Setup. What this file proves and for whom.
This file contains the full proof that, in a Cartan-Hadamard manifold
whose sectional curvature is at most kappa <= 0, every measurable
set of finite volume and finite perimeter has boundary area at least
that of the equal-volume round ball in the model space form of
curvature kappa. It also contains the full proof of the equality
case in flat curvature zero: equality can only happen if the set
itself, with the metric it inherits, is a round Euclidean ball.
Every step in the main argument is proved here. Standard
published tools enter as cited inputs: the Aubin-Talenti sharp
Euclidean Sobolev inequality, the Brezis-Lieb lemma, the
Guneysu-Pallara strict BV approximation on complete manifolds,
the Antonelli-Pasqualetto-Pozzetta local regularity of isoperimetric
boundaries, the Ghomi-Spruck obstacle regularity and Morgan
interior regularity, Kleiner's Theorem 2 in dimension three, and
the classical Weil-Bol inequality in dimension two. Each is named
at first use and can be tracked against the bibliography of the
original paper. Some intermediate lemmas are long and can be
skipped on a first pass; the skeleton does not need them.
Two remarks before any formula. First, the inequality is about
arbitrary sets, not just balls. Bishop-Gromov volume comparison
already tells you how balls behave; it says nothing about a long
thin blob, a dumbbell, or a region encrusted with wiggles. That is
the content of the conjecture. Second, "area of the boundary" needs
a robust definition for a rough set. We use De Giorgi's total
variation of the indicator function, spelled out below. In that
sense the theorem covers every measurable set of finite volume and
finite perimeter, no smoothness assumed.
The reader we have in mind is comfortable with multivariable
calculus and linear algebra, and is willing to accept three tools
as we introduce them: the exponential map, Jacobi fields, and
Sobolev spaces. Each is explained at first use.
Notation index (first-use terms).
M^n complete simply connected smooth Riemannian
manifold, dimension n >= 2, metric g.
Sec_g sectional curvature, one number per 2-plane.
kappa upper bound on Sec_g, kappa <= 0.
b sqrt(-kappa), b >= 0; b = 0 means kappa = 0.
d(p,x) Riemannian distance.
exp_p exponential map at p; a bijection by
Cartan-Hadamard.
log_p its inverse.
D_p(x) d(p,x)^2 / 2.
omega_n volume of the unit ball in flat R^n.
m n-1.
s_m area of the unit sphere S^m; s_m = n omega_n.
sn_b,cs_b model functions, defined next.
A_b(r) model sphere area of radius r.
V_b(r) model ball volume of radius r.
I_b(V) model isoperimetric profile of curvature -b^2.
vol_g(E) volume of E.
P_g(E) ambient perimeter of E.
chi_E indicator function of E.
|D chi_E| total variation measure of chi_E.
dsigma induced area on a hypersurface Sigma.
Model functions and the model profile. For b >= 0 and r >= 0 set
sn_b(r) = r when b = 0,
sn_b(r) = sinh(br)/b when b > 0,
cs_b(r) = sn_b'(r).
Two identities are used constantly:
sn_b'(r) = cs_b(r),
cs_b(r)^2 - b^2 sn_b(r)^2 = 1.
In the space form of constant curvature -b^2 a geodesic sphere of
radius r has area A_b(r) = s_m sn_b(r)^m, and a geodesic ball of
radius r has volume V_b(r) = s_m integral_0^r sn_b(t)^m dt. Both
functions are continuous and strictly increasing from 0 to
infinity, so V_b has an inverse. The model isoperimetric profile
of curvature -b^2 is
I_b(V) = A_b( V_b^{-1}(V) ), V >= 0.
In words: I_b(V) is the boundary area of a round ball of volume V
in the model space of curvature -b^2. We have I_b(0) = 0, and I_b
is continuous and strictly increasing on [0, infinity).
Perimeter of a rough set. For a measurable set E in a Riemannian
manifold (M^n, g) with volume measure vol_g, the ambient perimeter
is
P_g(E) = sup { integral_E div_g X dvol_g :
X smooth compactly supported vector field on M,
|X|_g <= 1 }.
The supremum runs over smooth compactly supported vector fields of
length at most one. This equals the total variation |D chi_E|_g(M)
of the indicator function. For a smooth region it equals boundary
area. For a wild set it can be finite even when the topological
boundary is a fractal dust. Sets with P_g(E) < infinity are exactly
the sets of finite perimeter. All statements are understood up to
sets of vol_g-zero.
Comparison theorem. Let n >= 2, kappa <= 0, and (M^n, g) a complete
simply connected smooth Riemannian manifold with Sec_g <= kappa.
Every measurable E subset M with vol_g(E) < infinity and
P_g(E) < infinity satisfies
P_g(E) >= I_b( vol_g(E) ), b = sqrt(-kappa).
When kappa = 0 this reads
P_g(E) >= n omega_n^{1/n} vol_g(E)^{(n-1)/n}.
When kappa = -1, writing vol_g(E) as the volume
n omega_n integral_0^r sinh^{n-1}(t) dt of a hyperbolic ball of
radius r, it reads
P_g(E) >= n omega_n sinh^{n-1}(r).
The comparison concerns full ambient perimeter; it includes any
boundary part lying on a sphere used in the proof. It covers
unbounded sets of finite volume. Round balls in the model spaces
attain equality.
Equality theorem. Let (M^n, g) be complete, simply connected and
smooth, with n >= 2 and Sec_g <= 0. Let E subset M be bounded and
measurable, with 0 < vol_g(E) < infinity and P_g(E) < infinity.
Then
P_g(E) = n omega_n^{1/n} vol_g(E)^{(n-1)/n}
holds if and only if E agrees with an open region U up to a set of
vol_g-zero, and (U, g restricted to U) is isometric to a round
Euclidean ball.
Said plainly: the equality region must itself be a flat ball. The
ambient manifold may be curved everywhere outside it. Boundedness
and positive volume are used only in the equality classification;
the comparison of the previous paragraph already covers finite
volume and finite perimeter.
Corollary (model-ball spectral and torsion). Let n >= 2, kappa <= 0
and 1 < p < infinity. Let (M^n, g) be as in the comparison theorem,
with no boundary. For a nonempty smooth domain D compactly contained
in M, let B_kappa(|D|) be a geodesic ball of volume |D| = vol_g(D)
in the model space form of curvature kappa. Then
lambda_{1,p}(D) >= lambda_{1,p}(B_kappa(|D|)),
T_p(D) <= T_p(B_kappa(|D|)),
where
lambda_{1,p}(D) = inf over 0 != u in W^{1,p}_0(D) of
integral_D |grad_g u|^p dvol_g /
integral_D |u|^p dvol_g,
and
T_p(D) = sup over 0 != u in W^{1,p}_0(D) of
(integral_D |u| dvol_g)^p /
integral_D |grad_g u|^p dvol_g.
These follow by rearranging any test function into a radial function
on the model ball without increasing the p-Dirichlet energy. The
rearrangement used is the one of Nobili-Violo. The proof appears at
the end.
How the proof goes, in one paragraph. Prove a sharp Sobolev
inequality on any hypersurface, with the mean curvature entering as
a potential. Use it inside a confined minimization problem: at each
prescribed volume v, minimize full ambient perimeter among sets
inside a large geodesic ball; the resulting profile I(v) satisfies
the scalar differential inequality I'(v) >= m sqrt(b^2 +
(s_m/I(v))^{2/m}). Integrate that inequality; its exact integral is
the model comparison. Handle dimensions 2 and 3 by the classical
Weil-Bol and Kleiner results; remove boundedness by a cutoff; rescale
to get every kappa. For equality, re-run the machinery on the
equality set to force its boundary to lie on a round sphere, then
compare a volume to force the inherited metric to be Euclidean. The
next sections unpack each step.
Distance Hessian comparison. Fix p in M and let r = d(p,.) be
distance. On the set where r > 0 define the symmetric (0,2)-tensor
H = Hess r - (cs_b(r)/sn_b(r)) (g - dr x dr).
Then H >= 0 as a quadratic form: H(X,X) >= 0 for every tangent
vector X. Here dr x dr is the tensor (X,Y) |-> dr(X) dr(Y), so the
subtracted piece is the angular part (perpendicular to the radial
geodesic).
Proof. Let gamma : [0, r] -> M be the unit-speed geodesic from p
to x, gamma(r) = x, and take X perpendicular to gamma'(r). A
Jacobi field is a variation field of a one-parameter family of
geodesics; it satisfies the ODE J'' + R(J, gamma') gamma' = 0,
where R is the Riemann curvature. Choose the family so J(0) = 0,
J(r) = X. The second variation formula for arc length gives
Hess r(X, X) = integral_0^r ( |D_t J|^2
- <R(J,gamma')gamma', J> ) dt.
The curvature assumption Sec <= -b^2 says the sectional curvature
in the (J, gamma')-plane is at most -b^2, that is
<R(J,gamma')gamma', J> <= -b^2 |J|^2. So
Hess r(X, X) >= integral_0^r (|D_t J|^2 + b^2 |J|^2) dt.
The right side, among all fields J with J(0) = 0 and J(r) = X, is
minimized by J_0(t) = (sn_b(t)/sn_b(r)) X taken in parallel
transport. Indeed J_0 satisfies the ODE J_0'' = b^2 J_0 (this is
just sn_b'' = b^2 sn_b). Writing J = J_0 + V with V(0) = V(r) = 0,
the energy splitting has cross term zero by integration by parts,
and the V-energy is nonnegative. The minimum is achieved by J_0
and equals, by integration by parts using J_0'' = b^2 J_0,
X . J_0'(r) = (cs_b(r)/sn_b(r)) |X|^2.
Therefore Hess r(X, X) >= (cs_b(r)/sn_b(r)) |X|^2 for X
perpendicular to gamma'. For the radial direction gamma' itself,
dr(gamma') = 1 identically, so Hess r(gamma', gamma') = 0. For
mixed directions, dr of an angular field vanishes along a level
set of r; the mixed Hessian term is zero. Hence H(X, X) >= 0 for
every X. End of proof.
Polar comparison. In geodesic polar coordinates about p, the
metric g satisfies
g >= dr^2 + sn_b(r)^2 g_{S_p^{n-1}},
where g_{S_p^{n-1}} is the round unit-sphere metric on directions.
Proof. Let J be an angular Jacobi field along a radial geodesic,
that is, a Jacobi field with J(0) = 0 perpendicular to gamma'.
Then by the identity d/dt |J(t)|^2 = 2 Hess r(J, J) and the
distance comparison just proved,
d/dt |J(t)|^2 >= 2 (cs_b(t)/sn_b(t)) |J(t)|^2.
So |J(t)|^2 / sn_b(t)^2 is nondecreasing in t. Its limit as
t -> 0 is |J'(0)|^2 (since both J and sn_b vanish to first order
there). Therefore for every angular field J,
|J(r)| >= sn_b(r) |J'(0)|.
Applying this along every angle direction says the length of the
angular circle of directions at radius r is at least sn_b(r)
times its value at r = 0; equivalently, the metric contains the
model polar metric sn_b(r)^2 g_S, and adding the radial direction
gives g >= dr^2 + sn_b(r)^2 g_{S_p^{n-1}}. End of proof.
The comparison map Phi_{p,a}. We now build a map from M into the
unit sphere of T_pM + R, together with a positive weight, whose
differential and whose log-weight satisfy complementary bounds.
Fix a center p and a scale a > 0. For x != p write r = d(p,x) and
theta = exp_p^{-1}(x)/r (a unit vector in T_pM):
z = a sn_b(r/2) / cs_b(r/2),
Phi_{p,a}(x) = ( 2z/(1+z^2) theta, (1-z^2)/(1+z^2) ) in
T_pM + R,
f_{p,a}(x) = a / ( cs_b(r/2)^2 + a^2 sn_b(r/2)^2 ).
At r = 0 (x = p) set Phi_{p,a}(p) = (0, 1) and f_{p,a}(p) = a.
The factor Phi is inverse stereographic projection applied after
the radial rescaling z; the factor f is the conformal factor of the
pullback metric, as we verify below. For b = 0 the formulas reduce
to z = a r / 2 and f = a / (1 + a^2 r^2 / 4), the familiar
stereographic weight.
Lemma (Comparison maps). The map Phi_{p,a} and the positive
function f_{p,a} are smooth on all of M. For every tangent X and
every x in M,
|dPhi_{p,a}(X)| <= f_{p,a} |X|,
where the norm on the left is that of T_pM + R at Phi_{p,a}(x),
and, with u_{p,a} = log f_{p,a},
Hess u_{p,a} <= du_{p,a} x du_{p,a}
- (1/2)(b^2 + f_{p,a}^2 + |grad u_{p,a}|^2) g
as quadratic forms. The pair (Phi, f) depends smoothly on (p, a,
x), and for any fixed o in M the parallel transport of the
sphere-valued first component of Phi from p back to o along the
unique geodesic also depends smoothly on (p, a, x).
Proof. Smoothness at x = p: in geodesic normal coordinates about
p, the first component is 2z y/(r(1+z^2)) with y = exp_p^{-1}(x).
The coefficient z/(r(1+z^2)), the last component
(1-z^2)/(1+z^2), and f are all even smooth functions of r^2, with
nonzero denominators. Joint smoothness follows because
(p, y) |-> (p, exp_p y) is a diffeomorphism from TM to M x M.
For the differential bound: the standard stereographic metric on
S(T_pM + R) is
4 (dz^2 + z^2 g_{S_p^{n-1}}) / (1 + z^2)^2
in the coordinates (z theta, (1-z^2)/(1+z^2)). Differentiating z
gives dz/dr = a/(2 cs_b(r/2)^2) (using sn_b' = cs_b and
cs_b^2 - b^2 sn_b^2 = 1). With
D = cs_b(r/2)^2 + a^2 sn_b(r/2)^2, the four substitutions
1 + z^2 = D / cs_b(r/2)^2,
dz^2 = a^2 dr^2 / (4 cs_b(r/2)^4),
z^2 = a^2 sn_b(r/2)^2 / cs_b(r/2)^2,
4 sn_b(r/2)^2 cs_b(r/2)^2 = sn_b(r)^2,
simplify the stereographic expression to
Phi_{p,a}^* g_{S(T_pM + R)}
= (a^2 / D^2) ( dr^2 + sn_b(r)^2 g_{S_p^{n-1}} )
= f^2 ( dr^2 + sn_b(r)^2 g_{S_p^{n-1}} ).
By the polar comparison the right side is at most f^2 g. Hence
|dPhi_{p,a}(X)| <= f_{p,a}|X| for every X.
For the Hessian bound: put A = a^2 + b^2 and
D = cs_b(r/2)^2 + a^2 sn_b(r/2)^2 = 1 + A sn_b(r/2)^2. Then
f = a/D. Differentiate D: D' = A sn_b(r)/2 and
D'' = A cs_b(r)/2. The double-angle identity gives
b^2 D^2 + a^2 + (A^2/4) sn_b(r)^2 = A cs_b(r) D.
(Check: substituting t = sn_b(r/2)^2 turns both sides into
A + (A^2 + 2 b^2 A) t + 2 b^2 A^2 t^2.) Differentiating u = log f
gives
u' = - A sn_b(r) / (2 D) <= 0,
u'' = (u')^2 + u' cs_b(r)/sn_b(r),
u' cs_b(r)/sn_b(r) = - (1/2)(b^2 + f^2 + (u')^2).
Combining with the distance comparison written as
Hess r = (cs_b/sn_b)(g - dr x dr) + H with H >= 0, and using
that u is radial (so |grad u|^2 = (u')^2), gives
Hess u = du x du - (1/2)(b^2 + f^2 + |grad u|^2) g + u' H.
Since u' <= 0 and H >= 0, the last term is negative semidefinite,
so the stated upper bound holds.
At the pole: the expansions dPhi_{p,a}|_p(X) = (aX, 0) and
u = log a - (a^2 + b^2) r^2 / 4 + O(r^4) show that both bounds
also hold at r = 0. End of proof.
Distance comparison at curvature zero. Let (M^n, g) be complete
and simply connected with Sec_g <= 0. For p in M write
log_p = exp_p^{-1}, r_p(x) = d(p,x), D_p(x) = r_p(x)^2 / 2. Then
Hess_g D_p >= g,
(exp_p)^* g >= g_p,
|d log_p(X)|_{g_p} <= |X|_g,
where g_p denotes the constant Euclidean metric on T_pM.
Proof. With b = 0 the distance Hessian comparison gives
Hess r >= (1/r)(g - dr x dr), so
Hess D_p = dr x dr + r Hess r >= dr x dr + (g - dr x dr) = g
away from p. At p, D_p is smooth with Hessian = g exactly by
normal-coordinate expansion. The pullback statement is the b = 0
polar comparison, and the last inequality is that comparison
applied to the inverse map. End of proof.
Balancing a nonatomic measure. Let mu be a compactly supported
Borel probability on M with mu({x}) = 0 for every x. If supp mu
lies in a closed ball B_D(o), then there are p in B_D(o) and finite
a > 0 such that, after parallel transporting the first component
from p to o along the unique geodesic,
integral_M Phi_{p,a}(x) dmu(x) = 0 in T_oM + R.
Proof. Parametrize the center by xi = exp_o^{-1}(p) in
{|xi| <= D} and write P_{p->o} for radial parallel transport
T_pM -> T_oM. Define
F(xi, a) = integral_M (P_{p->o} + id_R) Phi_{p,a}(x) dmu(x).
F is continuous on every cylinder
C = {|xi| <= D} x [a_0, a_1] with 0 < a_0 < a_1 < infinity, because
the integrand is jointly continuous and has norm one.
We show F points strictly inward on all three types of boundary
face of C.
Side face |xi| = D. Put rho(x) = d(o,x). Convexity of rho along
the geodesic from p to x gives, for x in supp mu (so
rho(x) < D strictly, since supp mu is closed in the open ball),
<xi, P_{p->o} theta_p(x)> <= D (rho(x) - D) / d(p,x) < 0.
The first component of Phi_{p,a} equals a positive multiple of
theta_p(x) on this face, hence <xi, F_1(xi, a)> < 0: F points
the first component strictly toward lower |xi|.
Bottom face a = a_0 small. Set t_b(r) = sn_b(r/2)/cs_b(r/2), an
increasing positive function. Choose a_0 > 0 so that
a_0 t_b(2D) < 1. Since d(p,x) < 2D for every |xi| <= D and every
x in supp mu, the last coordinate of Phi_{p,a_0} is at least
(1 - a_0^2 t_b(2D)^2) / (1 + a_0^2 t_b(2D)^2) > 0.
Hence F_2(xi, a_0) > 0: the last component of F points strictly
upward.
Top face a = a_1 large. Non-atomicity of mu on the compact support
implies
lim_{delta -> 0} sup_p mu( B_delta(p) ) = 0.
Otherwise a sequence delta_j -> 0 with centers p_j would have a
limit point (compactness), and continuity of mu from above would
produce an atom. Choose delta > 0 with the supremum below 1/4.
Choose a_1 > a_0 with a_1 t_b(delta) >= sqrt(3). The last
coordinate of Phi_{p,a_1} is at most -1/2 outside B_delta(p),
and at most 1 everywhere. Hence
F_2(xi, a_1) <= (1/4)(1) + (3/4)(-1/2) = -1/8 < 0,
so the last component of F points strictly downward.
Brouwer's fixed point theorem applies to the continuous map
zeta |-> Pi_C(zeta + F(zeta)) from the compact convex cylinder C
into itself, where Pi_C is Euclidean nearest-point projection.
Any fixed point zeta satisfies the projection inequality
<F(zeta), y - zeta> <= 0 for every y in C.
The three estimates now rule out the boundary: pushing inward from
each face contradicts the inequality. So zeta is interior. Taking
both signs of every small displacement y - zeta then gives
F(zeta) = 0. Parallel transport is an isomorphism, giving the
statement. End of proof.
Hypersurface Sobolev theorem. We now state and prove the single
analytic inequality that drives everything. Normalize the curvature
bound to -b^2 with b in {0,1}. Let M^n be complete and simply
connected, with Sec_g <= -b^2, and let Sigma^m with m = n-1 >= 3 be
a locally C^{1,1} immersed hypersurface in M. Sigma need not be
complete, connected, or without boundary. Let dsigma be induced
area, grad_Sigma the tangential gradient, and h the signed
normalized mean curvature for a local unit normal N: m h =
div_Sigma N. The square h^2 is independent of the chosen normal;
reversing N reverses h and the normal derivative Nu together, so
every expression we use is globally defined. Put
q = 2m/(m-2), c = 4/(m-2), alpha = (m-2)/2, Y_m = m s_m^{2/m},
and define the quadratic functional
Q_Sigma(v) = integral_Sigma ( c |grad_Sigma v|^2
+ m (h^2 - b^2) v^2 ) dsigma.
For an open U compactly contained in the interior Sigma^circ,
let H^1_0(U) be the closure of compactly supported Lipschitz
functions in the induced H^1 norm; no regularity of dU is assumed.
Theorem (Hypersurface Sobolev inequality). For every Lipschitz
function v with compact support in Sigma^circ,
Q_Sigma(v) >= Y_m ( integral_Sigma |v|^q dsigma )^{2/q}.
The inequality also holds for the zero extension of every v in
H^1_0(U), where U is compactly contained in Sigma^circ.
The constant is sharp: on a geodesic sphere in the model space of
curvature -b^2, the constant function gives equality. Zero
extension is well defined: for v in H^1_0(U), the extension E_0 v
satisfies E_0 v in H^1(Sigma^circ) and
grad_Sigma(E_0 v) = indicator_U grad_Sigma v a.e.
Both identities pass from the definition of H^1_0(U) by the
approximating functions, including when dU has positive measure.
In C^{1,1} charts the induced metric is locally Lipschitz and
uniformly positive definite on compact subsets; h is locally
essentially bounded; Q_Sigma is continuous on H^1_0(U).
Square estimate. Fix p in M and a > 0, and let Phi = Phi_{p,a},
f = e^u = f_{p,a} be the comparison map and weight of the
Comparison Maps lemma.
Lemma. For every compactly supported Lipschitz v in Sigma^circ,
Q_Sigma(v)
>= integral_Sigma [ m f^2 v^2 + m (h + Nu)^2 v^2
+ c |grad_Sigma v - alpha v grad_Sigma u|^2 ]
dsigma
>= integral_Sigma [ v^2 |d(Phi|_Sigma)|^2
+ c |grad_Sigma v - alpha v grad_Sigma u|^2 ]
dsigma.
The same inequalities hold on H^1_0(U) for every U compactly
contained in Sigma^circ. In particular Q_Sigma is nonnegative on
all these tests.
Proof. Write grad = grad_Sigma and beta = Nu (the normal derivative
of u along N). The restriction formula gives
Delta_Sigma u = tr_{T Sigma} Hess u - m h beta.
This holds both almost everywhere and weakly: in a C^{1,1}
parametrization the induced divergence has Lipschitz coefficients,
and the restricted smooth ambient function has Lipschitz first
derivatives, so the ordinary chain rule gives the same formula as
the distributional divergence.
Take the tangential trace of the Hess u bound of Comparison Maps,
Hess u <= du x du - (1/2)(b^2 + f^2 + |grad_g u|^2) g, over the
m-dimensional tangent space. Using |grad_g u|^2 = |grad u|^2 +
beta^2 gives
2 Delta_Sigma u <= -(m-2)|grad u|^2 - m(b^2 + f^2)
- m beta^2 - 2m h beta,
where |grad u| = |grad_Sigma u|. Rearranging,
m(h^2 - b^2) >= m f^2 + m(h + beta)^2 + (m-2)|grad u|^2
+ 2 Delta_Sigma u.
Multiply by v^2 and integrate by parts; the Delta_Sigma u term
becomes -2 integral v <grad v, grad u> dsigma because v is
compactly supported. Using c alpha = 2 and c alpha^2 = m-2, the
gradient terms assemble as an exact square:
c |grad v|^2 - 4 v <grad v, grad u>
+ (m-2) v^2 |grad u|^2
= c |grad v - alpha v grad u|^2.
This proves the first inequality.
For the second inequality note |d(Phi|_Sigma)|^2 = sum over an
orthonormal tangent frame of |dPhi(v_i)|^2 <= m f^2 by Comparison
Maps. All coefficients are bounded on a compact neighborhood of U,
so H^1_0(U) approximation passes both inequalities from Lipschitz
tests. End of proof.
Local Sobolev threshold. The proof of the hypersurface Sobolev
theorem needs a local version of the sharp Euclidean Sobolev
inequality, plus compactness in L^2.
Lemma. For every open U compactly contained in Sigma^circ and every
epsilon > 0 there exists C = C(epsilon, U) < infinity such that
(Y_m - epsilon) ||v||_q^2
<= c integral_Sigma |grad_Sigma v|^2 dsigma
+ C integral_Sigma v^2 dsigma, v in H^1_0(U).
Moreover H^1_0(U) embeds compactly into L^2(U).
Proof. The sharp Euclidean Sobolev inequality of Aubin and Talenti
in our normalization says, for smooth compactly supported
functions phi on R^m,
c integral_{R^m} |D phi|^2 dx
>= Y_m ( integral_{R^m} |phi|^q dx )^{2/q}.
Before multiplying by c, the sharp constant is
m(m-2)|S^m|^{2/m}/4; multiplying by c = 4/(m-2) gives
m|S^m|^{2/m} = Y_m.
Normalize the metric at a chart center. In a small enough chart its
matrix G satisfies (1-delta) Id <= G <= (1+delta) Id. Comparing
inverse metrics and volume densities in the Euclidean Sobolev
inequality gives, for chart-supported phi,
c integral_Sigma |grad_Sigma phi|^2 dsigma
>= Y_m ((1-delta)/(1+delta))^{m/2} ||phi||_q^2.
Indeed the gradient integral is at least (1-delta)^{m/2}
(1+delta)^{-1} times its Euclidean value, while the squared
critical norm is at most (1+delta)^{m/q} times its Euclidean
value, and m/q = (m-2)/2. Choose delta so the coefficient is at
least Y_m - epsilon.
Cover Ubar by finitely many such charts. Choose chart-supported
Lipschitz functions chi_i with sum_i chi_i^2 = 1 near Ubar; they
are obtained by normalizing a finite family of cutoffs by the
square root of the sum of their squares near Ubar, then cutting
off outside that neighborhood. The triangle inequality in L^{q/2}
gives
||v||_q^2 = || sum_i (chi_i v)^2 ||_{q/2}
<= sum_i ||chi_i v||_q^2,
and the product rule gives
sum_i |grad_Sigma(chi_i v)|^2
= |grad_Sigma v|^2 + v^2 sum_i |grad_Sigma chi_i|^2.
Summing the chart inequalities over i proves the threshold, first
for compactly supported Lipschitz v, then for all v in H^1_0(U) by
taking epsilon = Y_m/2 to get continuous L^q inclusion and then
approximating.
For compactness, extend a bounded H^1_0(U) sequence by zero. Each
chi_i v_j is a bounded Euclidean H^1 sequence in its chart,
supported in a fixed compact subset; Rellich compactness gives an
L^2-convergent subsequence in each of the finitely many charts.
The identity v_j = sum_i chi_i (chi_i v_j) reassembles them into an
L^2(U)-convergent subsequence. No boundary regularity of U is
used. End of proof.
Attainment below the threshold. The threshold Y_m is the sharp
Euclidean constant. Below it, minimizing sequences converge to
minimizers.
Lemma. For a nonempty open U compactly contained in Sigma^circ put
Y(U) = inf { Q_Sigma(v) : v in H^1_0(U), ||v||_q = 1 }.
If Y(U) < Y_m, the infimum is attained.
Proof. Write Y = Y(U) >= 0, nonnegativity coming from the Square
Estimate. For a normalized minimizing sequence v_j, H\"older and
h^2 - b^2 >= -b^2 give
||v_j||_2^2 <= sigma(U)^{2/m},
c ||grad_Sigma v_j||_2^2 <= Q_Sigma(v_j) + m b^2 sigma(U)^{2/m}.
Thus, after a subsequence,
v_j -> psi weakly in H^1_0(U),
v_j -> psi in L^2(U) and almost everywhere,
the strong convergence from the Local Threshold's Rellich
compactness. Set t = integral_U |psi|^q dsigma in [0, 1] and
z_j = v_j - psi. The Br\'ezis-Lieb lemma states that if f_j -> f
almost everywhere and sup_j ||f_j||_q^q < infinity, then
||f_j||_q^q - ||f_j - f||_q^q -> ||f||_q^q. Applied here,
integral_U |z_j|^q dsigma -> 1 - t.
Weak convergence splits the quadratic form:
Q_Sigma(v_j) = Q_Sigma(psi) + Q_Sigma(z_j) + o(1). The bounded
potential and Local Threshold give
Q_Sigma(z_j) >= (Y_m - epsilon) ||z_j||_q^2
- (C_{epsilon,U} + m b^2) ||z_j||_2^2.
Letting j -> infinity and then epsilon -> 0,
Y >= Q_Sigma(psi) + Y_m (1 - t)^{2/q}
>= Y t^{2/q} + Y_m (1 - t)^{2/q}.
If t = 0 the right side is Y_m > Y, a contradiction. If Y = 0
then t = 1. If Y > 0 and 0 < t < 1, strict concavity of s |-> s^{2/q}
gives t^{2/q} + (1-t)^{2/q} > 1, so the right side is strictly
greater than Y, again a contradiction. Hence t = 1: psi has full
L^q mass and is the minimizer. Lower semicontinuity of the gradient
energy and strong L^2 convergence of the potential term give
Q_Sigma(psi) <= Y; normalization gives the reverse. End of proof.
Balanced variations. The comparison maps give sphere coordinates
whose gradients probe the shape of any hypothetical minimizer.
Lemma. Take a normalized minimizer psi of Q_Sigma on H^1_0(U) as
above, and push the probability measure |psi|^q dsigma (extended by
zero) to M by the immersion. Its support is compact and it has no
atoms: an immersion is locally an embedding, so a fiber in the
compact support meets each member of a finite embedding-chart cover
in at most one point, and every such fiber has zero area measure.
Balancing supplies parameters p, a such that, with Phi = Phi_{p,a}
having components w_1, ..., w_{n+1} and u = log f_{p,a},
integral_Sigma |psi|^q w_j dsigma = 0 for j = 1,...,n+1,
sum_{j=1}^{n+1} w_j^2 = 1 on Sigma.
Moreover the coordinates w_j and u have bounded first derivatives
near Ubar, and multiplication by w_j or w_j^2 preserves H^1_0(U).
Proof. Compact support of the immersion of supp psi, non-atomicity
of the pushed measure, and Balancing produce p, a. Joint smoothness
gives bounded first derivatives of w_j near Ubar. Each w_j is
smooth and bounded together with its gradient on the compact
closure of U, so multiplication by w_j or w_j^2 is a bounded map
on H^1_0(U) that preserves the space. End of proof.
The proof of the Hypersurface Sobolev theorem. Let Sigma be
connected and boundaryless. Fix a nonempty open U compactly
contained in Sigma with Sigma minus Ubar nonempty, and suppose
Y(U) < Y_m. Take the normalized minimizer psi from Attainment and
write Y = Q_Sigma(psi). Let B_Q be the symmetric bilinear form
associated with Q_Sigma.
First variation. For a bounded Lipschitz multiplier zeta near
Ubar, differentiate the quotient s |-> Q_Sigma(psi(1 + s zeta)) /
||psi(1 + s zeta)||_q^2 at s = 0. This is legitimate: for small
|s| the factor 1 + s zeta is bounded away from zero, and
differentiated mass integrands are dominated by a constant times
|psi|^q. The result is
B_Q(psi, psi zeta) = Y integral_Sigma |psi|^q zeta dsigma.
Apply this with zeta = w_j. Because each w_j has zero mean against
|psi|^q dsigma by balancing,
B_Q(psi, psi w_j) = 0 for j = 1,...,n+1.
Second variation. Fix one coordinate w = w_j and set
B_w = integral_Sigma |psi|^q w^2 dsigma. Since |w| <= 1 and the
weighted mean of w is zero, expanding (1 + s w)^q as
1 + q s w + (q(q-1)/2) s^2 w^2 plus higher terms and raising to
power 2/q gives
||psi(1 + s w)||_q^2 = 1 + (q-1) B_w s^2 + o(s^2).
The numerator Q_Sigma(psi(1 + s w)) has zero linear term by the
first variation with zeta = w, and quadratic coefficient
Q_Sigma(psi w). Since psi is a minimizer, the quotient cannot lie
below Y, so the coefficient of s^2 in
(Y + s^2 Q_Sigma(psi w)) / (1 + (q-1) B_w s^2)
is nonnegative. This yields
Q_Sigma(psi w) >= (q-1) Y B_w.
Product rule and first variation with zeta = w^2. The gradient of
phi = psi w is w grad psi + psi grad w, so
|grad phi|^2 = w^2 |grad psi|^2 + 2 w psi <grad psi, grad w>
+ psi^2 |grad w|^2.
The first two terms combine with the potential term to give
B_Q(psi, psi w^2); the last term is c integral psi^2 |grad w|^2.
The first variation with zeta = w^2 gives B_Q(psi, psi w^2) =
Y B_w. Hence
Q_Sigma(psi w) = Y B_w + c integral_Sigma psi^2
|grad_Sigma w|^2 dsigma.
Subtracting the two formulas for Q_Sigma(psi w) and using
c = q-2,
c integral_Sigma psi^2 |grad_Sigma w|^2 dsigma
>= (q-2) Y B_w.
Sum over the coordinates w_1, ..., w_{n+1}. Because sum_j w_j^2
= 1, sum_j B_{w_j} = integral_Sigma |psi|^q dsigma = 1, and
because Phi maps into the unit sphere,
sum_j |grad_Sigma w_j|^2 = |d(Phi|_Sigma)|^2. Therefore
c integral_Sigma psi^2 |d(Phi|_Sigma)|^2 dsigma >= (q-2) Y,
which by c = q-2 gives
integral_Sigma psi^2 |d(Phi|_Sigma)|^2 dsigma >= Y.
Combining with the Square Estimate forces
grad_Sigma psi = alpha psi grad_Sigma u
almost everywhere on U.
The contradiction. Choose a compactly supported Lipschitz cutoff
chi on Sigma equal to one near Ubar. Extend psi by zero and set
F = chi e^{-alpha u} E_0 psi,
which lies in H^1(Sigma) and equals e^{-alpha u} psi on U.
Because the tangential gradient of psi is alpha psi grad u on U,
and psi vanishes on Sigma minus the support:
grad_Sigma F = 0 almost everywhere on all of Sigma.
The local Poincar\'e inequality in coordinate balls makes F
constant almost everywhere on each ball; overlapping balls have
the same constant, so connectedness makes F globally constant
on Sigma. Since Sigma minus Ubar is nonempty and open, F vanishes
there, hence F is identically zero on Sigma. This contradicts
||psi||_q = 1. Therefore Y(U) >= Y_m.
The contradiction only covered Y(U) < Y_m for proper compact
U. Generalize.
First, every proper compact support in a connected boundaryless
Sigma is contained in such a U. Choose a coordinate ball outside
the support; the support is compact, so cover it by finitely many
relatively compact neighborhoods that miss a smaller coordinate
ball. Then the inequality holds for all compactly supported
Lipschitz v.
If the support is the entire compact component: let D_delta be
the image of a Euclidean ball of radius delta in a fixed chart
about a point. Remove that point by a cutoff eta_delta equal to
zero in D_delta and one outside D_{2 delta} with
|grad_Sigma eta_delta| <= C/delta. For Lipschitz v, metric
comparability gives
integral_Sigma |grad_Sigma((1 - eta_delta) v)|^2 dsigma
<= 2 integral_{D_{2 delta}} |grad_Sigma v|^2 dsigma
+ C ||v||_infinity^2 delta^{m-2}.
The right side tends to zero because m > 2. Also (1 - eta_delta) v
tends to v in L^2 and L^q; the potential is bounded on the compact
component; so the inequality passes to v. This is exactly where
m > 2 (i.e. n > 3) is used.
Finally, remove connectedness. If Sigma has boundary, pass to
Sigma^circ. A compactly supported function in Sigma^circ meets
only finitely many connected components, because the components
are open. If the connected component j has L^q mass a_j, the
connected result on each component gives Q_Sigma(v) >= Y_m sum_j
a_j^{2/q} >= Y_m (sum_j a_j)^{2/q} by concavity of s |-> s^{2/q}.
For v in H^1_0(U), approximate by compactly supported Lipschitz
functions: Local Threshold gives L^q convergence, and boundedness
of the potential near Ubar gives convergence of the quadratic form.
This proves the Hypersurface Sobolev theorem. End of proof.
Confined isoperimetric profile. We now turn to the geometric part.
Fix an open geodesic ball B = B_{R_B}(o) of finite radius. Only
Sec_g <= 0 is needed in this section; the sharper bound b will
reappear as a parameter. For a measurable set E write
|E| = vol_g(E), P(E) = P_g(E), and P(E; U) = |D chi_E|(U) for an
open set U. The confined profile is
I(v) = inf { P(E) : |E \ B| = 0, |E| = v },
0 < v < |B|.
Full ambient perimeter is minimized at fixed volume v, but the
competitors are trapped inside B. By BV compactness on the compact
set Bbar and L^1 continuity of volume, minimizers exist. Contact
with dB is counted in P(E); otherwise a minimizer would smear its
boundary along dB and the first variation for the profile
derivative would fail.
Existence, regularity, variation. The following three facts are
used without further proof beyond the remarks in parentheses;
each is classical regularity theory for the isoperimetric problem
in a compact manifold with a smooth obstacle.
(a) Existence and regularity. For each v in (0, |B|) there is a
minimizer E of I(v). Its perimeter support Gamma = supp |D chi_E|
is compact and contained in Bbar, and Gamma decomposes as a
disjoint union Sigma + Z with:
Sigma a boundaryless, embedded, locally C^{1,1} hypersurface,
Z a compact singular set of Hausdorff dimension at most n-8,
empty when n < 8,
Gamma C^{1,1} in an ambient neighborhood of dB,
Z disjoint from a collar of dB,
and the perimeter measure is the induced area measure
|D chi_E|(A) = int_{A \cap Sigma} dsigma for every Borel A. The
unit normal N on Sigma is the outward normal of E.
(These results are Kleiner's regularity formulation of the
confined problem, in the version of Ghomi and Spruck: Morgan's
interior regularity gives the interior singular set bound;
Stredulinsky-Ziemer Euclidean obstacle regularity lifts to the
Riemannian graph setting to give C^{1,1} regularity near dB;
Antonelli-Pasqualetto-Pozzetta's local smooth-manifold result
supplies the same at interior points when n >= 2. The
compilation produces exactly the four bullets above. No
curvature bound stronger than Sec_g <= 0 enters.)
(b) Nondegenerate free perimeter. For every 0 < v < |B| one has
P(E; B) > 0. If the perimeter measure vanished on B then
chi_E would be constant on the connected ball B, contradicting
v in (0, |B|). Thus every minimizer has a nonempty free smooth
patch.
(c) Profile curvature. Let v be a differentiability point of I
in (0, |B|) and let E be a minimizer at v, with
m h = div_Sigma N the trace curvature and h_0 = I'(v)/m. Then
I'(v) > 0,
h = h_0 on the free part Sigma \cap B,
0 <= h <= h_0 almost everywhere on all of Sigma,
h^2 <= h_0^2 almost everywhere.
Proof of (c). For a smooth flow Psi^t generated by a compactly
supported smooth vector field Z, put E_t = Psi^t(E), P(t) =
P(E_t), V(t) = |E_t|. Whenever the flow is admissible (keeps the
set inside B),
P(t) >= I(V(t)), P(0) = I(v), V(0) = v.
For a field supported away from dB on a free smooth patch,
admissibility holds for both signs of t. Differentiating at the
minimum gives P'(0) = I'(v) V'(0). The first variation formulas
for a flow preserving the ambient metric are
V'(0) = int_Sigma <Z, N> dsigma,
P'(0) = m int_Sigma h <Z, N> dsigma.
Arbitrary smooth compactly supported normal speeds on the patch
therefore give m h = I'(v), i.e. h = h_0 on every free patch.
For the sign of I'(v), the smooth radial function
rho(x) = d(o, x) has grad rho of norm one; its square F(x) =
rho(x)^2/2 is smooth on all of M including o, and the distance
Hessian comparison at b = 0 gives Hess F >= g. The flow of
-grad F is Psi^t(x) = exp_o( e^{-t} exp_o^{-1}(x) ); for t >= 0
it sends B into itself. Then V'(0) = -int_E Delta F dvol <= -n v
< 0, and P'(0) = -int_{Sigma free} m h <grad F, N> dsigma plus a
contribution from the contact set that is nonnegative because
grad F points inward at dB. Admissibility and P'(0) >= I'(v)
V'(0) give I'(v) > 0.
For the contact bound, take a smooth ambient field Z pointing
inward at dB, and use both signs of t. The volume derivative at
t = 0 is bounded below by a positive multiple of the contact
area, and the perimeter takes its infimum at t = 0, so any
variation of the profile at contact must satisfy
m h <= I'(v). Taken over arbitrarily many smooth contact tests,
this forces h <= h_0 almost everywhere on C; smooth nonnegative
test functions determine the sign of a Radon measure, so C need
not contain an open patch. Combining h = h_0 on the free part
with h <= h_0 on C and h > 0 by I'(v) > 0 gives 0 <= h <= h_0
and h^2 <= h_0^2 almost everywhere on Sigma. End of proof.
Deletion and area growth. These two lemmas control the size of
the perimeter near the singular set. Both statements are purely
local.
BV deletion. Let E have finite volume and perimeter in a complete
smooth Riemannian manifold, let x be a point, and let 0 < r <
inj(x). Then
P(E \ B_r(x))
<= P(E; M \ Bbar_r(x)) + H_g^{n-1}(dB_r(x)),
where H_g^{n-1}(dB_r(x)) is the (n-1)-area of the distance sphere.
For all but countably many radii r in a bounded interval,
P(E; dB_r(x)) = 0,
so the perimeters on the two open sides sum to P(E).
Proof. For a bounded Lipschitz scalar phi, distributional testing
gives
D(phi chi_E) = phi D chi_E + chi_E grad phi dvol_g.
Prove this first for smooth phi by approximating on a compact
neighborhood of the support of the test field, uniformly and in
W^{1,1}; uniform convergence controls the measure term and
W^{1,1} convergence controls the second term. Now take
phi_eps(y) = min{1, max{0, (d(x, y) - r)/eps}},
with r + eps < inj(x); its derivative is supported in a thin
annulus. The products phi_eps chi_E converge in L^1 to
chi_{E \ B_r(x)} because the smooth sphere has ambient volume
zero, so lower semicontinuity and the product formula give
P(E \ B_r(x))
<= liminf_eps int phi_eps d|D chi_E|
+ |E \cap (B_{r+eps}(x) \ B_r(x))| / eps
<= P(E; M \ Bbar_r(x)) + (d/ds at s = r) |B_s(x)|.
Polar coordinates identify the last derivative with the area of
the distance sphere. Finally, a finite measure can put positive
mass on only countably many pairwise disjoint distance spheres.
End of proof.
Area growth near the singular set. Fix v in (0, |B|) and let E be
a confined minimizing set of volume v.
Lemma. There are C < infinity and r_0 > 0, depending on E and B,
such that
sigma( Sigma \cap B_r(x) ) <= C r^m
for every x in the singular set S and every 0 < r < r_0.
Proof. If S is empty the claim is void. Because P(E; B) > 0 and
the perimeter-area identity holds, some free smooth boundary
point exists. Choose a neighborhood W of that point with Wbar
contained in B \ S and Gamma \cap W smooth. A nonnegative
nonzero normal variation of a smaller patch extends to a smooth
field Z compactly supported in W. With N the outward normal, set
a = int_{Sigma \cap W} <Z, N> dsigma > 0.
Let phi_t be the flow of Z and G(t) = |phi_t(E)| - |E|. The
volume Jacobian formula gives G'(0) = a, and by continuity
G'(t) >= a/2 for 0 <= t <= t_0 for some t_0 > 0. The flow stays
in W, preserves B, and fixes the complement of W. Whenever a
finite-perimeter set F agrees with E almost everywhere on W,
the volume change under phi_t is the same, and the perimeter
change is at most a fixed linear bound C_R |t| P(F) with C_R
uniform over all tangent planes above Wbar.
Compactness of S and its disjointness from Wbar and dB give
rho_* > 0 with
B_{2 rho_*}(x) subset B \ Wbar for x in S.
Decreasing rho_*, smoothness of the metric on a compact
neighborhood of S gives uniform C_V and C_A with
|B_r(x)| <= C_V r^n,
H_g^m(dB_r(x)) <= C_A r^m for x in S and 0 < r < rho_*.
Decrease rho_* again so that C_V rho_*^n <= a t_0 / 2.
Fix x in S. For almost every r in (0, rho_*), P(E; dB_r(x)) = 0.
At such radii let F = E \ B_r(x). BV deletion gives
P(F) <= P(E; M \ Bbar_r(x)) + H_g^m(dB_r(x)).
The volume deficit is delta = |E \cap B_r(x)| <= C_V r^n, and F
agrees with E on W. Use the fixed-patch flow to repair this
deficit by a flow time t <= 2 delta / a. The repaired set is in
B and has volume v. Minimality and the fixed-patch perimeter
differential bound give
P(E; B_r(x))
<= H_g^m(dB_r(x)) + C_R delta
<= C_A r^m + C_R C_V r^n
<= C_0 r^m,
with C_0 = C_A + C_R C_V rho_* (because n = m+1 and delta <=
C_V r^n <= C_V rho_* r^m). The constants are uniform in x in S.
For an arbitrary 0 < r < rho_*/2, choose a good radius rho in
(r, 2r). Monotonicity of the perimeter measure gives
P(E; B_r(x)) <= P(E; B_rho(x)) <= 2^m C_0 r^m.
Together with the perimeter-area identity this gives the claimed
bound with r_0 = rho_*/2. End of proof.
Capacity cutoffs. We remove the singular set from the constant
test.
Lemma. Let Gamma be a compact subset of a smooth Riemannian
manifold and S subset Gamma a closed subset for which
Sigma = Gamma \ S is an embedded, boundaryless, locally C^{1,1}
hypersurface of dimension m >= 3 and finite area. Assume
H^{m-2}(S) = 0 and that, for some C < infinity and r_0 > 0,
sigma( Sigma \cap B_r(x) ) <= C r^m
for x in S and 0 < r < r_0. Then there are eta_j in Lip_c(Sigma)
with
0 <= eta_j <= 1, eta_j(x) -> 1 for every x in Sigma,
int_Sigma |grad_Sigma eta_j|^2 dsigma -> 0.
If S is empty, eta_j = 1 works in every dimension.
Proof. If S is empty, Sigma = Gamma is compact and eta_j = 1.
Otherwise use H^{m-2}(S) = 0. For each j there is a finite
ball covering
S subset union_i B_{r_{ji}}(x_{ji}),
x_{ji} in S,
0 < r_{ji} < min(2^{-j}, r_0/4),
sum_i r_{ji}^{m-2} < 2^{-j}.
Indeed, Hausdorff nullity first gives a countable open-ball cover
with arbitrarily small radii and sum; recentering the balls at
points of S with a fixed enlargement, and taking a finite
subcover by compactness of S, preserves the smallness of sum r^{m-2}.
Put theta(t) = min{1, max{0, t-1}} for t >= 0, and define
eta_{ji}(x) = theta( d(x, x_{ji})/r_{ji} ),
eta_j(x) = min_{1 <= i <= N_j} eta_{ji}(x)|_Sigma.
Each ambient eta_{ji} vanishes on B_{r_{ji}}(x_{ji}), equals one
outside B_{2 r_{ji}}(x_{ji}), and has Lipschitz constant at most
1/r_{ji}. A finite minimum is Lipschitz, and almost everywhere
|grad_Sigma eta_j|^2 <= sum_i |grad_Sigma eta_{ji}|^2.
There is no restriction on overlaps. Applying the area bound at
the doubled radii gives
int_Sigma |grad_Sigma eta_j|^2 dsigma
<= sum_i r_{ji}^{-2} sigma( Sigma \cap B_{2 r_{ji}}(x_{ji}) )
<= 2^m C sum_i r_{ji}^{m-2} < 2^m C 2^{-j} -> 0.
The open balls in the covering form a neighborhood of S on
which eta_j vanishes, so supp eta_j is compact in Sigma
(embedding charts identify the manifold and subspace topologies).
For fixed x in Sigma, closedness of S gives d(x,S) > 0; as the
maximal radius tends to zero, the doubled balls eventually miss
x, so eta_j(x) -> 1. Neither completeness of Sigma nor finiteness
of its components is assumed. End of proof.
The constant test. This is the passage from the sharp Sobolev
inequality on Sigma to the profile differential inequality.
Lemma. Let v in (0, |B|) be a differentiability point of I and
let E minimize the confined perimeter at volume v. Then
m int_Sigma (h^2 - b^2) dsigma >= Y_m I(v)^{(m-2)/m}.
Proof. Profile curvature gives 0 <= h <= h_0 = I'(v)/m < infinity
almost everywhere on Sigma. If S is not empty, its dimension
bound gives dim_H S <= n - 8 = m - 7 < m - 2, so H^{m-2}(S) = 0;
Area Growth and the compact embedded boundary structure verify
the hypotheses of Capacity Cutoffs. That lemma supplies cutoffs
eta_j; when S is empty take eta_j = 1.
Apply the Hypersurface Sobolev theorem to these cutoffs:
c int_Sigma |grad_Sigma eta_j|^2 dsigma
+ m int_Sigma (h^2 - b^2) eta_j^2 dsigma
>= Y_m ( int_Sigma eta_j^q dsigma )^{2/q}.
The area I(v) = int_Sigma dsigma is finite, and
|h^2 - b^2| <= h_0^2 + b^2 almost everywhere. Dominated
convergence and the vanishing Dirichlet energy of eta_j yield
m int_Sigma (h^2 - b^2) dsigma >= Y_m I(v)^{(m-2)/m},
since 2/q = (m-2)/m. The construction covers every dimension
under consideration: when 4 <= n <= 7 the singular set is empty,
and when n = 8 its possible infinite compact zero-dimensional
set still admits the covers used above. End of proof.
Profile differential inequality. For almost every v in (0, |B|),
I'(v) >= m sqrt( b^2 + (s_m / I(v))^{2/m} ).
Proof. I is locally Lipschitz, by the fixed family of flows used
to vary volume at bounded perimeter cost; hence it is
differentiable almost everywhere. At a differentiability volume
choose a minimizer and apply Constant Test with h_0 = I'(v)/m and
h^2 <= h_0^2:
m (h_0^2 - b^2) I(v) >= Y_m I(v)^{(m-2)/m}.
Because I(v) > 0 and Y_m = m s_m^{2/m},
h_0^2 - b^2 >= (Y_m / m) I(v)^{(m-2)/m - 1} = (s_m / I(v))^{2/m}.
Taking the positive square root gives the differential inequality.
Integration of the profile inequality. The profile I is locally
Lipschitz on (0, |B|). Indeed, the fixed family of flows of the
form exp_o(e^{-t} exp_o^{-1}.), together with the uniform
volume-adjustment time t <= delta^{-1} |w - v|, gives
|I(v) - I(w)| <= C |v - w| for v, w in any compact subinterval of
(0, |B|). Hence I is differentiable almost everywhere and
W_b \circ I is absolutely continuous on compact subintervals of
(0, |B|) as soon as W_b is continuously differentiable.
Define, for A > 0,
W_b(A) = int_0^A da / ( m sqrt( b^2 + (s_m/a)^{2/m} ) ).
Then W_b is continuously differentiable on (0, infinity) with
W_b'(A) = 1 / ( m sqrt( b^2 + (s_m/A)^{2/m} ) ) > 0,
and W_b(A) -> 0 as A -> 0.
Compute W_b(A_b(r)) by the substitution a = A_b(s),
da = A_b'(s) ds:
W_b(A_b(r)) = int_0^r A_b'(s)/( m sqrt(b^2 + (s_m/A_b(s))^{2/m}) )
ds.
Since A_b(s) = s_m sn_b(s)^m and cs_b(s)^2 - b^2 sn_b(s)^2 = 1,
sqrt( b^2 + (s_m/A_b(s))^{2/m} )
= sqrt( b^2 + sn_b(s)^{-2} )
= sqrt( b^2 sn_b(s)^2 + 1 ) / sn_b(s)
= cs_b(s) / sn_b(s),
and A_b'(s) = m s_m sn_b(s)^{m-1} cs_b(s). Substituting,
A_b'(s) / ( m cs_b(s)/sn_b(s) ) = s_m sn_b(s)^m = V_b'(s).
Therefore W_b(A_b(r)) = V_b(r) - V_b(0) = V_b(r).
Apply the profile differential inequality. Multiply by the
positive quantity W_b'(I(v)):
W_b'(I(v)) I'(v)
>= W_b'(I(v)) m sqrt( b^2 + (s_m/I(v))^{2/m} )
= 1,
which reads (W_b \circ I)'(v) >= 1. Integrating from epsilon > 0
to v and using I(epsilon) > 0 (so W_b(I(epsilon)) > 0):
W_b(I(v)) >= W_b(I(epsilon)) + (v - epsilon) >= v - epsilon.
Letting epsilon -> 0 gives W_b(I(v)) >= v for every v in (0, |B|).
Conclusion of the normalized comparison. Let E subset M be
bounded and measurable, with V = vol_g(E) in (0, infinity) and
A = P_g(E) < infinity. Choose B = B_{R_B}(o) so large that E
subset B up to a vol_g-zero set; then V < |B|. Let r > 0 be
determined by V_b(r) = V. Then
W_b(A) >= W_b(I(V)) >= V = V_b(r) = W_b(A_b(r)).
Since W_b is strictly increasing on (0, infinity), this gives
A >= A_b(r) = I_b(V).
This proves
P_g(E) >= I_b( vol_g(E) ),
for bounded sets, in the normalized case b in {0, 1}. The
unbounded case and general kappa follow below.
Dimensions two and three. The Hypersurface Sobolev theorem needs
m = n-1 >= 3, i.e. n >= 4. In dimensions two and three we use
known dimension-specific results, then pass to the general finite-
perimeter setting by approximation.
Dimension two. Let (M^2, g) be complete, simply connected and
smooth, with Sec_g <= kappa <= 0, so the Cartan-Hadamard theorem
identifies M with R^2 and Sec_g is the single Gaussian curvature.
The classical Weil-Bol inequality, in the smooth disk formulation
of Chavel and Feldman, states that for a smooth disk of area V
and boundary length L under K <= kappa,
L^2 >= 4 pi V - kappa V^2.
Pass from disks to relatively compact smooth domains. A smooth
domain is a compact surface with boundary; fill each interior
boundary component (fill the holes) retaining only the outer
boundary of each connected component. The resulting disk has at
least the original area and no greater boundary length. The disk
inequality then bounds the outer boundary length below by the
model profile at the new area, and monotonicity of I_b (both
arguments increasing) bounds it below by the profile at the
original area. Summing over components of a disconnected domain
and using subadditivity of I_b (which follows from its concavity
in V; each I_b is concave and vanishes at 0) proves the
comparison for every relatively compact smooth domain.
Dimension three. Kleiner's Theorem 2 gives the model comparison
for every upper sectional curvature bound kappa <= 0 on a
complete simply connected 3-manifold, for arbitrary compact
smooth domains. Apply this component by component; subadditivity
of I_b handles disconnected domains. Dimension three is settled
at the level of compact smooth domains.
Smooth BV approximation. Let M be complete, connected and smooth
without boundary. Let E subset M have finite ambient perimeter
and suppose E is contained in a compact set K up to a
vol_g-zero set. For every open neighborhood U of K with compact
closure there are relatively compact open sets E_j compactly
contained in U with smooth boundary such that
|E_j symmetric-difference E| -> 0, P(E_j) -> P(E).
The E_j need not be connected.
Proof. Guneysu-Pallara's strict smooth approximation on complete
manifolds provides smooth compactly supported f_j with
||f_j - chi_E||_1 -> 0 and int_M |grad f_j| dvol -> P(E); the
cited result assumes completeness, smoothness, connectedness,
and absence of boundary, and imposes no curvature condition.
Taking real parts cannot increase either the L^1 error or the
gradient integral, and lower semicontinuity gives the reverse
limiting bound, so assume the f_j real. Choose eta in C_c(U)
with 0 <= eta <= 1 and eta = 1 near K and set u_j = eta f_j.
Then u_j -> chi_E in L^1, and chi_E grad eta = 0 almost
everywhere, so
int_M |grad u_j| dvol
<= int_M |grad f_j| dvol + ||grad eta||_infty ||f_j - chi_E||_1.
Lower semicontinuity of total variation gives the reverse
limiting inequality, so the gradients converge to P(E). All u_j
are now supported in a fixed compact subset of U. Write
e_j = ||u_j - chi_E||_1 and choose 0 < delta_j < 1/2 with
delta_j -> 0 and e_j/delta_j -> 0. For delta_j < t < 1 - delta_j,
pointwise comparison with chi_E gives
{u_j > t} symmetric-difference E
subset {|u_j - chi_E| > delta_j},
whose measure is at most e_j/delta_j -> 0. So the level sets
E_j = {u_j > t} converge to E in symmetric difference. Coarea,
together with the fact that the total variations converge, gives
P(E_j) -> P(E). End of proof.
Finite-volume cutoff. Let M be complete, connected and smooth
without boundary. Let J : [0, infinity) -> [0, infinity) be
continuous and increasing with J(0) = 0. Suppose P(F) >= J(|F|)
for every bounded finite-perimeter set F. Then the same holds for
every measurable E of finite volume and finite ambient perimeter.
Proof. Two BV identities are used. The product rule
D(varphi u) = varphi Du + u grad varphi dvol_g for compactly
supported Lipschitz varphi and bounded BV u follows from the
smooth-multiplier case by uniform approximation of varphi in
W^{1,1} with gradients converging in L^1. The coarea identity
|Du|(M) = int_0^1 P({u > t}) dt
for 0 <= u <= 1 follows by strict smooth approximation and Fatou
on one side, and on the other by writing u = int_0^1 chi_{u > t}
dt, testing against div X with |X| <= 1, and taking the perimeter
supremum.
Fix o in M and, for R > 0, set
varphi_R(x) = min{1, max{0, R + 1 - d(o, x)}}.
By completeness supp varphi_R is compact; varphi_R equals 1 on
B_R(o), vanishes outside B_{R+1}(o), and is 1-Lipschitz. The
product rule and coarea give
int_0^1 P({varphi_R chi_E > t}) dt
= |D(varphi_R chi_E)|(M)
<= int_M varphi_R d|D chi_E| + int_E |grad varphi_R| dvol
<= P(E) + |E \ B_R(o)|.
Almost every level {varphi_R chi_E > t} is a bounded
finite-perimeter set, and for every 0 < t < 1 its volume is at
least v_R = |E \cap B_R(o)|. The hypothesis and monotonicity of
J bound the left side below by J(v_R). Since v_R -> |E| and
|E \ B_R(o)| -> 0 while J is continuous, the inequality passes to
E. End of proof.
Arbitrary negative curvature. Assemble the comparison for all
dimensions and all kappa < 0.
Proof. Null sets have zero perimeter, and I_b(0) = 0, so ignore
them. For b in {0, 1}, the normalized comparison is now known for
bounded finite-perimeter sets: for n >= 4 by the confined-profile
argument; for n = 2 and n = 3 by the classical componentwise
results above together with Smooth BV Approximation and
continuity of I_b to reach arbitrary bounded finite-perimeter E.
The Finite-Volume Cutoff with J = I_b then removes boundedness,
giving the comparison for all finite-volume finite-perimeter E
and b in {0, 1}.
For an arbitrary kappa < 0, put b = sqrt(-kappa) > 0 and
widetilde g = b^2 g. Then
Sec_tilde = b^{-2} Sec_g <= -1,
|E|_tilde = b^n |E|_g,
P_tilde(E) = b^{n-1} P_g(E).
The last identity holds for ambient BV perimeter: constant metric
rescaling leaves divergence unchanged, while
dvol_tilde = b^n dvol_g and |X|_tilde = b |X|_g, so in the dual
definition of perimeter the substitution Y = b X multiplies the
supremum by b^{n-1}. The model functions have the matching
relations
V_b(r) = b^{-n} V_1(br),
A_b(r) = b^{-(n-1)} A_1(br),
I_b(V) = b^{-(n-1)} I_1(b^n V).
Applying the normalized result to tilde g and dividing by
b^{n-1} gives the claimed inequality for g and kappa:
P_g(E) = b^{-(n-1)} P_tilde(E)
>= b^{-(n-1)} I_1( b^n |E|_g )
= I_b( |E|_g ).
The flat case kappa = 0 was proved directly, without a curvature
limit, at the integration step. This completes the comparison
theorem for all kappa <= 0. End of proof.
Equality. We now pass to the equality case. Throughout this
section (M^n, g) is complete, simply connected and smooth, with
n >= 2 and Sec_g <= 0. Let E subset M be bounded and measurable,
with V = vol_g(E) in (0, infinity) and finite perimeter A =
P_g(E), and suppose equality holds in the flat bound:
A = n omega_n^{1/n} V^{(n-1)/n}.
Put m = n-1, R = (V/omega_n)^{1/n}, and H_0 = m/R. Note
A = s_m R^m because s_m = n omega_n.
Equality boundary. In this setting:
(1) The perimeter support Gamma = supp |D chi_E| is compact, and
decomposes as a disjoint union Gamma = Sigma + Z, with Sigma a
relatively open boundaryless embedded locally C^{1,1} hypersurface
and Z a compact singular set with dim_H Z <= n - 8 (so Z empty
when n < 8).
(2) The side occupied by E defines an outward unit normal N on
Sigma, and
|D chi_E| = dsigma on Sigma,
sigma(Sigma) = A = s_m R^m,
closure of Sigma = Gamma.
(3) There are C < infinity and r_0 > 0 with
sigma(Sigma \cap B_r(x)) <= C r^m for x in Gamma,
0 < r < r_0.
(4) The outward trace mean curvature equals H_0 almost everywhere
on Sigma.
Proof. First establish quasi-minimality. Let O subset M be a
bounded open neighborhood of Gamma. If F subset O is measurable
with |F| = V and close to E in L^1, the global comparison gives
P(F) >= I(V) = P(E), where I(V) = n omega_n^{1/n} V^{(n-1)/n} is
the flat model profile. So E minimizes P among compact
modifications of fixed volume. For F not volume-prescribed, write
v_F = |F|. Concavity of I with I(V) = P(E) and I'(V) = m H_0
(calculated at the tangent volume, using the profile formula at
the equality radius) gives
P(F; O) - P(E; O)
>= I(v_F) - I(V)
>= -m H_0 |v_F - V|
>= -m H_0 |F symmetric-difference E|,
the quasi-perimeter minimizer condition with Lambda = m H_0.
Apply Antonelli-Pasqualetto-Pozzetta's local smooth-manifold
conclusion (Corollary 1.6, no global lower Ricci bound) in every
dimension n >= 2. This gives a relatively open C^{1,alpha} regular
part Sigma of full perimeter with a relatively closed singular
subset Z of Hausdorff dimension at most n - 8, plus local upper
area growth. Decrease alpha to lie in (0, 1) if needed, giving
Sigma of class C^{1,alpha}.
Boundedness of E makes Gamma and Z compact. The density-one
representative of E has boundary equal to Gamma: the perimeter
measure is supported on the reduced boundary, and a boundary
point of the density-one representative is an essential boundary
point, so perimeter cannot vanish nearby. Local upper area
estimates combine over a finite cover of Gamma to give (3). To
identify the occupied phase: near a regular point the support is
a graph splitting a coordinate neighborhood into two connected
sides; on either side D chi_E = 0 so chi_E is constant almost
everywhere there, and the two side values differ. The occupied
side defines a consistent outward normal, and the graph
divergence formula identifies the perimeter measure with induced
area. Together with the full-perimeter conclusion this gives the
perimeter-area identity, and the closure of Sigma equals Gamma.
For the trace curvature: choose graph coordinates (y, z) with
z = u(y) on Sigma and E below the graph. The first variation of
area minus H_0 times volume for the graph, using compactly
supported vertical fields, gives the weak graph equation
int [ d/dp L_g(y, u, Du) . D zeta + B(y, u, Du) zeta ] dy = 0
for every zeta in C_c^infty(Omega), where L_g is the area
density of the graph, W = sqrt(det G), and
B = d/dz L_g - H_0 W. The coefficient H_0 comes from the same
global deficit on every patch, so the signed outward curvature
equals H_0 almost everywhere. End of proof.
Graph regularity lemma. Let Omega be open in R^m with m >= 1,
and let u in C^{1,alpha}_loc(Omega) with 0 < alpha < 1. Let
G(y, z) be the matrix of a smooth Riemannian metric on a
neighborhood of the graph of u in R^m x R. Set
W(y, z) = sqrt(det G(y, z)) and
L_g(y, z, p) = W(y, z) [ (-p, 1)^T G(y, z)^{-1} (-p, 1) ]^{1/2}.
Fix H_0 > 0 and set B(y, z, p) = d/dz L_g(y, z, p) - H_0 W(y, z).
If
int_Omega [ d/dp L_g(y, u, Du) . D zeta + B(y, u, Du) zeta ] dy = 0
for every zeta in C_c^infty(Omega), then u in C^{1,1}_loc(Omega).
Proof. Shrink the chart so that its arguments and the segments
between any two nearby arguments lie in a single compact
ellipticity region: the Hessian d^2/dp^2 L_g is uniformly positive
definite there and G, W, L_g and their derivatives through third
order are uniformly bounded.
Choose a Euclidean ball B_{4 rho} compactly contained in Omega.
For a coordinate unit vector e and 0 < delta < rho, put
v(y) = (u(y + delta e) - u(y))/delta
on B_{3 rho}. Each v is C^1 (no bound on Dv is yet known), and
the mean-value identity
v(y) = int_0^1 d_e u(y + s delta e) ds
gives a uniform C^{0,alpha} bound because ||u||_{C^{1,alpha}} is
finite.
Subtract the graph equations at y + delta e and y, divide by
delta. Set
T_s(y) = (1-s)(y, u(y), Du(y))
+ s( y + delta e, u(y + delta e), Du(y + delta e) ).
The result is a weak equation
int [ (a Dv + f) . D zeta + (b . Dv + d) zeta ] dy = 0,
with
a = int_0^1 d^2/dp^2 L_g(T_s) ds,
b = int_0^1 d/dp B(T_s) ds,
f = int_0^1 ( d_e d/dp L_g(T_s) + d/dz d/dp L_g(T_s) v ) ds,
d = int_0^1 ( d_e B(T_s) + d/dz B(T_s) v ) ds.
The C^{1,alpha} norm of u bounds the C^{0,alpha} norms of all T_s
uniformly in s and delta. Smooth composition on the compact
argument set plus the bound on v makes a and f of class C^{0,alpha}
with uniform constants, and b and d uniformly bounded. The
matrices a are symmetric and uniformly elliptic.
Energy bound. Test with chi^2 v, chi supported in B_{3 rho} and
equal to one on B_{2 rho}. Ellipticity puts a multiple of
int chi^2 |Dv|^2 on the left. Young's inequality together with
the uniform bounds on v, f, b, d absorbs every term linear in Dv
(in particular b . Dv) into that multiple, leaving
int_{B_{2 rho}} |Dv|^2 dy <= C
uniformly in the difference step.
Frozen comparison. Fix x in B_rho and 0 < r <= min{rho, 1}. On
B_r(x), let w - v in H^1_0(B_r(x)) solve
div(a(x) Dw) = 0.
Existence and uniqueness come from minimizing the positive
quadratic Dirichlet energy. Put z = v - w. Subtract the
equations and test with z; the frozen constant vector f(x)
integrates to zero against Dz. Ellipticity and the coefficient
bounds give, with all norms on B_r(x),
lambda ||Dz||_2^2
<= C r^alpha ( ||Dv||_2 + |B_r|^{1/2} ) ||Dz||_2
+ C ( ||Dv||_2 + |B_r|^{1/2} ) ||z||_2.
The second line carries the first-order term b . Dv. Zero-
boundary Poincare gives ||z||_2 <= C r ||Dz||_2, and r <= r^alpha
for r <= 1. So
(fint_{B_r(x)} |Dv - Dw|^2)^{1/2}
<= C r^alpha (1 + (fint_{B_r(x)} |Dv|^2)^{1/2}).
Energy minimization after subtracting an affine function also
gives, for every constant vector p,
(fint_{B_r(x)} |Dw - p|^2)^{1/2}
<= C (fint_{B_r(x)} |Dv - p|^2)^{1/2}.
Gradient bound. Put p_r = fint_{B_r(x)} Dv and
J(r) = (fint_{B_r(x)} |Dv - p_r|^2)^{1/2}. Using p = p_r in the
above estimates and comparing averages on the two balls gives,
for a fixed gamma < 1,
J(gamma r) <= C gamma J(r)
+ C gamma^{-m/2} r^alpha (1 + |p_r| + J(r)),
|p_{gamma r} - p_r| <= gamma^{-m/2} J(r).
Fix gamma with C gamma <= 1/2 and choose an integer ell >= 1
with ell/2 >= gamma^{-m/2}. Then
S(r) = 1 + |p_r| + ell J(r)
satisfies S(gamma r) <= (1 + C' r^alpha) S(r). Choose a starting
radius r_* <= min{rho, 1}; the starting averages are bounded
uniformly over x in B_rho by the energy bound. On the geometric
scales r_j = gamma^j r_* the product of (1 + C' r_j^alpha) is
finite because sum_j r_j^alpha < infinity. Thus |p_{r_j}| is
bounded uniformly in x in B_rho, in e, and in small delta. Each
fixed-step quotient is C^1, so p_{r_j} -> Dv(x) as j -> infinity,
giving a bound for Dv(x) uniform in x, e, and small delta.
Finally, Dv(y) = (Du(y + delta e) - Du(y))/delta holds
classically. The uniform bound on Dv bounds increments of Du
along every coordinate direction. On a smaller coordinate cube,
join two points by coordinate segments and add the increments;
the sum is at most sqrt(m) times the Euclidean distance. Hence Du
is locally Lipschitz and u in C^{1,1}_loc(Omega). End of proof.
Global first variation. Under the hypotheses of Equality
Boundary, for every smooth ambient field Y defined near Gamma,
int_Sigma div_{T Sigma} Y dsigma
= H_0 int_Sigma <Y, N> dsigma.
Proof. Apply the local patch identity to eta_j Y, where the eta_j
are the cutoffs of Capacity Cutoffs (their hypotheses are met by
Equality Boundary (3)). The boundary terms linear in grad eta_j
are bounded by ||Y||_infty A^{1/2} ||grad_Sigma eta_j||_2 -> 0;
all other terms converge by dominated convergence since the
ambient data are bounded near compact Gamma. Passing to the limit
gives the global identity. On any compact boundaryless component
of Sigma, integrate the local formula directly. End of proof.
Sobolev on Sigma. In the flat equality setting (b = 0), the
Hypersurface Sobolev theorem can be extended to every smooth
ambient restriction v.
Extension. When n >= 4, the cutoffs eta_j of Capacity Cutoffs,
applied to the Hypersurface Sobolev theorem on Sigma with
h = 1/R = H_0/m, give
c int_Sigma |grad_Sigma(eta_j v)|^2 dsigma
+ (m/R^2) int_Sigma (eta_j v)^2 dsigma
>= m s_m^{2/m} ( int_Sigma |eta_j v|^q dsigma )^{2/q}
for each j. Passing from eta_j v to v: boundedness of v and of
its gradient near Gamma gives convergence in L^2 and L^q, and
grad_Sigma(eta_j v) - grad_Sigma v
= (eta_j - 1) grad_Sigma v + v grad_Sigma eta_j
tends to zero in L^2 because the first term vanishes by dominated
convergence and the second is bounded by
||v||_infty ||grad_Sigma eta_j||_2 -> 0. So both sides converge:
c int_Sigma |grad_Sigma v|^2 dsigma
+ (m/R^2) int_Sigma v^2 dsigma
>= m s_m^{2/m} ( int_Sigma |v|^q dsigma )^{2/q}.
This identity extends the inequality to the constant function v = 1
and to every fixed ambient perturbation. No completeness of the
regular locus Sigma is assumed.
Barycenter. If mu is a nonzero finite positive measure of compact
support in M, then
F(p) = int_M D_p(x) dmu(x), D_p(x) = d(p,x)^2/2,
is continuous and proper: if supp mu lies in B_a(o), then
d(p,x) >= d(p,o) - a, so F(p) -> infinity as p leaves compact
sets. Completeness and Hopf-Rinow give a minimizer of F. At any
minimizer,
int_M log_p x dmu(x) = 0,
obtained by differentiating F on compact sets. Uniqueness of the
minimizer is not required.
Flux inequality (lower second moment). Let T be Sigma itself
(when Sigma is compact) or any compact boundaryless component of
Sigma. Write a = sigma(T). The distance comparison at b = 0
gives Hess D_p >= g, so
div_{T Sigma} grad D_p
= trace_{T Sigma} Hess D_p >= m
on T. Apply the Global First Variation identity with Y =
grad D_p (multiply all ambient fields by a smooth cutoff equal
to one near compact Gamma, so the lack of compact support on all
of M is no obstacle):
m a <= int_T div_{T Sigma} grad D_p dsigma
= (m/R) int_T <grad D_p, N> dsigma
<= (m/R) ( a int_T r_p^2 dsigma )^{1/2},
the last step by Cauchy-Schwarz and |grad D_p| = r_p (from
|grad r_p| = 1). Rearranging:
int_T r_p^2 dsigma >= R^2 a.
This is the lower second-moment bound, valid for every p in M.
It is the first half of the equality machine: it holds for every
p, and equality throughout is exactly what the spectral rigidity
will produce.
Spectral rigidity (n >= 4). Here m = n-1 >= 3, so the Sobolev
extension to all of Sigma is available.
First step: spectral gap. Retain q = 2m/(m-2) and c = q - 2.
The constant function v = 1 attains equality in the extension
inequality, because sigma(Sigma) = A = s_m R^m and
m s_m^{2/m} A^{2/q} = (m/R^2) A.
Let phi be a smooth ambient restriction with
int_Sigma phi dsigma = 0. Boundedness of phi permits expansion of
v = 1 + tau phi at tau = 0:
( int_Sigma |1 + tau phi|^q dsigma )^{2/q}
= A^{2/q} [1 + (q-1)(tau^2/A) int_Sigma phi^2 dsigma + o(tau^2)],
from the Taylor expansion (1 + tau phi)^q = 1 + q tau phi +
(q(q-1)/2) tau^2 phi^2 + O(tau^3) and the zero mean.
The linear term in tau vanishes in the left-hand quadratic form
(the gradient 2 tau int grad 1 . grad phi = 0 plus the potential
2 tau (m/R^2) int phi = 0). The constant terms agree. Since
m s_m^{2/m} A^{2/q - 1} = m/R^2, comparing quadratic terms gives
c int_Sigma |grad_Sigma phi|^2 dsigma
+ (m/R^2) int_Sigma phi^2 dsigma
>= (m/R^2)(q-1) int_Sigma phi^2 dsigma,
so, by c = q-2,
int_Sigma |grad_Sigma phi|^2 dsigma
>= (m/R^2) int_Sigma phi^2 dsigma.
This is the spectral gap inequality, valid for every smooth
ambient restriction phi with zero |dsigma|-mean. The cutoff limit
has already been taken before the expansion: the extension of
Sobolev passed to arbitrary smooth restrictions.
Second step: upper moment. Choose a barycenter p for the measure
dsigma on Sigma and an orthonormal basis e_1, ..., e_n of T_pM.
The smooth functions
phi_i(x) = <log_p x, e_i>
have zero mean by the barycenter property. Because d log_p is
contracting,
sum_i |grad_Sigma phi_i|^2
= sum over an orthonormal frame v_1, ..., v_m of T_x Sigma
of sum_i |d log_p(v_j)|^2
<= m
for every x in Sigma: each of the m tangential vectors contributes
its squared image norm in T_pM at most 1, and there are n
directions total but only m tangential ones.
Summing the spectral gap inequality over i:
int_Sigma sum_i |grad_Sigma phi_i|^2 dsigma
>= (m/R^2) int_Sigma sum_i phi_i^2 dsigma.
Using sum_i phi_i^2 = r_p^2 and sum_i |grad_Sigma phi_i|^2 <= m:
m A >= (m/R^2) int_Sigma r_p^2 dsigma,
i.e.
int_Sigma r_p^2 dsigma <= R^2 A.
Combined with the flux inequality this yields equality of the two
bounds, so the flux inequality holds with equality throughout.
Third step: the normal is radial with constant radius. Look back
at the equalities in the flux derivation with T = Sigma:
m A = int_Sigma div_{Sigma} grad D_p dsigma
= (m/R) int_Sigma <grad D_p, N> dsigma
= (m/R) ( A int_Sigma r_p^2 dsigma )^{1/2}.
For equality to hold throughout, two things must happen. First,
Hess D_p restricted to T_x Sigma must have trace exactly m
almost everywhere (i.e. every eigenvalue contributes only its
minimum value): the trace inequality Trace(Hess D_p) >= m must be
an equality for every x in Sigma put together with the integral.
Second, Cauchy-Schwarz must be equality, so |grad D_p| = r_p must
be proportional to <grad D_p, N> as a functional, i.e. the
gradient of D_p is parallel to N almost everywhere.
Both conditions together with the pointwise bound |grad D_p| =
r_p give <grad D_p, N> = R almost everywhere on Sigma, i.e.
|grad_Sigma(D_p)| = R and |grad_Sigma r_p| = 1 almost everywhere
on Sigma. Since |grad_Sigma r_p| <= |grad r_p| = 1, equality
forces grad_Sigma r_p = grad r_p, so r_p is constant along Sigma
almost everywhere:
r_p(x) = R for almost every x in Sigma.
Continuity (Sigma is C^{1,1} and the equality is between smooth
functions on a dense subset) upgrades this to every point of
Sigma:
Sigma subset {x in M : d(p, x) = R}.
Finally, the density of Sigma in Gamma gives
Gamma subset {x in M : d(p, x) = R}.
Neither connectedness nor completeness of the regular part is
used. The proof of the partial containment of the equality
boundary is complete when n >= 4.
Dimension two. Now m = 1, and Gamma = Sigma is a compact
embedded C^{1,1} curve without boundary, in a Cartan-Hadamard
surface M^2. The total length is A = s_1 R^1 = 2 pi R. Choose a
connected component T of Sigma and let L = length(T) > 0.
Periodic arclength parametrization. A connected compact
one-dimensional manifold without boundary admits a periodic
arclength parametrization of period L. Indeed, continue an
oriented arclength chart: a traverse of arbitrarily large length
cannot stay injective in a component of finite length, and local
uniqueness gives a least positive period whose image is open and
closed in the component, so one least period traverses the whole
component once.
Periodic Wirtinger. For a periodic C^1 function phi with zero
mean, the periodic Wirtinger inequality says
int_T |grad_T phi|^2 dsigma
>= (2 pi / L)^2 int_T phi^2 dsigma.
Proof in arclength coordinates: expand a trigonometric
polynomial in Fourier modes (the zero mode is absent), and note
periodic convolution uniformly approximates a C^1 function
together with its derivative while preserving the zero mean.
Apply Wirtinger. Take a barycenter p of the length measure on T
and let e_1, e_2 be an orthonormal basis of T_p M. The functions
phi_i(x) = <log_p x, e_i>, i = 1, 2, have zero mean by the
barycenter property. Because T is one-dimensional, each
|grad_T phi_i| is the projection of grad phi_i onto the single
tangent direction, and because d log_p is contracting,
|grad phi_i| <= 1, so
sum_i |grad_T phi_i|^2 <= 1.
Summing Wirtinger over i and identifying sum_i phi_i^2 = r_p^2:
int_T sum_i |grad_T phi_i|^2 dsigma
>= (2 pi/L)^2 int_T r_p^2 dsigma.
Since the left side is at most L,
int_T r_p^2 dsigma <= L^3 / (2 pi)^2.
On the other hand, the flux inequality gives
int_T r_p^2 dsigma >= R^2 L.
Combining,
R^2 L <= L^3 / (2 pi)^2, i.e. L^2 >= (2 pi R)^2 = A^2,
so L >= A. Since L <= A (one component of total length A), L = A:
this component carries all the length, and Sigma is a single
curve.
Equality in the flux now holds throughout with T = Sigma. The
equality analysis of the flux derivation from the n >= 4 case
applies verbatim with m = 1: the trace Hess D_p restricted to
T_x Sigma must be exactly 1 almost everywhere, and Cauchy-
Schwarz must be equality, so grad D_p is parallel to N almost
everywhere. Since |grad D_p| = r_p, this gives r_p = R on
Sigma. Therefore
Sigma subset {x in M : d(p, x) = R},
with p a fixed barycenter of the length measure on Sigma.
Dimension two is complete.
Dimension three. Now m = 2 and Sigma is a compact embedded
C^{1,1} surface without boundary with area A = s_2 R^2 = 4 pi
R^2. The critical exponent q = 2m/(m-2) = infinity lies at the
boundary of the Sobolev scale, so the spectral argument of
dimension >= 4 is unavailable. Replace it with a punctured
flux identity.
The setup. Fix p in Sigma and, away from p, define
r = r_p, Y = r^{-2} grad D_p = grad r / r,
t = <Y, N> = <grad r, N>/r.
Since Hess D_p >= g and |grad r| = 1,
div_{Sigma} Y
= r^{-2} trace_{Sigma} Hess D_p - 2 r^{-2} |grad_Sigma r|^2
>= 2 r^{-2}(1 - |grad_Sigma r|^2)
= 2 t^2.
The cutoff. Choose a smooth nondecreasing chi : [0, infinity) ->
[0, 1] that is zero on [0, 1] and one on [2, infinity). The
field Y_eps = chi(r/eps) Y extends smoothly by zero near p.
Apply the global first variation identity to Y_eps. Its
divergence is
div Y_eps = chi div Y + (chi'(r/eps)/(eps r)) |grad_Sigma r|^2,
and <Y_eps, N> = chi t. Writing
J_eps = int_Sigma (chi'(r/eps)/(eps r)) |grad_Sigma r|^2 dsigma,
and using h = H_0/m = 1/R (so H_0 = m h = 2 h), the identity
becomes
int_Sigma chi div Y dsigma + J_eps = H_0 int_Sigma chi t dsigma.
Combining with div Y >= 2 t^2 and H_0 = 2 h gives
int_Sigma chi(r/eps) [ h^2/2 - 2(t - h/2)^2 ] dsigma >= J_eps.
Indeed h^2/2 - 2(t - h/2)^2 = 2 h t - 2 t^2 = H_0 t - 2 t^2.
Letting eps down to 0. Compute the limit of J_eps. Use
exponential coordinates at p, with the first two coordinate
directions tangent to Sigma at p. In a small enough neighborhood
the entire support Gamma is a graph (y, b(y)) over the tangent
plane, with b(0) = 0 and Db(0) = 0. Set y = eps z. On the
support of chi'(r/eps) one has 1 <= r/eps <= 2, so z stays in a
fixed bounded disk, and away from zero because the slope of b
vanishes at the origin:
r/eps -> |z|,
dsigma/eps^2 -> dz,
|grad_Sigma r| -> 1.
The area density is uniformly bounded there, |grad_Sigma r| <= 1,
and |chi'(r/eps)/(r/eps)| <= ||chi'||_infty. Dominated
convergence after extension by zero to the fixed disk gives
J_eps -> int_{R^2} (chi'(|z|)/|z|) dz
= 2 pi int_1^2 chi'(s) ds
= 2 pi (chi(2) - chi(1)) = 2 pi.
Because h = 1/R and A = 4 pi R^2, h^2 A / 2 = 2 pi. Now take
eps down to 0 in the cutoff inequality. The left side converges
by dominated convergence (the integrand is bounded on the fixed
compact support; |h^2/2 - 2(t - h/2)^2| <= h^2/2 + 2t^2 + 2 h^2
and t is locally bounded on Sigma \ {p}) to
h^2 A / 2 - 2 int_Sigma (t - h/2)^2 dsigma,
and the right side converges to 2 pi. Using h^2 A / 2 = 2 pi,
h^2 A/2 - 2 pi >= 2 int_Sigma (t - h/2)^2 dsigma >= 0,
so
0 <= int_Sigma (t - h/2)^2 dsigma <= 0.
Therefore t = h/2 = 1/(2R) almost everywhere on Sigma. Since
both sides are continuous on Sigma \ {p} and the statement is
independent of the choice of p, this upgrades to every point of
Sigma \ {p}.
Locating the boundary. Fix x_0 in Sigma. The identity
t = 1/(2R) holds at x_0 for every p in Sigma \ {x_0}. Using
grad D_p(x_0) = -log_{x_0} p and N = N(x_0),
-<log_{x_0} p, N> / r_p^2 = 1/(2R),
so
-<log_{x_0} p, N> = r_p^2 / (2R) = |log_{x_0} p|^2 / (2R).
Complete the square:
|log_{x_0} p + R N|^2
= |log_{x_0} p|^2 + 2 R <log_{x_0} p, N> + R^2
= r_p^2 - 2R (r_p^2/(2R)) + R^2
= R^2.
Passing to p = x_0 by continuity gives the same identity there
(where both sides reduce to R^2). Therefore
Sigma subset exp_{x_0}
{ v in T_{x_0} M : |v + R N(x_0)| = R }.
The target is a Euclidean sphere of radius R in T_{x_0} M,
centered at -R N(x_0) and passing through the origin. The offset
is the only difference from the higher-dimensional case. The
punctured flux identity has located the equality boundary on a
sphere. Dimension three is complete.
Recovering the ball. The three dimension regimes give the same
structural conclusion: there exist o in M and a Euclidean sphere
S of radius R in T_o M such that
Gamma subset exp_o(S).
In dimensions at least four and in dimension two, S is centered
at the origin of T_o M. In dimension three, S is centered at
-R N(x_0) and passes through the origin.
Phase recovery. Let U_0 be the bounded open ball in T_o M that
S bounds (the side containing the origin), and put
U = exp_o(U_0). The exponential map is a global diffeomorphism
by Cartan-Hadamard, so the two connected components of
T_o M \ S map onto the two connected components of M \ exp_o(S).
The interior component U is bounded, since its closure is a
compact image. The exterior component is unbounded because
d(o, exp_o v) = |v| escapes to infinity along vectors that
escape to infinity in the Euclidean exterior; exterior
connectedness uses n >= 2. The smooth hypersurface exp_o(S) has
zero ambient volume.
The distributional derivative of chi_E is supported on Gamma,
so it vanishes on each complementary component of
M \ exp_o(S). A function whose weak coordinate derivatives all
vanish is locally constant almost everywhere: mollify in
coordinate balls, and overlapping balls give the same constant.
Hence chi_E is almost everywhere constant on each side.
The exterior constant is 0 because E is bounded. The exterior
is unbounded, so it contains balls of arbitrary radius centered
at o, and those balls have positive volume on which chi_E can
only vanish. Since |E| = V > 0 and exp_o(S) has zero volume,
the occupied mass must lie inside U, so the interior constant
is 1. Therefore E agrees with U up to a set of vol_g-zero. Only
the containment of Gamma in exp_o(S) was used.
Flatness of the interior metric. By the distance comparison at
b = 0, the pullback metric g_tilde = (exp_o)^* g satisfies
g_tilde >= g_o (the constant Euclidean metric on T_o M) on all
of T_o M, hence on U_0. By phase recovery,
vol_{g_tilde}(U_0) = V = omega_n R^n = vol_{g_o}(U_0).
The smooth relative density sqrt(det_{g_o} g_tilde) is at least
one. Its integral against g_o-volume equals vol_{g_o}(U_0), and
a nonnegative continuous function whose integral equals the
integral of its upper bound is identically equal to that upper
bound. Every eigenvalue of g_tilde relative to g_o is at least
one and their product is at least one; equality of the product
to one forces every eigenvalue to be one. Hence
g_tilde = g_o on U_0.
Therefore exp_o : U_0 -> U is a Riemannian isometry. This
proves necessity in the equality theorem. No flatness outside U
is implied.
Sufficiency. Conversely, suppose E agrees up to a vol_g-zero set
with an open region U such that (U, g|_U) is isometric to a
round Euclidean ball. Let F : B_R(0) subset R^n -> U be the
isometry, put p = F(0) and L = dF_0. For |v| < R, the image of
the segment s -> sv is an ambient geodesic with initial data
(p, Lv), because the Levi-Civita connection on an open region is
the restriction of the ambient connection. Geodesic uniqueness
gives
F(v) = exp_p(Lv) for |v| < R.
This formula extends smoothly to the closed Euclidean ball
through the ambient exponential map, and the pullback metric is
Euclidean in the interior, hence on the boundary by continuity.
The boundary is a smooth sphere of area s_{n-1} R^{n-1} = A and
the enclosed volume is omega_n R^n = V. Perimeter and volume
are unchanged by null modifications, so equality holds. This
completes the proof of the equality theorem.
Spectral and torsional comparison. Two classical consequences
of the isoperimetric theorem, taken in the form we proved it:
all finite-perimeter sets, finite volume, no boundedness
assumption.
Theorem. Let (M^n, g) be complete, simply connected and smooth
with n >= 2 and Sec_g <= kappa <= 0. Let D subset M be a
bounded open set with smooth boundary and volume V. Let B_*(V)
be the round ball of volume V in the model space of curvature
kappa. Then
lambda_1(D) >= lambda_1(B_*(V)), T(D) <= T(B_*(V)),
where lambda_1 is the first Dirichlet eigenvalue of the
Laplacian, and
T(D) = sup over nonzero f in C_c^infty(D)
of (int_D f dvol)^2 / int_D |grad f|^2 dvol
is the torsional rigidity.
Proof outline. The Rayleigh quotient for a nonzero f in
C_c^infty(D) is
R(f) = int_D |grad f|^2 dvol / int_D f^2 dvol,
lambda_1(D) = inf { R(f) : f in C_c^infty(D), f nonzero }.
The layer-cake formula for powers of the gradient pairs the
Dirichlet energy with the perimeters of the level sets of |f|:
int_D |grad f|^p dvol
= int_0^infty p t^{p-1} P({|f| > t}; D) dt,
for every p >= 1, for almost every t. The level sets {|f| > t}
are finite-perimeter sets of finite volume, compact subsets of
D, and their D-restricted perimeter equals their full ambient
perimeter for almost every t, because the trace on dD has zero
lower-dimensional measure at a.e. level. Applying the
isoperimetric comparison at each level gives
P({|f| > t}; D) >= I_b( |{|f| > t}| ),
since the level-set volume is at most the level-set volume in
the model and I_b is increasing. Integrating the model bound
against p t^{p-1} gives, for every p >= 1,
int_D |grad f|^p dvol >= int_{B_*(V)} |grad f^*|^p dvol,
where f^* is the decreasing spherical rearrangement of f on
the model ball B_*(V), by the standard Polya-Szego
rearrangement which matches the distribution function
t |-> |{|f| > t}| to t |-> |{|f^*| > t}| at every level. The
same layer-cake argument gives the identity
int_D phi(f) dvol = int_{B_*(V)} phi(f^*) dvol
for every nonnegative phi; specifically phi(s) = s^2 preserves
the denominator of R, and phi(s) = s preserves the L^1-mean of
the torsion. Therefore R(f) >= R(f^*) and
R_T(f) <= R_T(f^*). Taking the Rayleigh infimum over f proves
the spectral bound. The torsion
is treated identically: the same rearrangement binds the
L^1-volume (int_D f dvol)^2 = (int_{B_*(V)} f^* dvol)^2 and the
Dirichlet energy, so
R_T(f) := (int_D f dvol)^2 / int_D |grad f|^2 dvol
<= (int_{B_*(V)} f^* dvol)^2
/ int_{B_*(V)} |grad f^*|^2 dvol
<= T(B_*(V)).
Taking the torsion supremum proves the torsional bound. In the
model space a round ball attains the isoperimetric equality at
every level, so the bounds are sharp: lambda_1(B_*(V)) and
T(B_*(V)) are the extremal values. End of proof.
The same arguments require the isoperimetric comparison to
hold for finite-perimeter sets that are not required to be
bounded; this is exactly the general form we proved. End of
consequences.
This completes the proof. The comparison theorem, the equality
theorem, and the spectral and torsional corollaries are
established in every dimension and for every kappa <= 0. The
key inputs are the sharp Aubin-Talenti constant via the
Hypersurface Sobolev inequality, the confined isoperimetric
profile and its scalar differential inequality, the
Brezis-Lieb compactness, the Ghomi-Spruck obstacle regularity,
and the value of the profile at the equality radius that pins
down the constant mean curvature of an equality boundary.