal H_g^1$ is Hausdorff measure for the Riemannian distance.
The same constant applies to every eigenfunction in every positive
eigenspace.
\end{theorem}

Yau's conjecture asks for upper and lower bounds of order
$\sqrt\lambda$ for the $(n-1)$-dimensional measure of the nodal set
on a smooth closed $n$-dimensional manifold \cite{Yau1982}.
Donnelly and Fefferman proved both bounds for real analytic manifolds
and metrics \cite[Theorem~1.2]{DonnellyFefferman1988}.
For smooth surfaces, Br\"uning established the lower bound
\cite[Satz~2]{Bruning1978}; Logunov subsequently proved the lower bound
in every dimension \cite[Theorem~1.1]{Logunov2018Lower}.
For the surface upper bound with a smooth metric, Donnelly and
Fefferman proved $C\lambda^{3/4}$ \cite{DonnellyFefferman1990},
and Dong gave a different proof at the same order \cite{Dong1992}.
Logunov and Malinnikova then obtained
$C\lambda^{3/4-\beta}$ for a universal $\beta\in(0,1/4)$
\cite[Equation~(1.1)]{LogunovMalinnikova2018}.
Together with the known lower bound, Theorem~\ref{thm:main}
resolves Yau's conjecture positively for smooth closed surfaces.

\subsection*{Mean growth as the target}

Set $k=\sqrt\lambda$, the frequency. In local smooth isothermal
coordinates the metric is $g=p(x)(dx_1^2+dx_2^2)$, with $p>0$ smooth,
as in \cite[Section~1]{DeTurckKazdan1981}, and the eigenfunction
equation becomes
\[
 \Delta u+k^2p(x)u=0.
\]
Nazarov, Polterovich, and Sodin advocated a statistical approach to
local doubling and sketched its relation to nodal length, building on
Nadirashvili's approach
\cite[Section~7.3]{NazarovPolterovichSodin2005}.
Roy-Fortin established precise comparisons between growth and nodal
length \cite{RoyFortin2015,RoyFortinLq}. We prove the uniform mean
growth bound needed for the sharp upper estimate and use his planar
theorem for the final transfer.

Here is why an average suffices. For a coordinate square $Q$ of side
$s$ and center $x_Q$, consider the unweighted logarithmic growth
\[
 E(Q,0)=\log
 \frac{\|u\|_{L^2(\D_{Rs}(x_Q))}}
      {\|u\|_{L^2(\D_s(x_Q))}},
\]
where $R$ is the fixed large disk ratio chosen below and $\D_r(x)$ is
the Euclidean disk of radius $r$. The norms are unnormalized.
Subdivide a fixed coordinate square into congruent squares of side
$s$ comparable to $k^{-1}$, stopping once $ks$ is sufficiently small;
call these the terminal squares.
The estimate proved below gives
\[
 \av_{Q\text{ terminal}} E(Q,0)\le C.
\]
On each such square, Roy-Fortin's theorem, after rescaling and a
change from supremum norms to $L^2$ norms, gives
\[
 \mathcal H^1(Q\cap Z_u)\le Cs\bigl(1+E(Q,0)\bigr).
\]
There are $O(s^{-2})$ squares in a fixed coordinate region.
Summing the local estimates therefore gives $O(s^{-1})=O(k)$
nodal length. The precise grids, radii and constants are specified
in Sections~\ref{sec:subdivision} and~\ref{sec:length}.
The task is thus to control the average growth while allowing large
values on individual squares.

\subsection*{Affine correction and persistent growth}

The analytic step extracts logarithmic profiles from norms measured
on an exponential scale. With the required separation from the rescaled
frequency $ks$, a localized Carleman estimate makes these profiles
subharmonic, even after arbitrarily large affine tilts. Its decisive
feature is two-dimensional: when the normalized derivative energy
is perpendicular to the weight gradient, the commutator sees the
full trace of the weight Hessian.

A local affine part of a profile presents an obstruction to decay:
its oscillation and $ks$ decrease by the same factor
under subdivision. We subtract that affine part and carry its slope
separately. The remaining profile growth decreases faster than the
side length on average, while the changes of slope have bounded
average cost. Section~\ref{sec:subdivision} quantifies both estimates;
Section~\ref{sec:descent} sums their costs along paths that may stop.

To use this improvement through many scales, we divide a grid path
into excursions of high growth. A newly entered excursion produces
a nearby set of positive area on which high growth persists to the
terminal scale. These witness sets may lie outside their entry
squares. A packing argument controls them by counting discrepancies
between spatial membership and grid ancestry; those discrepancies
are confined to summable boundary collars. This geometric argument
is stated separately in Lemma~\ref{lem:packing}. It permits the
analytic persistence statement to be used without requiring its
witnesses to lie inside the original squares.

Section~\ref{sec:profiles} constructs the subharmonic limits and
establishes the initial growth bound. Section~\ref{sec:subdivision}
proves the affine subdivision estimate, and
Section~\ref{sec:descent} controls stopped descents and persistent
slopes. Section~\ref{sec:persistence} proves persistence at entry,
packs the excursions, and bounds the terminal mean growth.
Section~\ref{sec:length} completes the nodal-length estimate.
All coordinate constructions are local, so no orientability
assumption is needed.
==== END Sharp-nodal-length-on-smooth-surfaces//introduction.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//length.tex ====
\begin{samepage}
\section{Nodal length at the wavelength scale}\label{sec:length}

We first state the planar estimate that relates nodal length to
growth. The disks in this section are Euclidean.

\begin{theorem}[Roy-Fortin]\label{thm:local-length}
There are constants $\eps_0>0$ and $C>0$ with the following property.
Let $F$ be a nonzero real solution of
\[
 \Delta F+qF=0\quad\text{on }\D_3,
 \qquad q\in C^\infty(\D_3),\quad \norm{q}_\infty<\eps_0.
\]
If
\[
 \beta=\log\frac{\norm{F}_{L^\infty(\D_{5/2})}}
                         {\norm{F}_{L^\infty(\D_{1/4})}},
\]
then
\[
 \mathcal H^1(Z_F\cap\D_{1/60})\le C\max\{1,\beta\}.
\]
\end{theorem}
\end{samepage}

This is \cite[Theorem~2.1.1]{RoyFortin2015}. The potential need not
have a constant sign. We use the following consequence, keeping
the norm and length scales explicit.

\begin{lemma}[Nodal length from local $L^2$ growth]\label{lem:wavelength-length}
Fix the coefficient bounds and coordinate neighborhood from
Section~\ref{sec:subdivision}. There is an $\eps>0$ such that,
for every admissible square $I$ of side $s$ with $ks\le\eps$,
\begin{equation}\label{eq:wavelength-length}
 \mathcal H^1(I\cap Z_u)\le Cs\bigl(1+E(I,0)\bigr).
\end{equation}
The constants are independent of $u$, $k$, and $s$.
\end{lemma}

\begin{proof}
Let $x_I$ be the center of $I$, and put
\[
 F(z)=u(x_I+60sz),\qquad q(z)=(60ks)^2p(x_I+60sz).
\]
The disk of physical radius $180s$ is available because $R=1000$.
Choose $\eps$ so small that
$(60\eps)^2\norm{p}_\infty<\eps_0$ uniformly over the coordinate
regions in use. Then $F$ satisfies Theorem~\ref{thm:local-length}.

An interior $H^2$ estimate for $\Delta F=-qF$, followed by the
two-dimensional Sobolev embedding, gives
\[
 \norm{F}_{L^\infty(\D_{5/2})}
 \le C\norm{F}_{L^2(\D_{11/4})}.
\]
Only the uniform bound on $q$ is needed here. On the smaller disk,
\[
 \norm{F}_{L^2(\D_{1/4})}\le
 |\D_{1/4}|^{1/2}\norm{F}_{L^\infty(\D_{1/4})}.
\]
Consequently
\[
 \beta\le C+
 \log\frac{\norm{F}_{L^2(\D_{11/4})}}
               {\norm{F}_{L^2(\D_{1/4})}}.
\]
The same $L^2$ version of the local length bound is recorded directly
in \cite[Theorem~3.1.1]{RoyFortinLq}.

In physical coordinates the two norm radii are $165s$ and $15s$.
Their common change-of-variables factor cancels, and nesting gives
\[
 \log\frac{\norm{F}_{L^2(\D_{11/4})}}
               {\norm{F}_{L^2(\D_{1/4})}}
 =\log\frac{\norm{u}_{L^2(\D_{165s}(x_I))}}
                {\norm{u}_{L^2(\D_{15s}(x_I))}}
 \le E(I,0).
\]
All these norms are nonzero by Proposition~\ref{prop:initial-growth}.
The length disk becomes $\D_s(x_I)$ and contains the closed square
$I$, whose corner distance is $s/\sqrt2$. Hausdorff length scales
by $60s$. Theorem~\ref{thm:local-length} therefore yields
\eqref{eq:wavelength-length}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Choose finitely many smooth isothermal neighborhoods and root
squares $Q_0$ whose interiors cover $M$. Make their sides
$s_0\le1$ small enough that each closed disk of radius $2Rs_0$
about a root center is contained in its coordinate neighborhood.
The conformal factors and their first derivatives have uniform
bounds on the finitely many regions used below, and the conformal
factors are bounded away from zero there. Proposition~\ref{prop:initial-growth}
gives a uniform initial bound $N(Q_0)\le C_{\mathrm{init}}$.

Choose the subdivision constants from
Proposition~\ref{prop:subdivision}, and then the small
$\eps$ of Lemma~\ref{lem:wavelength-length}. Write
$k_{\min}=\sqrt{\lambda_1(M,g)}>0$. A single permissible lower
scale bound for all roots is
\[
 a_0=\min\left\{\frac{\eps}{A},\,
           k_{\min}\min_{Q_0}s(Q_0)\right\}>0.
\]
Now fix the descent and persistence constants with this $a_0$.
This order of choices makes every constant depend only on the
fixed metric and cover.

For a given eigenfunction, subdivide each root to the first level
at which $ks\le\eps$, where $s$ is that level's common side.
If this level is below the root, its parent has $kAs>\eps$, so
$ks>\eps/A$. If it is the root, then $ks\ge k_{\min}s_0$.
In both cases, every visited square satisfies $K_Q\ge a_0$.
Theorem~\ref{thm:mean-growth} therefore gives
\[
 \av_{I\text{ terminal in }Q_0}E(I,0)\le Cks.
\]
There are $(s_0/s)^2$ terminal squares. Applying
Lemma~\ref{lem:wavelength-length} and summing their lengths yields
\begin{align*}
 \mathcal H^1(Q_0\cap Z_u)
 &\le Cs\left(\frac{s_0}{s}\right)^2(1+Cks)\\
 &\le Cs_0^2(1/s+k)
 \le Cs_0^2(a_0^{-1}+1)k.
\end{align*}
Subadditivity suffices even if a nodal arc lies on a grid edge.

On each convex coordinate square, the inverse coordinate map is
Lipschitz into $(M,d_g)$ with constant at most
$\sqrt{\sup p}$: the coordinate line segment is an admissible
path for the Riemannian distance. Hence the preceding Euclidean
Hausdorff-length bound gives the same bound for $\mathcal H_g^1$
up to a fixed factor. Sum over the finite root cover to obtain
$\mathcal H_g^1(Z_u)\le C(M,g)k$.
The coordinate choices and all constants were independent of
the eigenfunction and of its eigenspace. Since $k=\sqrt\lambda$,
this proves the theorem.
\end{proof}
==== END Sharp-nodal-length-on-smooth-surfaces//length.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Sharp nodal length on smooth surfaces}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that a nonzero real Laplace eigenfunction with eigenvalue $\lambda>0$
on a fixed smooth closed connected Riemannian surface has nodal length at most
$C\sqrt\lambda$. Together with the known lower bound, this proves
Yau's conjecture in this setting.
\end{abstract}
\input{introduction}
\input{profiles}
\input{subdivision}
\input{descent}
\input{persistence}
\input{length}
\input{bibliography}
\end{document}
==== END Sharp-nodal-length-on-smooth-surfaces//paper.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//persistence.tex ====
\section{Persistence and packing}
\label{sec:persistence}

We now bound the average of $N$ at the terminal level. The stopped
descent estimate controls the contribution of one excursion of high
growth, but a path may enter several excursions. We will bound the
total area of their entry squares. The argument has two parts: an
entry produces a nearby region on which few later entries can occur,
and a geometric packing estimate converts these regions into the
required area bound.

Fix the root square $Q_0$ and the constants $A,C_0,C_1$ from
Section~\ref{sec:subdivision}. We consider finite subdivision trees whose
terminal squares have a common side length and for which $K_Q\ge a_0>0$
at every level. All constants in this section may depend on these fixed
data, including $a_0$, but are independent of the eigenfunction and the
depth of the tree. Let $L$ be the threshold in
Lemma~\ref{lem:descent}, and let $P,\sigma$ be the constants in
Proposition~\ref{prop:stability}. Proposition~\ref{prop:initial-growth}
and the fixed coordinate volume comparisons give a constant
$C_{\mathrm{init}}$ such that $N(Q_0)\le C_{\mathrm{init}}$.

\subsection{Entries and persistent nearby regions}

\begin{definition}[Excursions]
\label{def:excursions}
Choose a threshold $H>\max\{L,C_{\mathrm{init}}\}$. Along each descending
grid path, an excursion starts the first time $N(Q)>H$ and ends the first
subsequent time $N(Q)\le L$. After an excursion has ended, apply the same
rule again. A square at which an excursion starts is called an
\emph{entry}. An excursion is \emph{active} at a square if it has started
and has not yet ended there.
\end{definition}

These decisions depend only on a square and its ancestors, so each entry
is a whole grid square. Write $\mathcal E$ for the family of entries.
The root is not an entry. In addition to the
lower bound defining an entry, we have a uniform upper bound on its
growth.

\begin{lemma}[Growth at an entry]
\label{lem:entry-bound}
For all sufficiently large fixed $H$, every entry $Q$ and its grid
parent $Q^p$ satisfy
\begin{equation}
\label{eq:entry-bound}
 N(Q^p)\le H,
 \qquad
 H<N(Q)\le
 A^2\bigl[\tfrac12+(R+1)C_1\bigr]
       \left(H+\frac{C_0}{a_0}\right).
\end{equation}
\end{lemma}

\begin{proof}
Immediately before an entry the path is inactive. An inactive square
cannot have $N>H$, since that would start an excursion, so
$N(Q^p)\le H$. Apply Proposition~\ref{prop:subdivision} at $Q^p$ with
$b=0$ and $S=HK_{Q^p}+C_0$. Indeed, $E(Q^p,0)\le HK_{Q^p}\le S$, and
\[
 \frac{K_{Q^p}}{S}\le\frac1H,
 \qquad
 \frac{K_{Q^p}^2s(Q^p)}{S^2}\le\frac1{H^2},
\]
since $s(Q^p)\le1$. Thus
\eqref{eq:subdivision-hypotheses} holds when $H$ is sufficiently large.
All terms in the averages in \eqref{eq:subdivision-estimate} are
nonnegative. There are $A^2$ children, so the tilt $b_Q$ assigned to the
particular child $Q$ satisfies
\[
 m(Q,b_Q)\le \frac{A^2S}{2K_{Q^p}},
 \qquad
 |b_Q|\le \frac{A^2C_1S}{K_{Q^p}}.
\]
Combining these inequalities with
\eqref{eq:untilted-comparison} and $K_{Q^p}\ge a_0$ proves the upper
bound in \eqref{eq:entry-bound}.
\end{proof}

For a square $Q$ and a number $D\ge1$, write $DQ$ for its concentric
dilation by the factor $D$. The following proposition supplies a region
of controlled future entry count near each entry. Its location need not
be inside the entry square.

\begin{proposition}[Persistence near an entry]
\label{prop:persistence}
There are a finite threshold $H$, a number $\delta>0$, and an integer
$J\ge0$ such that every entry $Q$ admits a measurable set
$Y_Q\subset D_*Q$, where $D_*=20A$, with
\begin{equation}
\label{eq:persistence}
 \begin{split}
 |Y_Q|&\ge\delta|Q|,\\
 \#\{F\in\mathcal E:\ s(F)<s(Q),\ y\in F\}&\le J
 \qquad\text{for almost every }y\in Y_Q.
 \end{split}
\end{equation}
Only entries of the given tree are counted, including when $Y_Q$
extends outside the root.
\end{proposition}

\begin{proof}
The contradiction argument below extracts a nonconstant profile,
finds a nearby square with small affine excess relative to a nonzero
slope, and transfers that square to Proposition~\ref{prop:stability}.

We may require $H$ to exceed any fixed threshold needed above. Suppose
there is no triple $(H,\delta,J)$ with the asserted uniform property.
Choose $H_j\to\infty$, all above these fixed thresholds and at least
$2$, and apply this negation with
$\delta_j=1/H_j$ and $J_j=\lceil H_j\rceil$. For each $j$ there are an
eigenfunction, a finite tree, and an entry $Q_j$ for which
\eqref{eq:persistence} fails with these parameters.

There must be more than $J_j$ levels strictly below $Q_j$. Otherwise the
finer-entry count would be at most $J_j$ everywhere, and $Y_{Q_j}=Q_j$
would satisfy \eqref{eq:persistence}. In particular, every fixed number
of further levels is eventually available.

Rescale about the center $x_j$ of the parent $Q_j^p$, using its side
$s_j=s(Q_j^p)$ as unit length. Write $k_j$ for the eigenfrequency of
$u_j$, and put
\[
 K_j=k_js_j,\qquad S_j=H_jK_j,\qquad
 U_j(\xi)=u_j(x_j+s_j\xi),\qquad
 p_j(\xi)=p(x_j+s_j\xi).
\]
The rescaled equation is
$\Delta U_j+K_j^2p_jU_j=0$. Its coefficients satisfy
$\|p_j\|_\infty\le C$ and $\|\nabla p_j\|_\infty\le Cs_j$.
Moreover,
\[
 S_j\ge H_ja_0\longrightarrow\infty,
 \qquad \frac{K_j}{S_j}=\frac1{H_j}\longrightarrow0,
 \qquad
 \frac{K_j^2\|\nabla p_j\|_\infty}{S_j^2}
 \le\frac{Cs_j}{H_j^2}\longrightarrow0.
\]
Normalize $|U_j|^2\,d\xi$ to have mass one on $\D_R$.
Lemma~\ref{lem:profile-extraction} and
Theorem~\ref{thm:subharmonic-profile}, with zero tilt, give a
subharmonic profile $V$ on $\D_R$ satisfying
\[
 V\le0,\qquad \sup_{\overline{\D_1}}V\ge-1.
\]
For the second assertion, the parent inequality gives
$E(Q_j^p,0)/S_j\le1$, and the compact upper bound in
\eqref{eq:profile-bounds} applies to the closed unit disk. Thus $V$ is
not identically $-\infty$. Its supremum on every nonempty open disk in
$\D_R$ is therefore finite.

There are only $A^2$ possible positions of the entry inside its parent.
After a subsequence its rescaled center is a fixed point $y_e$. The
entry inequality and the compact upper and open lower bounds in
\eqref{eq:profile-bounds} imply
\[
 \frac1A
 \le \liminf_j\frac{E(Q_j,0)}{S_j}
 \le \limsup_j\frac{E(Q_j,0)}{S_j}
 \le
 \sup_{\overline{\D_{R/A}(y_e)}}V
       -\sup_{\D_{1/A}(y_e)}V.
\]
The finiteness of the second supremum justifies the subtraction of
these bounds. Since $A>10R$, both disks are contained in $\D_2$.
Consequently $V$ is nonconstant on $\D_2$.

The representation in Lemma~\ref{lem:potential-decomposition} implies
$V\in W^{1,1}(\D_2)$: the gradient of the logarithmic kernel is locally
integrable in two dimensions, and the representing measure has finite
mass. There is a Lebesgue point $z\in\D_2$ of $\nabla V$ for which
$c=\nabla V(z)\ne0$. Indeed, otherwise the weak gradient would vanish
almost everywhere on the connected disk, forcing $V$ to be constant
almost everywhere; its subharmonic representative, which is determined
by local averages, would then be constant everywhere.

Extend the fine subdivision grids in the rescaled parent coordinates
to the whole plane. For $r=A^{-m}$, choose a grid square with center
$y_m$ such that $|y_m-z|\le r$. These grids have edges
$-\tfrac12+nA^{-m}$ in each coordinate and are the same for every $j$.
We claim that
\begin{equation}
\label{eq:affine-lebesgue}
 \sup_{\overline{\D_{Rr}(y_m)}}(V(x)-c\cdot x)
 -\sup_{\D_r(y_m)}(V(x)-c\cdot x)=o(r)
 \qquad(m\longrightarrow\infty).
\end{equation}
To prove this, let $W(x)=V(x)-c\cdot x$, and let $a_r$ be its average
on $B_r=\D_{(2R+2)r}(z)$. For large $m$ these disks are contained in
$\D_2$. Poincare's inequality and the Lebesgue-point property give
\[
 \av_{B_r}|W-a_r|
 \le C_Rr\,\av_{B_r}|\nabla V-c|=o(r).
\]
If $x\in\overline{\D_{Rr}(y_m)}$, the disk
$\D_{(R+1)r}(x)$ lies in $B_r$. The submean inequality for the
subharmonic function $W$ therefore gives, uniformly in such $x$,
\[
 W(x)\le a_r+4\av_{B_r}|W-a_r|=a_r+o(r).
\]
On the other hand,
\[
 \sup_{\D_r(y_m)}W
 \ge\av_{\D_r(y_m)}W
 \ge a_r-(2R+2)^2\av_{B_r}|W-a_r|=a_r-o(r).
\]
This proves \eqref{eq:affine-lebesgue}. Subharmonicity is essential
here: it turns the average control into a uniform upper bound.

Now fix $m\ge2$ sufficiently large that $Rr\le1/4$ and the quotient of
the left side of \eqref{eq:affine-lebesgue} by $r|c|$ is strictly less
than $\sigma$. In the original coordinates let $I_j$ be the square
with center $x_j+s_jy_m$ and side $s_jr$, and assign it the tilt
$b_{I_j}=H_jc$. The additional weight in parent coordinates is
$e^{-2S_jc\cdot\xi}$, up to a constant canceled in the norm ratio.
The weighted bounds of Lemma~\ref{lem:profile-extraction} therefore yield
\begin{equation}
\label{eq:persistence-transfer}
 \limsup_j\frac{m(I_j,b_{I_j})}{|H_jc|}
 \le
 \frac{
 \sup_{\overline{\D_{Rr}(y_m)}}(V(x)-c\cdot x)
 -\sup_{\D_r(y_m)}(V(x)-c\cdot x)}{r|c|}
 <\sigma.
\end{equation}
The factor is exactly $1/(r|c|)$, since $K_{I_j}=rK_j$ and
$S_j=H_jK_j$. Because $c\ne0$, we also have $|H_jc|\ge P$ for all
large $j$. Thus \eqref{eq:stability-hypotheses} holds at $I_j$.

We verify that Proposition~\ref{prop:stability} can be used on the
whole subtree of $I_j$ down to the original bottom. The chosen $m$ is
fixed before $j$ tends to infinity, so that level is eventually
available. All squares of a given extended-grid level have the same
physical scale as the original tree, and hence have $K\ge a_0$.
The center $y_m$ stays in $\D_3$, and $Rr\le1/4$. The disks required
for $I_j$ and all its descendants are consequently contained in
$\D_{5s_j}(x_j)$: even the root margin has rescaled radius $2Rr$,
and $|y_m|+2Rr<3+1/2<5$. Since $Q_j^p\subset Q_0$, we have
$|x_j-x_{Q_0}|\le s_0/\sqrt2$ and $s_j\le s_0$, so this disk lies in
$\D_{6s_0}(x_{Q_0})\subset\D_{2Rs_0}(x_{Q_0})$, within the fixed
coordinate patch. This also covers the larger disk margins required
when regarding $I_j$ as the root of a subtree.

Proposition~\ref{prop:stability} now gives a measurable subset of
$I_j$ of area at least $|I_j|/2$ on which every visited square from
$I_j$ through the bottom has $N>L$. The extended grids align with the
original root: a root edge differs from the corresponding parent edge
by an integer multiple of $s_j$, and the extended parent grid has
rescaled edges $-1/2+nA^{-m}$. Thus a fine grid square is either
contained in $Q_0$ or has interior disjoint from it. In the latter case
its points belong to no
entry of the original tree. In the former case its descendants are
exactly those of the original tree. From the level of $I_j$ onward
there can be at most one entry along any of these paths, since two
starts require an intervening value $N\le L$. Before that level there
are at most $m$ relevant levels below the tested entry. Thus the number
of strictly finer entries is at most $m+1$ on this subset.

Finally, the subset lies in $20A Q_j$: in parent units that dilated
square has half-side $10$, whereas $y_m\in\D_3$ and the entry center
lies in the unit parent square. Its area relative to the entry is at
least
\[
 \frac{|I_j|}{2|Q_j|}=\frac12 A^{2-2m}>0.
\]
Both this positive number and $m+1$ are fixed independently of $j$.
They eventually satisfy the tested requirements
$\delta_j=1/H_j$ and $J_j=\lceil H_j\rceil$, contradicting the choice
of $Q_j$. This proves the proposition.
\end{proof}

\subsection{Packing with witnesses outside the squares}

The next geometric lemma converts the nearby regions just constructed
into a bound on the sum of entry areas. We state it for an arbitrary
finite family in a subdivision tree.

\begin{lemma}[Packing from nearby witnesses]
\label{lem:packing}
Let $\mathcal F$ be a finite family of squares in the $A$-adic
subdivision tree of a square $Q_0$, where $A\ge2$. Suppose there are
fixed $D\ge1$, $\delta>0$, and an integer $J\ge0$ such that every
$Q\in\mathcal F$ admits a measurable set $Y_Q\subset DQ$ satisfying
\[
 |Y_Q|\ge\delta|Q|,
 \qquad
 \#\{F\in\mathcal F:s(F)<s(Q),\ y\in F\}\le J
 \quad\text{for almost every }y\in Y_Q.
\]
Then
\begin{equation}
\label{eq:packing}
 \sum_{Q\in\mathcal F}|Q|
 \le C(A,D,\delta,J)|Q_0|.
\end{equation}
\end{lemma}

\begin{proof}
Ignore grid boundaries and the boundaries of the finitely many dilates
$DQ$ with $Q\in\mathcal F$; these form a null set. Define
\[
 n(Q)=1+\#\{F\in\mathcal F:Q\subsetneq F\}.
\]
Squares of a fixed rank $n$ have disjoint interiors. At each rank,
process the squares in decreasing order of side length, retaining a
square if its $D$-dilate has interior disjoint from the previously
retained dilates. Let $\mathcal S$ be the union of these selected
families. Every discarded square $Q$ has its dilate meeting that of a
selected square $P$ with $s(P)\ge s(Q)$; hence
$Q\subset(2D+1)P$. Covering the original disjoint squares at that rank
by these larger dilates and then summing over ranks gives
\[
 \sum_{Q\in\mathcal S}|Y_Q|
 \ge\delta\sum_{Q\in\mathcal S}|Q|
 \ge\alpha\sum_{Q\in\mathcal F}|Q|,
 \qquad \alpha=\frac{\delta}{(2D+1)^2}.
\]
The selected witness sets at a single rank are disjoint up to null
sets. It remains to control overlap between different ranks. For a
point $y$ outside the ignored boundaries, put
\[
 n_{\mathrm{tot}}(y)=\#\{F\in\mathcal F:y\in F\}.
\]
If a witness point $y$ lies inside its square $Q$, every larger square
containing $y$ is an ancestor of $Q$, so the witness assumption makes
$n_{\mathrm{tot}}(y)$ differ from $n(Q)$ by at most $J$. An exterior
witness point can also disagree with the ancestry of $Q$ at larger
scales. For $Q\in\mathcal S$ and $y\in Y_Q$, let
\[
 d_Q(y)=
 \sum_{\substack{F\in\mathcal F\\s(F)>s(Q)}}
 \left|\mathbf 1_F(y)-\mathbf 1_{\{Q\subset F\}}\right|
\]
count the mismatches between membership of $y$ in a larger square and
that square's being an ancestor of $Q$. If $b$ is the number of larger
squares in $\mathcal F$ containing $y$, then
\[
 |n(Q)-1-b|\le d_Q(y).
\]
At most one square of side $s(Q)$ contains $y$, and the witness
assumption bounds the number of smaller squares containing it by $J$.
Decomposing $n_{\mathrm{tot}}(y)$ into these three size ranges gives
\begin{equation}\label{eq:rank-mismatch}
 |n(Q)-n_{\mathrm{tot}}(y)|\le d_Q(y)+J+1
 \quad\text{for almost every }y\in Y_Q.
\end{equation}
Thus a uniform bound for $d_Q(y)$ restricts the ranks that can contribute
at $y$ to a fixed interval. We next show that the total witness area
weighted by $d_Q$ is controlled:
\begin{equation}
\label{eq:mismatch}
 \sum_{Q\in\mathcal S}\int_{Y_Q}d_Q(y)\,dy
 \le B\sum_{F\in\mathcal F}|F|,
\end{equation}
where $B$ depends only on $A,D$.

Fix $F\in\mathcal F$ and a finer side length
$t=A^{-\ell}s(F)$, $\ell\ge1$. The grid is laminar: each square $Q$ of
side $t$ is either contained in $F$ or interior-disjoint from it. If it
can contribute a mismatch, a point of $DQ$ lies on the opposite side
of $\partial F$ from $Q$. It follows that the whole square $Q$ lies
in the collar
\[
 \{x:\dist_\infty(x,\partial F)\le(D+1)t\},
\]
where $\dist_\infty$ is distance for the maximum norm on $\R^2$.
Figure~\ref{fig:boundary-mismatch} shows the two possible directions
of this mismatch near one face of $F$.
This collar has area at most
\[
 8(D+1)s(F)t+4(D+1)^2t^2.
\]
All squares of side $t$, across all ranks, have disjoint interiors.
Since $|Y_Q|\le D^2|Q|$, their total contribution for this fixed $F$
and $t$ is at most $D^2$ times that collar area. Summing the geometric
series over $\ell\ge1$ and then summing over $F$ proves
\eqref{eq:mismatch}. For example, one may take
\[
 B=D^2\left(\frac{8(D+1)}{A-1}
            +\frac{4(D+1)^2}{A^2-1}\right).
\]

\begin{figure}[tbp]
\centering
\setlength{\unitlength}{0.9mm}
\begin{picture}(156,57)
\small
\put(35,54){\makebox(0,0){$Q\subset F,\quad y\notin F$}}
\put(112,54){\makebox(0,0){$Q\cap F=\varnothing,\quad y\in F$}}
\color{gray!35}
\put(3,13){\line(0,1){31}}
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\put(35,47){\makebox(0,0){$\partial F$}}
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\put(11,42){\makebox(0,0){$F$}}
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\put(16,13){\dashbox{1}(24,24){}}
\put(108,13){\dashbox{1}(24,24){}}
\put(20,40){\makebox(0,0){$DQ$}}
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\put(24,21){\framebox(8,8){$Q$}}
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\put(108,25){\makebox(0,0)[r]{$y$}}
\put(24,18){\vector(1,0){8}}
\put(32,18){\vector(-1,0){8}}
\put(28,15){\makebox(0,0){$t$}}
\put(116,18){\vector(1,0){8}}
\put(124,18){\vector(-1,0){8}}
\put(120,15){\makebox(0,0){$t$}}
\put(35,9){\vector(1,0){32}}
\put(67,9){\vector(-1,0){32}}
\put(51,5){\makebox(0,0){$(D+1)t$}}
\put(112,9){\vector(1,0){32}}
\put(144,9){\vector(-1,0){32}}
\put(128,5){\makebox(0,0){$(D+1)t$}}
\end{picture}
\caption{Ancestry and spatial membership can disagree in either
direction. Each panel shows one witness point $y\in Y_Q\subset DQ$
and a fine square of side $t=s(Q)$ near a face of the larger square
$F$. The dashed square is $DQ$; the pale lines mark the boundary of
the collar of width $(D+1)t$ on each side of the face.}
\label{fig:boundary-mismatch}
\end{figure}

Choose an integer $M\ge1$ with $M\ge2B/\alpha$, and set
$Z_Q=\{y\in Y_Q:d_Q(y)\le M\}$. Remove also the null sets where the
witness assumption fails. Markov's inequality and
\eqref{eq:mismatch} give
\[
 \sum_{Q\in\mathcal S}|Z_Q|
 \ge\frac\alpha2\sum_{Q\in\mathcal F}|Q|.
\]
For $y\in Z_Q$, Equation~\eqref{eq:rank-mismatch} now gives
\[
 |n(Q)-n_{\mathrm{tot}}(y)|\le M+J+1.
\]
Only $2(M+J+1)+1$ ranks can therefore contribute at $y$. At a given
rank the selected witness sets are disjoint, so this is also a bound
on the total overlap of the $Z_Q$.

For $D\ge1$, the inclusions $Q\subset Q_0$ imply $DQ\subset DQ_0$.
Integrating the overlap bound over this common region yields
\[
 \frac\alpha2\sum_{Q\in\mathcal F}|Q|
 \le\bigl(2(M+J+1)+1\bigr)D^2|Q_0|.
\]
This proves \eqref{eq:packing}.
\end{proof}

Apply Lemma~\ref{lem:packing} to the family of entries, using
Proposition~\ref{prop:persistence} with $D=D_*$. We obtain a constant
$C$ such that the sum of all entry areas is at most $C|Q_0|$.

\subsection{The terminal mean growth}

\begin{theorem}[Bounded mean growth]
\label{thm:mean-growth}
For the finite trees considered in this section, let
$\mathcal T_{\mathrm{term}}$ be the collection of terminal squares and
let $s_{\mathrm{term}}$ be their common side length. There is a
constant $C$, independent of the eigenfunction and the number of
levels, such that
\begin{equation}
\label{eq:mean-growth}
 \av_{q\in\mathcal T_{\mathrm{term}}}N(q)\le C,
 \qquad
 \av_{q\in\mathcal T_{\mathrm{term}}}E(q,0)
      \le Ck s_{\mathrm{term}}.
\end{equation}
\end{theorem}

\begin{proof}
Fix the threshold $H$ supplied by
Proposition~\ref{prop:persistence}. A terminal path on which no
excursion is active has terminal growth $N\le H$. Every other terminal
path has exactly one final active entry. For an entry $Q$, let
$\Omega_Q\subset Q$ be the union of terminal squares on which that
entry's excursion is still active. These sets are disjoint as $Q$
varies.

Starting separately at each entry $Q$, set $b_Q=0$ and perform the
descent of Lemma~\ref{lem:descent}, stopping at the first square with
$N\le L$ or at the bottom, whichever comes first. Every continuing
square has $N>L$, so the descent is available regardless of its current
tilt. The low-growth stopping square, if any, is included as the last
produced value; no further step is needed there. These separate choices
of tilts do not affect excursion membership, which is defined entirely
by the untilted quantity $N$.

Along a random descendant path from $Q$, denote the stopping time by
$T$ and the successive values by $m_i,b_i$. Since $b_0=0$,
\eqref{eq:untilted-comparison} gives
\[
 N_T\le m_T+(R+1)|b_T|
 \le m_T+(R+1)\sum_{i<T}|b_{i+1}-b_i|.
\]
The uniform distribution on paths agrees with normalized planar area
on $Q$. Lemma~\ref{lem:stopped-descent} and
Lemma~\ref{lem:entry-bound} therefore imply
\[
 \begin{split}
 \int_{\Omega_Q}N_{\mathrm{term}}(x)\,dx
 &\le |Q|\,\E\left[m_T+(R+1)
                    \sum_{i<T}|b_{i+1}-b_i|\right]\\
 &\le C|Q|\bigl(N(Q)+1\bigr)
 \le C'|Q|.
 \end{split}
\]
Here $N_{\mathrm{term}}$ is constant on each terminal square, with
value $N$ of that square. The first inequality merely restricts a
nonnegative integral to the paths still active at the bottom; it does
not condition on that event or divide by its probability.

Sum over all entries and use \eqref{eq:packing}, then include the
inactive terminal paths. This gives
\[
 \int_{Q_0}N_{\mathrm{term}}(x)\,dx
 \le H|Q_0|+C'\sum_{Q\text{ entry}}|Q|
 \le C|Q_0|.
\]
Equal terminal areas turn this integral bound into the first inequality
of \eqref{eq:mean-growth}. The second follows because every terminal
square has $K_q=ks_{\mathrm{term}}$ and $E(q,0)=K_qN(q)$.
\end{proof}
==== END Sharp-nodal-length-on-smooth-surfaces//persistence.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//preamble.tex ====
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==== END Sharp-nodal-length-on-smooth-surfaces//preamble.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//profiles.tex ====
\section{Subharmonic limits of tilted norms}\label{sec:profiles}

We record exponential growth through logarithms of squared local
$L^2$ norms.  Pointwise logarithms are singular at zeros; the compactness
statement below instead controls masses on all compact and open sets,
which is what later comparisons of norms require.  Its compatibility
with affine weights will allow us to change the tilt as we subdivide.
Throughout the planar arguments,
norms and integrals are Euclidean and are not normalized by area.
We write $\D_r(a)$ for the open Euclidean disk of radius $r$ centered
at $a$, and $\D_r=\D_r(0)$; a bar denotes closure. For a measurable
planar set $E$, the notation $|E|$ denotes its Euclidean area.

\begin{lemma}[Extraction of a logarithmic profile]\label{lem:profile-extraction}
Let $X$ be an open subset of $\R^2$ or a compact smooth surface.  Suppose
that $S_j\to\infty$ and that $\mu_j$ are nonnegative Borel measures on
$X$ with $\mu_j(X)\le 1$.  There are a subsequence and an upper
semicontinuous function $V\colon X\to[-\infty,0]$ such that, for every
compact $F\subset X$ and every open $G\subset X$,
\begin{equation}\label{eq:profile-bounds}
 \limsup_{j\to\infty}\frac{\log\mu_j(F)}{2S_j}\le\sup_F V,
 \qquad
 \liminf_{j\to\infty}\frac{\log\mu_j(G)}{2S_j}\ge\sup_G V.
\end{equation}
Here $\log 0=\sup\varnothing=-\infty$.  More generally, if continuous
real functions $f_j$ converge locally uniformly to $f$, then
\[
 \begin{aligned}
 \limsup_{j\to\infty}\frac{1}{2S_j}
       \log\int_F e^{2S_j f_j}\,d\mu_j&\le\sup_F(V+f),\\
 \liminf_{j\to\infty}\frac{1}{2S_j}
       \log\int_G e^{2S_j f_j}\,d\mu_j&\ge\sup_G(V+f).
 \end{aligned}
\]
The same subsequence works for all these sets and weights.
\end{lemma}

\begin{proof}
Choose a countable basis $\{G_i\}_{i\ge1}$ consisting of relatively
compact open sets.  A diagonal subsequence makes all the limits
\[
 \ell_i=\lim_{j\to\infty}\frac{\log\mu_j(G_i)}{2S_j}
        \in[-\infty,0]
\]
exist.  Define $V(x)=\inf_{i:x\in G_i}\ell_i$.  If $V(x)<a$, one of
these basis neighborhoods has $\ell_i<a$, and $V<a$ throughout that
neighborhood.  Thus $V$ is upper semicontinuous.

For any finite $a>\sup_F V$, every point of $F$ has a basis
neighborhood with exponent less than $a$.  A finite subcover and
$\mu_j(F)\le\sum_i\mu_j(G_i)$ give the compact upper bound by $a$.
Letting $a\downarrow\sup_F V$ proves the first inequality, also when
this supremum is $-\infty$.  If $x\in G$, a basis neighborhood
$G_i$ with $x\in G_i\subset G$ gives a lower bound by
$\ell_i\ge V(x)$, proving the second inequality.

For the weighted upper bound, take $a>\sup_F(V+f)$ and choose the
neighborhood at each point small enough that its limiting measure
exponent plus the supremum of $f$ on it is less than $a$.  This is
possible by continuity of $f$ and the definition of $V$, including
at points where $V=-\infty$.  A finite cover and uniform convergence
of $f_j$ on its compact closure give the claimed upper bound.  For
the lower bound at a point $x\in G$ with $V(x)>-\infty$, choose
$G_i$ compactly contained in $G$ so small that
$f\ge f(x)-\eps$ there.  The weighted integral over $G_i$ has lower
exponent at least $\ell_i+f(x)-\eps$.  Let $\eps\downarrow0$ and
then take the supremum over $x$.
\end{proof}

We call $V$ a \emph{logarithmic profile}.  Multiplying the measures
by positive densities uniformly bounded above and below on compact
sets does not change the profile bounds: the logarithms of these
bounds disappear after division by $2S_j$.  We may also use the
compact upper bound for a relatively compact set by taking its
closure.  Normalizing mass on an open disk alone does not assert
that the profile has a finite value inside the disk; the
identically $-\infty$ case remains possible.

\begin{theorem}[Subharmonicity of logarithmic profiles]
\label{thm:subharmonic-profile}
On a fixed planar disk $\Omega$, let real smooth functions $U_j,p_j$
satisfy
\[
 \Delta U_j+K_j^2p_jU_j=0,
\]
where $K_j\ge0$.  Let $S_j>0$, $B_j\in\R^2$, and
$D_j=S_j+|B_j|$.  Assume
\begin{equation}\label{eq:profile-hypotheses}
 \begin{gathered}
 S_j\longrightarrow\infty,\qquad
 \frac{K_j}{D_j}\longrightarrow0,\\
 \sup_j\|p_j\|_\infty<\infty,\qquad
 \frac{K_j^2\|\nabla p_j\|_\infty}{S_jD_j}\longrightarrow0.
 \end{gathered}
\end{equation}
Local versions of the coefficient assumptions on relatively compact
subdisks suffice.  For positive constants $c_j$, suppose the measures
\[
 d\mu_j=c_j e^{-2B_j\cdot x}|U_j(x)|^2\,dx
\]
have mass at most one on $\Omega$.  Every upper semicontinuous
logarithmic profile with the compact, open, and weighted bounds of
Lemma~\ref{lem:profile-extraction} is subharmonic on $\Omega$,
with the identically $-\infty$ function allowed.
\end{theorem}

The main estimate concerns weights whose normalized gradient is
nonzero.  It is essential that the next lemma allows $D_j/S_j$ to
be arbitrarily large.

\begin{lemma}[A local Carleman estimate]\label{lem:carleman}
Assume the coefficient and parameter hypotheses of
Theorem~\ref{thm:subharmonic-profile}.  After passing to a subsequence,
write
\[
 S_j/D_j\to d,\qquad B_j/D_j\to b,
 \qquad d\ge0,\quad d+|b|=1.
\]
Let $\chi$ be a real smooth function near $x_0\in\Omega$ such that
$\Delta\chi(x_0)<0$ and $b+d\nabla\chi(x_0)\ne0$.  Put
$T_j(x)=-B_j\cdot x-S_j\chi(x)$.  There are a neighborhood
$\Omega_0$ of $x_0$, a constant $c>0$, and an index $j_0$ such that
\begin{equation}\label{eq:carleman}
 \|e^{T_j}(\Delta+K_j^2p_j)w\|_2
 \ge c\sqrt{S_j}\,D_j\,\|e^{T_j}w\|_2
\end{equation}
for all $j\ge j_0$ and all real $w\in C_c^\infty(\Omega_0)$.
\end{lemma}

The two parameters reflect different derivatives of the weight:
$\nabla T_j=-B_j-S_j\nabla\chi$ has scale $D_j$, whereas
$T_j''=-S_j\chi''$ has scale $S_j$.  The affine tilt can therefore
enlarge the gradient without enlarging the curvature.

\begin{proof}
Suppress the index and write $v=e^T w$.  The conjugated operator is
$P_++P_-$, where
\[
 P_+=\Delta+|\nabla T|^2+K^2p,
 \qquad P_-=-2\nabla T\cdot\nabla-\Delta T.
\]
On compactly supported functions, $P_+$ is symmetric and $P_-$ is
skew-symmetric.  Integration by parts gives
\begin{equation}\label{eq:commutator}
 \begin{split}
 2\langle P_+v,P_-v\rangle
  ={}&4\int T''(\nabla v,\nabla v)\\
    &+\int\bigl(4T''(\nabla T,\nabla T)-\Delta^2T
             +2K^2\nabla p\cdot\nabla T\bigr)v^2.
 \end{split}
\end{equation}
For clarity, the contribution from $\Delta$ is
$4\int T''(\nabla v,\nabla v)-\int\Delta^2T\,v^2$.
Multiplication by a real function $a$ contributes
$2\int\nabla a\cdot\nabla T\,v^2$; applying this with
$a=|\nabla T|^2+K^2p$ gives the remaining terms in
Equation~\eqref{eq:commutator}.

If Equation~\eqref{eq:carleman} failed on every sufficiently small
neighborhood, a diagonal choice would give a subsequence and real
smooth $v_j$ with supports shrinking to $x_0$, such that
\[
 \|v_j\|_2=1,
 \qquad
 \|(P_++P_-)v_j\|_2=o(\sqrt{S_j}D_j).
\]
All constants below are uniform on one fixed small neighborhood of
$x_0$.  There, $|\nabla T|\le CD$, $|T''|+|\Delta^2T|\le CS$,
and the last hypothesis in Equation~\eqref{eq:profile-hypotheses}
controls the potential term in Equation~\eqref{eq:commutator}.
Set $X=\|P_+v\|_2$, $Y=\|P_-v\|_2$, and
$G=\|\nabla v\|_2$.  Expanding the squared norm and using the
identity gives
\[
 X^2+Y^2\le CS(G^2+D^2).
\]
Testing $P_+$ against $v$ gives $G^2\le CD^2+X$, since $K=o(D)$.
Combining the two inequalities first yields
$X\le C(S+\sqrt S D)$ and then
\[
 G\le CD,
 \qquad X+Y\le C\sqrt S D.
\]
Here $D\ge S\to\infty$ absorbs the term $S$.

Uniformly on the shrinking supports,
\[
 \nabla T_j/D_j\to a=-b-d\nabla\chi(x_0)\ne0,
 \qquad T_j''/S_j\to H=-\chi''(x_0).
\]
The preceding bounds give
\[
 \frac{\sqrt{S_j}}{D_j}\le\frac1{\sqrt{S_j}}\longrightarrow0.
\]
Dividing the identity obtained by testing $P_+$ against $v_j$ by
$D_j^2$ therefore shows
\[
 \int|\nabla v_j/D_j|^2\longrightarrow |a|^2.
\]
Likewise,
\[
 \left\|\frac{\nabla T_j}{D_j}\cdot
                \frac{\nabla v_j}{D_j}\right\|_2
 \le \frac{\|P_-v_j\|_2+\|\Delta T_j\,v_j\|_2}{2D_j^2}
 \longrightarrow0.
\]
Replacing $\nabla T_j/D_j$ by its uniform limit consequently gives
$\|a\cdot\nabla v_j/D_j\|_2\to0$.

Let $e$ be a unit vector perpendicular to $a$.  In dimension two,
the energy perpendicular to $a$ has only this one direction.
Decomposing $\nabla v_j/D_j$ in the orthonormal basis
$\{a/|a|,e\}$, the parallel component tends to zero in $L^2$,
whereas the squared norm of the perpendicular component tends to
$|a|^2$.  Hence the first term in
Equation~\eqref{eq:commutator}, divided by $S_jD_j^2$, tends to
$4|a|^2H(e,e)$.  The other main Hessian term tends to $4H(a,a)$.
The two remaining terms tend to zero: their absolute values after
this division are bounded by
\[
 \frac{C}{D_j^2}
 \quad\hbox{and}\quad
 \frac{CK_j^2\|\nabla p_j\|_\infty}{S_jD_j},
\]
respectively.  We have thus proved
\[
 \begin{aligned}
 \frac{2\langle P_+v_j,P_-v_j\rangle}{S_jD_j^2}
 &\longrightarrow 4\bigl(|a|^2H(e,e)+H(a,a)\bigr)\\
 &=4|a|^2\operatorname{tr}H>0.
 \end{aligned}
\]
The squared norms of $P_+v_j$ and $P_-v_j$ are nonnegative, so this
contradicts the assumed smallness of $\|(P_++P_-)v_j\|_2^2$.
\end{proof}

We also need to remove one possible exceptional gradient from the
upper-test criterion.  The following elementary perturbation does
so without any initial integrability assumption.  An upper test at
a finite point $x_0$ is a smooth function agreeing with the tested
function at $x_0$ and dominating it in a neighborhood of $x_0$.
This is an upper-semicontinuous subharmonic variant of the standard
viscosity exceptional-gradient-removal principle; compare
\cite[Section~5, Lemma~6]{ImbertSilvestre2013}.  We give the local
proof for functions allowed to take the value $-\infty$, rather than
apply that lemma directly.

\begin{lemma}[Removal of the exceptional gradient]
\label{lem:exceptional-gradient}
Let $W\colon\Omega\to[-\infty,\infty)$ be upper semicontinuous and
locally bounded above on a connected planar domain.  Suppose every
smooth function $\psi$ touching $W$ from above at a finite point
$x_0$, with $\nabla\psi(x_0)\ne0$, satisfies
$\Delta\psi(x_0)\ge0$.  Then $W$ is subharmonic, with the
identically $-\infty$ function allowed.
\end{lemma}

\begin{proof}
Suppose first that an upper test at $0$ has zero gradient and
negative Laplacian; translate the point to $0$ if necessary.  A
quadratic Taylor polynomial with its Hessian increased by a
sufficiently small positive multiple of the identity gives a
quadratic upper test $h$ with
\[
 \begin{gathered}
 W(0)=h(0),\qquad \nabla h(0)=0,\qquad
 \operatorname{tr}h''<0,\\
 W(x)<h(x)\quad(0<|x|\le r)
 \end{gathered}
\]
for some small $r>0$.  Choose a unit eigenvector $e$ of $h''$ with
eigenvalue $\alpha<0$.  For small $t>0$, maximize
$W(x)-h(x)+t e\cdot x$ on $\overline{\D_r}$.  Upper
semicontinuity and the strict boundary gap ensure that a maximizer
$x_t$ is interior.  Its value is at least the value $0$ at the
origin, and $W-h\le0$, so $e\cdot x_t\ge0$.  At $x_t$ an upper
test is $h(x)-t e\cdot x$, shifted by its contact constant.  Its
gradient has component
\[
 e\cdot\nabla(h-t e\cdot x)(x_t)
 =\alpha\, e\cdot x_t-t<0,
\]
while its Laplacian is still negative.  This contradicts the
hypothesis.  All smooth upper tests at finite points therefore
have nonnegative Laplacian.

We obtain harmonic comparison directly.  Let
$\overline{\D_r(a)}\subset\Omega$, let $h$ be harmonic in
$\D_r(a)$ and continuous on its closure, and suppose $W\le h$ on
the boundary.  If $W-h$ were positive somewhere inside, then for
sufficiently small $\eps>0$ the function
\[
 h_\eps(x)=h(x)+\eps(r^2-|x-a|^2)
\]
would still lie below $W$ at some interior point.  A positive
maximum of $W-h_\eps$ is attained at an interior finite point.
Adding this maximum to $h_\eps$ supplies an upper test with
Laplacian $-4\eps<0$, again a contradiction.  Thus $W$ satisfies
harmonic comparison.  For upper semicontinuous functions this is
the usual characterization of subharmonicity, including the
identically $-\infty$ case; see \cite[Chapter~2]{Ransford1995}.
In particular, local integrability
and the submean property of a nontrivial subharmonic function are
consequences here, not assumptions used in the argument.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:subharmonic-profile}]
Fix such a profile $V$ and pass to a further subsequence so
that $S_j/D_j\to d$ and $B_j/D_j\to b$ as in
Lemma~\ref{lem:carleman}.  The profile bounds remain valid on this
subsequence.  Suppose a smooth $\chi$ touches $V$ from above at
a finite point $x_0$, and suppose
\[
 \Delta\chi(x_0)<0,
 \qquad b+d\nabla\chi(x_0)\ne0.
\]
By adding a small positive multiple of $|x-x_0|^4$, we make the
touching strict on a sufficiently small closed disk without
altering the gradient or Hessian at $x_0$.  Take this disk inside
the neighborhood in Lemma~\ref{lem:carleman}.

Choose fixed radii $0<r_0<r_1<r_2<r_3$ inside that neighborhood.
Let $\zeta$ be a smooth cutoff equal to one on $\D_{r_1}(x_0)$
and supported in $\D_{r_2}(x_0)$.  Its derivatives are supported
in the annulus between these two disks.  Set
\[
 G=\D_{r_0/2}(x_0),\qquad
 F=\{x:r_0\le |x-x_0|\le r_3\}.
\]
We may choose the radii so that the strict touching holds on the
whole closed outer disk.  Then $F$ is a compact annulus,
$\sup_F(V-\chi)<0$, and $\sup_G(V-\chi)=0$.

For completeness, the cutoff error can be controlled with a
prefactor independent of the ratio $D_j/S_j$.  Choose a smooth
$\theta$ supported in the interior of $F$ and equal to one near
the derivatives of $\zeta$.  Testing the equation for $U_j$
against $\theta^2e^{2T_j}U_j$ and integrating by parts bounds
\[
 \begin{aligned}
 \int\theta^2e^{2T_j}|\nabla U_j|^2
 &\le C\int_F e^{2T_j}
       \bigl(|\nabla\theta|^2+
       \theta^2|\nabla T_j|^2+K_j^2|p_j|\theta^2\bigr)|U_j|^2\\
 &\le CD_j^2\int_F e^{2T_j}|U_j|^2.
 \end{aligned}
\]
The first inequality absorbs half of the gradient integral by
Cauchy--Schwarz; the second uses $K_j=o(D_j)$ and $D_j\to\infty$.
Since
\[
 (\Delta+K_j^2p_j)(\zeta U_j)
   =(\Delta\zeta)U_j+2\nabla\zeta\cdot\nabla U_j,
\]
Lemma~\ref{lem:carleman} gives
\[
 c\sqrt{S_j}D_j\|e^{T_j}U_j\|_{L^2(G)}
 \le CD_j\|e^{T_j}U_j\|_{L^2(F)}.
\]
Cancel $D_j$ before comparing exponential rates.  Squaring and
multiplying by the common normalization constant $c_j$ yields
\[
 S_j\int_G e^{-2S_j\chi}\,d\mu_j
       \le C\int_F e^{-2S_j\chi}\,d\mu_j.
\]
The weighted bounds of Lemma~\ref{lem:profile-extraction}, together
with $\log S_j/S_j\to0$, now give the contradiction
\[
 0=\sup_G(V-\chi)\le\sup_F(V-\chi)<0.
\]
In particular, no bound on $\log D_j/S_j$ was required.

We have established the upper-test inequality whenever
$b+d\nabla\chi(x_0)\ne0$.  If $d=0$, then $|b|=1$ and there is
no exceptional gradient; harmonic comparison follows as in
Lemma~\ref{lem:exceptional-gradient}.  If $d>0$, put
$W(x)=V(x)+(b/d)\cdot x$.  An upper test $\psi$ for $W$ corresponds
to the upper test $\chi=\psi-(b/d)\cdot x$ for $V$, and
$b+d\nabla\chi=d\nabla\psi$.  Thus $W$ satisfies the hypothesis
of Lemma~\ref{lem:exceptional-gradient}.  It is subharmonic, and
subtracting the affine function shows that $V$ is subharmonic too.
\end{proof}

The next estimate is a fixed-disk form of quantitative unique
continuation.  For the classical doubling estimate of Donnelly and
Fefferman and the corresponding propagation bound, see
\cite[Theorem~4.1 and Lemma~4.14]{NazarovPolterovichSodin2005}.
We give a proof using logarithmic profiles, which also supplies the
local nonvanishing needed below.

\begin{proposition}[Initial growth on a fixed disk]
\label{prop:initial-growth}
Let $(M,g)$ be a fixed smooth closed connected surface, and let
$G\subset M$ be a fixed nonempty open coordinate disk.  There is
a finite constant $C=C(M,g,G)$ such that every nonzero real
eigenfunction $-\Delta_g u=k^2u$ with $k>0$ satisfies
$\|u\|_{L^2(G)}>0$ and
\begin{equation}\label{eq:initial-growth}
 \log\frac{\|u\|_{L^2(M)}}{\|u\|_{L^2(G)}}\le Ck.
\end{equation}
Both norms use Riemannian volume.  The constant is uniform over
all eigenfunctions in every eigenspace.
\end{proposition}

\begin{proof}
The positive spectral gap gives $k\ge k_1>0$.  Interpret the log
ratio in Equation~\eqref{eq:initial-growth} as $+\infty$ when the
local norm is zero.  If the assertion failed, we could choose
eigenfunctions $u_j$, normalized by $\|u_j\|_{L^2(M)}=1$, such
that their log ratios $R_j$ obey $R_j\ge j^2 k_j$.  This also
covers a zero local norm by repeating that eigenfunction.  Set
$S_j=j k_j$.  Then $S_j\to\infty$ and $S_j/k_j\to\infty$,
whether or not $k_j$ tends to infinity.

Apply Lemma~\ref{lem:profile-extraction} on $M$ to the probability
measures $|u_j|^2\,d\operatorname{vol}_g$.  Compactness of $M$ and
the compact upper bound with $F=M$ give
$0\le\sup_M V\le0$.  Upper semicontinuity ensures that this
maximum is attained.

In a relatively compact isothermal chart,
$g=p(x)(dx_1^2+dx_2^2)$ with $p>0$ smooth, and the coordinate
functions satisfy $\Delta U_j+k_j^2pU_j=0$.  The positive
volume factor $p$ is bounded above and below on this chart.
Multiplying $|U_j|^2\,dx$ by a fixed positive constant if
necessary makes its mass at most one; the resulting logarithmic
profile is the restriction of $V$.  Theorem~\ref{thm:subharmonic-profile}
applies with $B_j=0$, $K_j=k_j$, and $D_j=S_j$, since
\[
 \frac{k_j}{S_j}=\frac1j\longrightarrow0,
 \qquad
 \frac{k_j^2\|\nabla p\|_\infty}{S_j^2}
       =\frac{\|\nabla p\|_\infty}{j^2}\longrightarrow0.
\]
Thus $V$ is subharmonic in every such chart.  The strong maximum
principle makes the set where $V=0$ open; it is also closed by
upper semicontinuity and nonempty by the preceding maximum
argument.  Connectedness gives $V\equiv0$ on $M$.

On the other hand, the normalized mass of the given disk satisfies
\[
 \frac{1}{2S_j}\log\int_G |u_j|^2\,d\operatorname{vol}_g
   =-\frac{R_j}{S_j}\le-j,
\]
with the same conclusion when the integral is zero.  This
contradicts the open lower bound in
Equation~\eqref{eq:profile-bounds}, whose right-hand side is
$\sup_GV=0$.  The argument proves both positivity of the local
norm and Equation~\eqref{eq:initial-growth}.  It uses only local
isothermal charts and therefore imposes no orientability
assumption.
\end{proof}
==== END Sharp-nodal-length-on-smooth-surfaces//profiles.tex ====

==== BEGIN Sharp-nodal-length-on-smooth-surfaces//subdivision.tex ====
\section{Affine improvement under subdivision}
\label{sec:subdivision}

The subharmonic profiles of Section~\ref{sec:profiles} admit a useful
local affine correction.  Averaged over a sufficiently fine square
grid, the remaining growth decreases faster than the side length.
We first establish this fact for profiles and then transfer it to
eigenfunctions.

\begin{samepage}
\begin{definition}[Growth and affine excess]
\label{def:growth}
Fix an isothermal coordinate region in which
\(\Delta u+k^2p u=0\), and work in a relatively compact subset on which
\(p\) and \(\nabla p\) have fixed uniform bounds.  Set \(R=1000\).
Choose a root square \(Q_0\), with side \(s_0\le1\) and center
\(x_{Q_0}\), such that
\(\overline{\D_{2Rs_0}(x_{Q_0})}\) lies in this subset.
For an integer \(A\ge2\), subdivide each square into \(A^2\) congruent
squares and write \(q\prec Q\) when \(q\) is an immediate child of
\(Q\).  For a square \(Q\) of side \(s=s(Q)\) and center \(x_Q\), put
\(K_Q=ks\).  For a real vector \(b\in\R^2\), define
\begin{equation}
\begin{aligned}
 E(Q,b)
 &=\log
 \frac{\norm{e^{-kb\cdot(x-x_Q)}u}_{L^2(\D_{Rs}(x_Q))}}
      {\norm{e^{-kb\cdot(x-x_Q)}u}_{L^2(\D_s(x_Q))}},\\
 m(Q,b)&=\frac{E(Q,b)}{K_Q},
 \qquad N(Q)=m(Q,0).
\end{aligned}
\label{eq:growth}
\end{equation}
All planar norms here and below are unnormalized Euclidean norms.
An average over children assigns weight \(A^{-2}\) to each child.
\end{definition}
\end{samepage}

A square \(Q\) is called \emph{admissible} if \(s(Q)\le1\) and
\(\overline{\D_{Rs(Q)}(x_Q)}\) lies in the fixed coordinate region
where the coefficient bounds hold.  All descendants of the chosen root
are admissible.  The stronger \(2R\) margin at the root reserves room
for nearby squares outside its subdivision tree.

Proposition~\ref{prop:initial-growth} ensures that the local norms in
\eqref{eq:growth} are nonzero.  In particular, \(E\), \(m\), and \(N\)
are finite and nonnegative, since the disks are nested.  Comparing the
exponential weight on the two disks gives
\begin{equation}
 N(Q)\le m(Q,b)+(R+1)\abs{b}.
\label{eq:untilted-comparison}
\end{equation}
Indeed the outer untilted norm is at most \(e^{KR\abs b}\) times its
tilted norm, whereas the inner untilted norm is at least
\(e^{-K\abs b}\) times its tilted norm, where \(K=K_Q\).
The same definitions and estimates apply to finite subtrees of an
extended translated grid consisting of admissible squares.

The normalization by \(K_Q\) explains why subdivision alone is
insufficient.  Put \(\rho=A^{-1}\).  An affine profile with slope
\(c\ne0\) has a difference of suprema equal to
\((R-1)\rho|c|\) between concentric disks of radii \(R\rho\) and
\(\rho\).  Division by the child frequency \(K_q=\rho K_Q\)
cancels this first-order gain.  We will subtract the local affine
part to obtain an average profile excess of order
\(\rho^2(1+|\log\rho|)\), while keeping the average correction slope
bounded.  The latter bound controls the cost of changing weights
through successive subdivisions.

\begin{lemma}[Uniform potential decomposition]
\label{lem:potential-decomposition}
Let \(V\) be subharmonic on \(\D_R\), with
\[
 V\le0,
 \qquad \sup_{\overline{\D_1}}V\ge-1.
\]
Then \(\norm{V}_{L^1(\D_8)}\) and the mass of its Riesz measure
\((2\pi)^{-1}\Delta V\) on \(\D_5\) are uniformly bounded.
There are a nonnegative measure \(\nu\) on \(\D_4\), of uniformly
bounded mass, and a harmonic function \(h\) on \(\D_3\) such that
\[
 V(x)=h(x)+\int_{\D_4}\log\abs{x-z}\,d\nu(z),
 \qquad x\in\D_3.
\]
The first and second derivatives of \(h\) on \(\D_2\) are uniformly
bounded.  All bounds are independent of \(V\), and the representation
holds for the pointwise subharmonic representatives, including values
equal to \(-\infty\).
\end{lemma}

\begin{proof}
Upper semicontinuity supplies a point
\(x_*\in\overline{\D_1}\) with \(V(x_*)\ge-1\).  The disk
\(\D_{10}(x_*)\) contains \(\D_8\) and is contained in \(\D_R\).
In particular, \(V\) is nontrivial and locally integrable.
Since \(V\le0\), the submean inequality gives
\[
 \int_{\D_8}\abs{V}
 \le -\int_{\D_{10}(x_*)}V
 \le100\pi.
\]
Its distributional Laplacian is a nonnegative Radon measure.
Choose a fixed nonnegative
smooth cutoff \(\psi\), supported in \(\D_8\) and equal to one on
\(\D_5\).  If \(\eta=(2\pi)^{-1}\Delta V\), then
\[
 \eta(\D_5)
 \le\int\psi\,d\eta
 =\frac1{2\pi}\int V\Delta\psi
 \le C\norm{V}_{L^1(\D_8)}.
\]
Let \(\nu\) be the restriction of \(\eta\) to \(\D_4\), and set
\(P(x)=\int_{\D_4}\log\abs{x-z}\,d\nu(z)\).
The elementary bound
\[
 \sup_{z\in\D_4}\int_{\D_3}\bigl|\log\abs{x-z}\bigr|\,dx<\infty
\]
shows that \(\norm{P}_{L^1(\D_3)}\le C\nu(\D_4)\).
The logarithmic fundamental solution identity
\(\Delta\log\abs{x-z}=2\pi\delta_z\) implies
\(\Delta(V-P)=0\) on \(\D_3\).  Thus \(V-P\) agrees almost everywhere
with a harmonic function \(h\).  Interior harmonic estimates, applied
to its bounded \(L^1(\D_3)\) norm, bound \(\nabla h\) and \(\nabla^2h\)
on \(\D_2\).  Finally, the logarithmic potential of a nonnegative
measure has its usual subharmonic representative.  Both \(V\) and
\(h+P\) are recovered pointwise as limits of their disk averages.
Their almost-everywhere equality therefore gives the asserted
pointwise representation.
\end{proof}

\begin{lemma}[Affine correction of a profile]
\label{lem:affine-profile}
Let \(V\), \(h\), and \(\nu\) be as in
Lemma~\ref{lem:potential-decomposition}.  Subdivide the unit square
\([-1/2,1/2]^2\) into \(A^2\) squares, where \(A>10R\), put
\(\rho=A^{-1}\), and denote their centers by \(y\).  Define
\[
 c_y=\nabla h(y)
   +\int_{\substack{z\in\D_4\\\abs{z-y}>2R\rho}}
        \frac{y-z}{\abs{y-z}^2}\,d\nu(z).
\]
Then
\begin{equation}
\begin{aligned}
 &\sup_{\overline{\D_{R\rho}(y)}}(V(x)-c_y\cdot x)
     -\sup_{\D_\rho(y)}(V(x)-c_y\cdot x)\\
 &\hspace{1cm}\le
 C\left(\rho^2+\int_{\D_4}
      \min\left\{1,\frac{\rho^2}{\abs{z-y}^2}\right\}\,d\nu(z)
   \right).
\end{aligned}
\label{eq:affine-error}
\end{equation}
At \(z=y\) the minimum in this formula is interpreted as one.
Moreover,
\[
 \av_y\abs{c_y}\le C,
 \qquad
 \av_y\left[\text{right-hand side of \eqref{eq:affine-error}}\right]
 \le C\rho^2(1+\abs{\log\rho}).
\]
The constants may depend on \(R\), but are independent of \(A\) and
\(V\).
\end{lemma}

\begin{proof}
All disks \(\overline{\D_{R\rho}(y)}\) lie in \(\D_2\).  The vector
\(c_y\) is finite because its kernel is truncated away from its
singularity.  Put \(W_y(x)=V(x)-c_y\cdot x\), and lower bound its
supremum on the inner disk by its disk average there.  For the
harmonic term, this average is its value at \(y\), after the affine
part is accounted for.  Taylor's theorem bounds the corresponding
outer deviation by \(C\rho^2\).

For a far source \(\abs{z-y}>2R\rho\), the logarithm is harmonic
on the outer disk and its Hessian is bounded there by
\(C\abs{z-y}^{-2}\).  Its inner disk average is
\(\log\abs{y-z}\), and the retained gradient in \(c_y\) removes
its first-order term.  Its contribution to the outer deviation is
therefore at most \(C\rho^2/\abs{z-y}^2\).

For a near source \(\abs{z-y}\le2R\rho\), rescale by \(\rho\).
Writing \(t=(z-y)/\rho\), the disk-average formula
\[
 \frac1\pi\int_{\D_1}\log\abs{\xi-t}\,d\xi
 =
 \begin{cases}
  (\abs{t}^2-1)/2,&\abs{t}\le1,\\
  \log\abs{t},&\abs{t}\ge1
 \end{cases}
\]
has lower bound \(-1/2\).  On the larger disk the corresponding
logarithm is at most \(\log(3R)\), since \(\abs t\le2R\).
The common term \(\log\rho\) cancels between these two quantities.
Thus the difference between the outer upper bound and the inner
average is at most \(\log(3R)+1/2\), uniformly in the source position.
This calculation permits an atom at \(y\), or anywhere else in
the inner disk: the logarithm remains integrable in the averaging
variable.  At an outer point where a potential is \(-\infty\), an
upper bound causes no difficulty.  Because the measure is
nonnegative, the kernel bounds may be integrated.  On the near
region,
\[
 \min\left\{1,\frac{\rho^2}{\abs{z-y}^2}\right\}
 \ge\frac1{4R^2}.
\]
Combining the near and far bounds proves
\eqref{eq:affine-error}.

For completeness, fix any \(z\in\D_4\).  The number of grid centers
within distance \(r\) of \(z\) is at most
\(C(1+r^2/\rho^2)\).  There are \(\rho^{-2}\) centers in total.
The centers at distance at most \(\rho\) contribute \(C\rho^2\)
to the averaged minimum kernel.  In each dyadic annulus
\(2^\ell\rho<\abs{z-y}\le2^{\ell+1}\rho\), the number of centers
is at most \(C2^{2\ell}\), whereas the kernel is at most
\(2^{-2\ell}\).  All distances are bounded by five, so only
\(C(1+\abs{\log\rho})\) annuli occur.  Hence, uniformly in \(z\),
\[
 \av_y\min\left\{1,\frac{\rho^2}{\abs{z-y}^2}\right\}
 \le C\rho^2(1+\abs{\log\rho}).
\]
The same annuli, for the truncated gradient kernel, give
\[
 \rho^2\sum_{\substack{y\\\abs{z-y}>2R\rho}}
 \frac1{\abs{z-y}}
 \le C.
\]
Indeed an annulus of radius comparable to \(2^\ell\rho\)
contributes at most \(C2^\ell\rho\), and these radii sum to a
fixed constant.  Integrating the two bounds against the bounded
measure \(\nu\), and using the bound for \(\nabla h\), proves
both averaged estimates.
\end{proof}

\begin{proposition}[One subdivision step]
\label{prop:subdivision}
There are an integer \(A>10R\) and constants \(C_0,C_1>0\), depending
only on the fixed coordinate coefficient bounds and \(R\), with the
following property.  Let \(Q\) be an admissible square of side
\(s\le1\), let \(K=ks>0\), and let \(b\in\R^2\) and \(S>0\).
Put \(B=Kb\).  If
\begin{equation}
 S\ge C_0,\qquad E(Q,b)\le S,\qquad
 \frac{K}{S+\abs B}\le\frac1{C_0},\qquad
 \frac{K^2s}{S(S+\abs B)}\le\frac1{C_0},
\label{eq:subdivision-hypotheses}
\end{equation}
then real vectors \(b_q\) can be chosen for the children \(q\prec Q\)
so that
\begin{equation}
 \av_{q\prec Q}\abs{b_q-b}\le C_1\frac SK,
 \qquad
 \av_{q\prec Q}m(q,b_q)\le\frac12\frac SK.
\label{eq:subdivision-estimate}
\end{equation}
The constants are independent of the eigenfunction, the scale,
the depth of a tree, and any lower bound for \(K\).
\end{proposition}

\begin{proof}
First choose \(A>10R\), using only the uniform estimates of
Lemma~\ref{lem:affine-profile}, so large that, with \(\rho=A^{-1}\),
\[
 C\rho^2(1+\abs{\log\rho})<\frac\rho4.
\]
Choose \(C_1\) strictly larger than the uniform bound for
\(\av_y\abs{c_y}\) in that lemma.  We prove that some finite
\(C_0\) works for these fixed choices.

Suppose otherwise.  There is a sequence of failures of the
conclusion for which the hypotheses hold with \(C_0=j\).
Use subscripts \(j\) for the associated quantities and rescale each
parent square to the unit square centered at zero:
\[
 U_j(X)=u_j(x_{Q_j}+s_jX),
 \qquad p_j(X)=p(x_{Q_j}+s_jX).
\]
The rescaled equation is
\(\Delta U_j+K_j^2p_jU_j=0\), and
\(\norm{p_j}_\infty\le C\),
\(\norm{\nabla p_j}_\infty\le Cs_j\).
With \(B_j=K_jb_j\) and \(D_j=S_j+\abs{B_j}\), the failure
hypotheses imply
\[
 S_j\longrightarrow\infty,\qquad
 \frac{K_j}{D_j}\longrightarrow0,\qquad
 \frac{K_j^2\norm{\nabla p_j}_\infty}{S_jD_j}
 \longrightarrow0.
\]
No restriction on \(\abs{B_j}/S_j\) has been imposed.

Normalize the tilted measures on \(\D_R\) by setting
\[
 d\mu_j(X)=
 \frac{e^{-2B_j\cdot X}\abs{U_j(X)}^2\,dX}
 {\displaystyle\int_{\D_R}e^{-2B_j\cdot X}\abs{U_j(X)}^2\,dX}.
\]
Lemma~\ref{lem:profile-extraction} and
Theorem~\ref{thm:subharmonic-profile} give, after passage to a
subsequence, a subharmonic profile \(V\), possibly identically
\(-\infty\), on \(\D_R\).  It satisfies \(V\le0\), while
\[
 \mu_j(\D_1)=e^{-2E(Q_j,b_j)}\ge e^{-2S_j}.
\]
The compact upper bound in \eqref{eq:profile-bounds}, applied to
\(\overline{\D_1}\), therefore gives
\(\sup_{\overline{\D_1}}V\ge-1\).  In particular, \(V\) is
nontrivial and the two preceding lemmas apply.

For each of the fixed, finitely many child centers \(y\), take
the vector \(c_y\) furnished by Lemma~\ref{lem:affine-profile} for
this limiting profile.  On the corresponding child \(q_j\), set
\[
 b_{q_j}=b_j+\frac{S_j}{K_j}c_y.
\]
In parent coordinates the additional weight is
\(e^{-2S_jc_y\cdot X}\).  To see the cancellation of recentering
constants, write
\[
 -K_jb_{q_j}\cdot(X-y)
 =-B_j\cdot X-S_jc_y\cdot X
   +(B_j+S_jc_y)\cdot y.
\]
The last term, as well as the measure normalization and the
coordinate Jacobian, cancels in the ratio defining \(E(q_j,b_{q_j})\).
Consequently this excess is one half the logarithm of
\[
 \frac{\displaystyle\int_{\D_{R\rho}(y)}
                  e^{-2S_jc_y\cdot X}\,d\mu_j(X)}
      {\displaystyle\int_{\D_\rho(y)}
                  e^{-2S_jc_y\cdot X}\,d\mu_j(X)}.
\]
Apply the weighted compact upper bound to the numerator, using
\(\overline{\D_{R\rho}(y)}\), and the weighted open lower bound
to the denominator.  This yields
\[
 \limsup_j\frac{E(q_j,b_{q_j})}{S_j}
 \le
 \sup_{\overline{\D_{R\rho}(y)}}(V(X)-c_y\cdot X)
 -\sup_{\D_\rho(y)}(V(X)-c_y\cdot X).
\]
The inner supremum is finite: a nontrivial subharmonic function is
locally integrable, so it cannot equal \(-\infty\) throughout an
open disk, and it is bounded above on compact subsets.  Both
suprema in the displayed difference are therefore finite.

There are only \(A^2\) children, and \(A\) has already been fixed.
Averaging the last inequality and invoking
Lemma~\ref{lem:affine-profile} gives
\[
 \limsup_j\av_{q_j\prec Q_j}
       \frac{E(q_j,b_{q_j})}{S_j}<\frac\rho4.
\]
Since \(K_{q_j}=\rho K_j\), it follows, for all sufficiently large
\(j\), that
\[
 \av_{q_j\prec Q_j}m(q_j,b_{q_j})
 \le\frac12\frac{S_j}{K_j}.
\]
At the same time,
\[
 \av_{q_j\prec Q_j}\abs{b_{q_j}-b_j}
 =\frac{S_j}{K_j}\av_y\abs{c_y}
 \le C_1\frac{S_j}{K_j}.
\]
Thus these choices satisfy both required inequalities, contrary
to the defining failure of the sequence.  This proves the
proposition.  In particular, \(A\) and \(C_1\) have been fixed
before \(C_0\), and no positive lower bound for \(K\) was used.
\end{proof}
==== END Sharp-nodal-length-on-smooth-surfaces//subdivision.tex ====