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WEAK HESSIAN BOUNDS ALONG EVERY GEODESIC IN RCD SPACES
A self-contained proof for a curious high-school reader
========================================================
1. WHAT THIS DOCUMENT DOES.
===========================
Suppose (M, d, m) is a metric space with distance d, reference
measure m, and a lower Ricci-curvature bound K and an upper
dimension bound N, all in a sense made precise in section 2.
Suppose F: M -> R is bounded and globally Lipschitz, and
G: M -> R is bounded and continuous. Suppose the DISTRIBUTIONAL
Hessian of F is bounded above by G, in the sense that whenever
you test it against a compactly supported nonnegative weight h,
H_F(nabla g, nabla g)(h) <= int_M h G Gamma(g) dm. (HB)
This inequality "looks at" typical points of M, because it is a
statement about integrals against dm. But a single geodesic
usually has m-measure zero. Our theorem says:
THEOREM 1. Under the hypotheses above, for every constant-speed
minimizing geodesic sigma: [0, 1] -> M of length ell, the
one-variable function F o sigma: [0, 1] -> R obeys
(F o sigma)''(t) <= ell^2 G(sigma(t)) (CONC)
as an inequality between distributions on (0, 1), i.e.
int_0^1 F(sigma(t)) phi''(t) dt
<= ell^2 int_0^1 G(sigma(t)) phi(t) dt
for every nonnegative phi in C_c^infty(0, 1).
In the special case where G = c is a constant, (CONC) says that
the function t |-> F(sigma(t)) - c ell^2 t^2 / 2 is concave on
[0, 1].
This document gives a complete proof, with all the technical
pieces either reduced to a small lemma or cited from the modern
literature.
2. VOCABULARY (WITH PICTURES).
==============================
Metric measure space. A triple (M, d, m): M a set, d a distance,
m a nonnegative measure. We assume M is complete and separable,
has the geodesic property (any two points are joined by a shortest
path), has FULL SUPPORT (m(U) > 0 for every nonempty open U),
and is LOCALLY FINITE (m of any bounded ball is finite).
RCD(K, N) with 1 < N < infinity. Briefly: the triple behaves
like a Riemannian manifold of dimension at most N with Ricci
curvature at least K, but we are not assuming any smooth atlas.
Equivalently, Wasserstein warp of measures by t |-> mu_t
satisfies a lower-convexity inequality for the RCD entropy with
strength K. The full analytic consequences (differential
calculus, Bochner inequality, Sobolev-to-Lipschitz) are in the
references; we cite them as we use them.
Minimizing geodesic. A path sigma: [0, 1] -> M of LENGTH ell
satisfying d(sigma(s), sigma(t)) = ell |t - s| for all s, t.
Constant-speed minimizing. "Velocity" |sigma-dot| means ell.
Sobolev space W^{1,2}(M), weak gradient, and Laplacian. A
function u in L^2(m) lies in W^{1,2}(M) iff it has a weak
gradient |nabla u| in L^2(m) in the sense of Cheeger energy
Ch(u) := (1/2) int_M |nabla u|^2 dm.
Its L^2 generator Delta u is defined by the integration-by-parts
identity
int_M <nabla u, nabla v> dm = -int_M (Delta u) v dm.
The domain D(Delta) is the set of u in W^{1,2} with Delta u in
L^2; it is contained in W^{1,2}.
Carre du champ. Gamma(u, v) = <nabla u, nabla v>, and
Gamma(u) = Gamma(u, u).
Second-order object. The Hessian Hess g of a test function g is
a symmetric bilinear form on the cotangent bundle, packaged in
metric-measure language. It satisfies
nabla_{nabla g} nabla g = (1/2) nabla Gamma(g).
The measure-valued Bochner Laplacian is
Gamma_2(g) := Delta(Gamma(g)/2) - Gamma(g, Delta g),
and the BOCHNER INEQUALITY for RCD(K, N) is
Gamma_2(g) >= K Gamma(g) m + |Hess g|^2 m.
RCD(K, N) for finite N. One consequence: every Sobolev function
with |nabla u| <= L (a.e.) has a globally L-Lipschitz
representative. Another consequence: finite-dimensional RCD(K, N)
spaces are NONBRANCHING (Deng 2020).
CD(k, infty) with variable k. A weaker, but equivalent-when-k is
constant version: with k: M -> R bounded and lower semicontinuous,
the Boltzmann entropy Ent_m is k-convex along Wasserstein geodesics
in the correction form
Ent_m(mu_t) <= (1 - t) Ent_m(mu_0) + t Ent_m(mu_1) (CD-k)
- int_{Geo(M)} int_0^1 (min(s, t) - st) k(gamma_s)
|gamma-dot|^2 ds dpi(gamma).
Here pi is a DYNAMICAL OPTIMAL PLAN (a probability measure on
the path space Geo(M) of constant-speed minimizing paths, whose
endpoint coupling is optimal for quadratic cost) and mu_t =
(e_t)_# pi with e_t(gamma) = gamma(t). For constant k, (CD-k)
is the standard CD(K, infty) condition; the equivalence with
RCD(K, infty) is AGS 2015 and BHS 2021.
Test class. Let
Test(M) := { g in D(Delta) : g has a bounded globally
Lipschitz representative and Delta g in W^{1,2}(M) }.
The extreme case is Test^infty(M), which demands Delta g in
L^infty(M) as well.
Weak Hessian of an arbitrary W^{1,2}_{loc} function. For F in
W^{1,2}_{loc}(M), g in Test(M) with compact support, and h in
Lip_c(M),
H_F(nabla g, nabla g)(h)
:= -int_M div(h nabla g) <nabla F, nabla g> dm
-int_M h <nabla F, nabla_{nabla g} nabla g> dm.
Here div(h nabla g) = Gamma(h, g) + h Delta g. When F in D(Delta)
with an L^2 Hessian, this agrees with the standard formula.
3. STRUCTURE OF THE PROOF.
==========================
We follow three steps.
Step A. Weight m by e^{lambda F}, with lambda > 0 large. Show
that the resulting measure m_lambda = e^{lambda F} m
keeps Sobolev space and Laplacian domain unchanged, and
that a bounded function whose V-Laplacian is also
bounded is globally Lipschitz. This is Lemma 2.
Step B. Under the assumed Hessian bound (HB), show that
m_lambda satisfies CD(K - lambda G, infty) for every
lambda > 0. This is Proposition 3.
Step C. Fix a strict subsegment of the geodesic, put shrinking
balls at its endpoints, and let lambda -> infinity fast
enough to absorb endpoint entropies. The CD inequality
for m_lambda, rearranged and pushed to the limit, yields
a Green-kernel concavity of F o sigma on every strict
subinterval of (0, 1). Concavity gives (CONC).
4. STEP A: BOUNDED DRIFT.
=========================
LEMMA 2 (Bounded drift). Let (M, d, m) be a full-support RCD(K, N)
space, 1 < N < infinity. Let V: M -> R be bounded and globally
Lipschitz. Set m_V = e^V m, and let Delta_V be the L^2 generator
of the V-Cheeger energy with respect to m_V. Then:
(a) D(Delta_V) = D(Delta), and
Delta_V u = Delta u + Gamma(V, u).
(b) The two graph norms
||u||_{L^2(m)} + ||Delta u||_{L^2(m)} and
||u||_{L^2(m_V)} + ||Delta_V u||_{L^2(m_V)}
are equivalent.
(c) If xi in D(Delta_V) satisfies xi, Delta_V xi in L^infty,
then xi has a globally Lipschitz representative.
Proof. (a) and (b). Since e^V is bounded above and below by
positive constants, the class of admissible test plans (the
"Wasserstein-language" definition of weak upper gradient) is
the same for m and for m_V. Thus weak gradients agree and the
W^{1,2} norms are equivalent. In the weak definition of the
generator, substitute eta -> e^V eta to pass between Delta and
Delta_V: this gives Delta_V u - Delta u = Gamma(V, u) as
distributions. All drift terms are in L^2, because
|Gamma(V, u)| <= |nabla V| |nabla u| and |nabla V| is bounded.
For (b), the energy identity
||nabla u||^2_{L^2(m)} = -int_M u Delta u dm,
together with its weighted analogue and the identity in (a),
gives bounds each way.
(c). Let P_t = e^{t Delta} be the ordinary heat semigroup
(for V = 0). For h in L^2 cap L^infty and a > 0 large, put
R_a h := int_0^infty e^{-a t} P_t h dt = (a - Delta)^{-1} h.
Standard estimates for the RCD heat flow (Bakry gradient estimate
and reverse Poincare) give, uniformly in h and a >= 1:
||R_a h||_p <= a^{-1} ||h||_p, p = 2, infinity,
||nabla R_a h||_2 <= C_1 a^{-1/2} ||h||_2,
||nabla R_a h||_infty <= C_2 a^{-1/2} ||h||_infty.
The last one follows by integrating the reverse Poincare
inequality
||nabla P_t h||_infty <= C_3 (t^{-1/2} + 1) e^{C_3 t} ||h||_infty
over t in (0, infinity) with weight e^{-a t}.
Consider the Banach space
E := { z in W^{1,2} : z and |nabla z| are in L^infty },
||z||_E := ||z||_2 + ||nabla z||_2 + ||z||_infty + ||nabla z||_infty.
Completeness follows because L^infty is complete and the weak
differential is closed.
For a large enough, the map
z |-> R_a (h + Gamma(V, z))
is a strict contraction on E. Indeed, the drift term
Gamma(V, z) is bounded in each L^p norm by
||nabla V||_infty ||nabla z||_p <= ||nabla V||_infty ||z||_E,
while R_a contracts in L^2, L^infty, and in the gradient norms
as listed. Its fixed point z lies in E and satisfies
(a - Delta_V) z = h. Uniqueness of solutions of
(a - Delta_V) z = h in D(Delta_V) (from the energy identity)
then forces every other solution to coincide with this fixed
point.
Now take xi in D(Delta_V) with xi, Delta_V xi in L^infty. Put
h := (a - Delta_V) xi in L^2 cap L^infty. By uniqueness, xi IS
the fixed point above, hence nabla xi is in L^infty, and by
Sobolev-to-Lipschitz, xi is globally Lipschitz. []
5. STEP B: CURVATURE CHANGE.
============================
PROPOSITION 3. Let (M, d, m) be a full-support RCD(K, N) space,
K in R, 1 < N < infinity. Let F: M -> R be bounded and globally
Lipschitz, and let G: M -> R be bounded and continuous. Assume
(HB). Then for every lambda > 0, the measure
m_lambda := e^{lambda F} m
satisfies CD(K - lambda G, infty) in the same metric d.
Proof. Fix lambda > 0. Write V = lambda F, w = e^V, and
k = K - lambda G. The weak Hessian is linear in its first
argument, so H_V = lambda H_F.
Define the weighted Bochner pairing, for g in Test(M) with
compact support and a nonnegative bounded Lipschitz function xi
on supp g:
B_V(g, xi)
:= -int_M w Gamma(xi, q) dm
+ int_M w [xi Delta_V g + Gamma(xi, g)] Delta_V g dm,
where q = Gamma(g)/2. The formula with V = 0 defines B_0(g, eta)
for an arbitrary multiplier eta; it is meaningful for
eta = w xi after localizing around supp g.
Step 1. The original measure-valued Bochner inequality,
integrated against the multiplier w xi, gives
B_0(g, w xi) >= K int_M w xi Gamma(g) dm. (*)
Step 2. Using the identity Delta_V u = Delta u + Gamma(V, u)
and the product rule
Delta(uv) = u Delta v + v Delta u + 2 Gamma(u, v),
a direct computation gives the comparison identity
B_V(g, xi) - B_0(g, w xi)
= int_M w xi Gamma(V, q) dm
+ int_M w Gamma(V, g)
[Gamma(xi, g) + xi Delta_V g] dm
= -H_V(nabla g, nabla g)(w xi). (**)
The second equality is the definition of the weak Hessian
applied to V against weight w xi, because
div(w xi nabla g) = w (Gamma(xi, g) + xi Delta_V g).
Step 3. Since V = lambda F, we have H_V = lambda H_F, so by (HB)
-H_V(nabla g, nabla g)(w xi)
>= -lambda int_M w xi G Gamma(g) dm.
Adding (*) and (**) yields, after collecting the
curvature contribution,
B_V(g, xi) >= int_M k xi Gamma(g) dm_lambda. (***)
Step 4. Close (***) in the Delta_V graph norm.
On the left, integrate by parts in the first term:
B_V(g, xi) =
(1/2) int_M (Delta_V xi) Gamma(g) dm_lambda
+ int_M (Delta_V g) Gamma(xi, g) dm_lambda
+ int_M xi (Delta_V g)^2 dm_lambda.
Every term on both sides of (***) is continuous under
convergence in the Delta_V graph norm. The compactly
supported Test^infty functions form a graph-norm core
for Delta_V (Gigli-Tamanini 2021, Appendix A). So (***)
in the rewritten form holds for every g in D(Delta_V).
When moreover Delta_V g is in W^{1,2}, an integration by
parts combines the last two terms and gives the weighted
Bakry-Emery inequality
(1/2) int_M (Delta_V xi) Gamma(g) dm_lambda
- int_M xi Gamma(g, Delta_V g) dm_lambda
>= int_M k xi Gamma(g) dm_lambda. (BE)
Step 5. From BE on the graph domain to CD.
Replace k by the constant K_V := inf_M k > -infinity.
Weak gradients are unchanged under m_lambda, so
(M, d, m_lambda) with Cheeger energy 2 Ch_{m_lambda} is
a strongly local symmetric Dirichlet form with the same
carre du champ Gamma.
Completeness, separability, full support, and local
finiteness give structure condition (MD) of AGS 2015.
Sobolev-to-Lipschitz gives (ED.a) and (ED.b) of AGS
2015. Theorem 3.9 of AGS 2015 identifies the intrinsic
distance as d. Theorem 3.14 preserves Gaussian volume
growth (bounded equivalence preserves it) and gives
conservativeness, upper regularity, and Riemannian
structure. Theorem 4.17 of AGS 2015 applied to (BE) with
constant K_V gives RCD(K_V, infty) for m_lambda.
On this RCD space, (BE) with the bounded continuous k is
precisely the BE_2(k, infty) condition; Theorem 1.1 and
Definition 1.4 of BHS 2021 give CD(k, infty). []
6. STEP C: LOCALIZING TO ONE GEODESIC.
======================================
Proof of Theorem 1. Fix a constant-speed minimizing geodesic
sigma: [0, 1] -> M of length ell. If ell = 0 the claim is
trivial; assume ell > 0.
Convention. By a STRICT subsegment of sigma we mean sigma
restricted to a closed subinterval [a, b] with 0 < a < b < 1,
then reparametrized affinely onto [0, 1]. Call the resulting
path tau: [0, 1] -> M. It is again a constant-speed minimizing
geodesic, with velocity |tau-dot| = ell (b - a). We will prove
(CONC) on each strict subsegment; the union of all such
subsegments is (0, 1), so this proves (CONC) on (0, 1).
Step 1. Uniqueness of the subsegment.
A finite-dimensional RCD(K, N) space is nonbranching
(Deng 2020). Therefore every strict subsegment tau of
sigma is the UNIQUE minimizing geodesic between its
endpoints. Indeed, if another minimizing geodesic
tau' joined tau(0) to tau(1), inserting it between the
two pieces sigma|_{[0, a]} and sigma|_{[b, 1]} would
produce a competing minimizing geodesic from sigma(0)
to sigma(1), agreeing with sigma on a nontrivial
initial interval. That is branching, which is
forbidden.
Step 2. Concentration at the endpoints.
Pick eps_j -> 0 with eps_j > 0. Set
mu_i^j := m|_{B_{eps_j}(tau(i))} / m(B_{eps_j}(tau(i))),
E_i^j := Ent_m(mu_i^j) = -log m(B_{eps_j}(tau(i))),
for i = 0, 1. Full support and local finiteness make
each quantity finite. Choose
lambda_j >= j (1 + |E_0^j| + |E_1^j|).
By Proposition 3, m_{lambda_j} satisfies
CD(K - lambda_j G, infty), so it supplies a dynamical
optimal plan pi_j with endpoints mu_0^j, mu_1^j and
intermediate marginals mu_t^j := (e_t)_# pi_j
satisfying (CD-k) with k = K - lambda_j G.
Step 3. Compactness of paths.
All pi_j have geodesics in a common compact set Q (a
big ball containing sigma, by properness of M), with
length bounded by L := 2 ell (for j large, since
eps_j -> 0). Arzela-Ascoli gives
compactness of {pi_j} in the space of probability
measures on the path space. Every limit of the plans
pi_j must be the delta-measure at tau, by Step 1.
Hence pi_j -> delta_tau weakly.
Step 4. Rearranging the entropy inequality.
For any probability measure mu with finite entropy
relative to m (hence to m_{lambda_j} on the bounded
region Q), the shift identity holds:
Ent_{m_{lambda_j}}(mu) = Ent_m(mu)
- lambda_j int_M F dmu.
Using it at t = 0, t = 1, and interior t in (CD-k),
collecting terms, dividing by lambda_j > 0, and
defining the path-space averages
A_t^j := int_M F d mu_t^j,
E_t^j := Ent_m(mu_t^j),
C_t^j := int_{Geo(M)} int_0^1 (min(s, t) - st)
|gamma-dot|^2 ds dpi_j(gamma),
D_t^j := int_{Geo(M)} int_0^1 (min(s, t) - st)
G(gamma_s) |gamma-dot|^2 ds dpi_j(gamma),
one obtains, for each t in [0, 1],
A_t^j >= (1 - t) A_0^j + t A_1^j - D_t^j
+ [E_t^j - (1 - t) E_0^j - t E_1^j
+ K C_t^j] / lambda_j. (R1)
Step 5. Take j -> infinity.
Since all paths in pi_j are contained in the compact set
Q, Jensen's inequality gives E_t^j >= -log m(Q). Since
int_0^1 (min(s, t) - st) ds = t(1 - t)/2 <= 1/8, one
has 0 <= C_t^j <= L^2 / 8. So the bracket in (R1) is
bounded below by
-log m(Q) + min(K L^2 / 8, 0),
independent of j and t. With lambda_j -> infinity, the
last term of (R1) has liminf >= 0 uniformly in t.
Continuity of F and G on the compact Q, and the weak
convergence pi_j -> delta_tau from Step 3, give, for
each fixed t in [0, 1],
F(tau_t) >= (1 - t) F(tau_0) + t F(tau_1)
- |tau-dot|^2 int_0^1 (min(s, t) - st)
G(tau_s) ds. (R2)
Step 6. Concavity, and the distributional inequality.
Write g(s, t) := min(s, t) - st for the Dirichlet Green
function of d^2/dt^2 on [0, 1] with zero boundary. Its
second t-derivative is d^2 g / dt^2 (s, t) = -delta(s - t)
in the distributional sense. Define
a(t) := -|tau-dot|^2 int_0^1 g(s, t) G(tau_s) ds,
for t in [0, 1]. Then a(0) = a(1) = 0 (because g(s, 0) =
g(s, 1) = 0), and by the second-derivative of g,
a''(t) = |tau-dot|^2 G(tau_t),
distributionally on (0, 1).
Now (R2) says precisely
F(tau_t) - a(t)
>= (1 - t) (F(tau_0) - a(0))
+ t (F(tau_1) - a(1)). (R3)
And Step 5 can be applied to ANY subinterval [c, d] of
[0, 1]: restrict tau to [c, d], reparametrize onto
[0, 1], and re-run the argument. Let a_cd denote the
analog of a on [c, d], satisfying a_cd'' = |tau-dot|^2
G(tau) there and a_cd(c) = a_cd(d) = 0. Then
u_cd(s) := F(tau_s) - a_cd(s) is above its chord:
u_cd((1 - theta) c + theta d)
>= (1 - theta) u_cd(c) + theta u_cd(d).
The difference a - a_cd is affine on [c, d] (both have
the same second derivative |tau-dot|^2 G(tau) there).
Subtracting an affine function preserves "above the
chord", so u(s) := F(tau_s) - a(s) satisfies the same
chord inequality on [c, d]. As [c, d] and theta are
arbitrary, u is CONCAVE on [0, 1]. A continuous concave
function on an interval has nonpositive second
distributional derivative, so u'' <= 0 distributionally
on (0, 1).
Therefore
(F o tau)''(t) <= a''(t) = |tau-dot|^2 G(tau_t),
distributionally on (0, 1).
Step 7. Back to sigma.
Recall that tau(t) = sigma(a + t(b - a)), so |tau-dot| =
ell(b - a), and |tau-dot|^2 = ell^2 (b - a)^2. The
change of variables t = (s - a)/(b - a) from s in (a, b)
to t in (0, 1) scales second derivatives by (b - a)^2:
(F o sigma)''(s) = (b - a)^{-2} (F o tau)''(t)
<= (b - a)^{-2} ell^2 (b - a)^2
G(tau(t))
= ell^2 G(sigma(s)).
This holds distributionally on every strict subinterval
(a, b) with 0 < a < b < 1, hence on the whole (0, 1).
That is the asserted inequality. []
7. WHAT IS GENUINELY DIFFICULT.
===============================
(a) Bounded drift. Bounded above-and-below equivalence of
measures is not enough to preserve the graph domain when V
is only Lipschitz; the resolvent contraction in Lemma 2(c)
is what bridges the gap. Without it, the weighted Bochner
inequality has no legitimate domain to close in.
(b) Variable-curvature equivalence. The passage from BE_2(k, infty)
to CD(k, infty) for bounded and continuous k, in the AGS 2015
and BHS 2021 setting, is the technical core of Step B. It
gives a WEAKER form of curvature than the constant RCD
structure; the trade is exactly the price we pay for a
non-smooth reference measure.
(c) Pointwise from measure. The entropy inequality of Step B is
measure-theoretic and lets a whole ensemble of geodesics
"smear" out. Step C uses two ingredients to pin the ensemble
down to a single curve: the CONCENTRATION of the endpoint
measures into small balls, and the UNIQUENESS of the strict
subsegment from nonbranching. Without either, the limit
would not be a single delta at tau.
8. A WORKED EXAMPLE (EUCLIDEAN, ONE LINE OF CALCULUS).
======================================================
Take M = R^n with the Euclidean metric, and m the standard
Gaussian measure on R^n. Then (M, d, m) is a full-support
RCD(1, n) space.
Example A. Let F(x) = <a, x> for a fixed vector a in R^n. Then
Hess F = 0, so (HB) holds with G = 0 (or any nonnegative
constant). Every minimizing geodesic is a straight line
sigma(t) = x + t v with |v| = ell. Then
(F o sigma)(t) = <a, x> + t <a, v>,
whose second derivative is 0, matching ell^2 G = 0.
Example B. Let F(x) = (1/2) |x|^2. Then Hess F is the identity
tensor, so (HB) holds with G = 1. Along sigma(t) = x + t v,
(F o sigma)(t) = (1/2) |x|^2 + t <x, v> + (1/2) t^2 |v|^2,
(F o sigma)''(t) = |v|^2 = ell^2 = ell^2 * 1.
Equality holds on every Euclidean geodesic: (F o sigma)''(t) =
ell^2 G(sigma(t)), matching the theorem.
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==============
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