Two theorems about a complete, simply connected smooth Rieman-
nian manifold M^n of dimension n >= 2, with sectional curvature
bounded above by some constant kappa <= 0. Such a manifold is
called Cartan-Hadamard. Its exponential map at every point is a
global diffeomorphism onto M, and that is the only global fact
about its geometry needed.
Theorem 1 (isoperimetric comparison). For every measurable set
E in M with finite volume V and finite perimeter A,
A >= I_b(V), where b = sqrt(-kappa),
and I_b(V) is the boundary area of the round ball of volume V
in the space form of constant curvature -b^2. Equality is
attained by those round balls.
Theorem 2 (equality rigidity). In flat curvature (kappa = 0),
equality
A = n * omega_n^(1/n) * V^((n-1)/n)
holds if and only if E coincides, up to a set of measure zero,
with an open set U that is isometric, under the metric it
inherits from M, to a round Euclidean ball. The ambient manifold
may be curved everywhere outside U; the theorem says nothing
there.
Why the comparison is not a consequence of volume comparison.
Bishop-Gromov says a geodesic ball in M has at least the volume
of a geodesic ball of the same radius in the model space. That
controls one family of shapes. It does not control a long thin
blob, a dumbbell, or a region whose boundary is encrusted with
wiggles, all of which can have the same volume as a round ball
and a far larger boundary. Theorem 1 says no shape beats the
round ball. That is genuinely new information, not a restate-
ment of Bishop-Gromov.
How the proof runs, in one paragraph. Prove a sharp Sobolev
inequality on any hypersurface, with the mean curvature entering
as a potential. Insert this inequality inside a confined mini-
mization problem: at each prescribed volume v, minimize full
ambient perimeter among sets inside a large geodesic ball. The
resulting profile I(v) obeys a scalar differential inequality
I'(v) >= m sqrt(b^2 + (s_m/I(v))^(2/m)), whose exact integral is
the model comparison of Theorem 1. Dimensions two and three
supply their own input (Weil-Bol and Kleiner). Removing the
boundedness restriction and passing to arbitrary kappa use a
cutoff and a metric rescaling. For Theorem 2, the machinery is
re-run on an equality set itself: it forces the boundary to lie
on a round sphere, then a volume comparison forces the inherited
metric to be Euclidean.
Notation.
M^n a Cartan-Hadamard manifold.
g its Riemannian metric.
Sec_g sectional curvature, one number per 2-plane.
kappa an upper bound on Sec_g, with kappa <= 0.
b sqrt(-kappa) >= 0. (b = 0 means kappa = 0.)
d(p, x) Riemannian distance between p and x in M.
exp_p the exponential map at p; a bijection.
log_p its inverse.
D_p(x) d(p, x)^2 / 2. Smooth on all of M including at
p. Its gradient away from p has norm d(p, x),
and its Hessian is at least the metric when
Sec <= 0.
omega_n the volume of the unit ball in flat R^n.
m n - 1.
s_m the area of the unit sphere S^m. One has
s_m = n * omega_n.
vol_g(E) the volume of E; also written |E| when g is
clear from context.
P_g(E) the perimeter of E.
chi_E the indicator function of E.
dsigma induced area measure on a hypersurface.
Volume and perimeter are unchanged by adding or removing sets
of vol_g-zero, and such sets are routinely identified.
Model functions. For b >= 0 and r >= 0 define
sn_b(r) = r when b = 0,
sn_b(r) = sinh(b*r) / b when b > 0,
cs_b(r) = sn_b'(r).
These are the analogues of sine and cosine in the space form of
curvature -b^2. The identities
sn_b'(r) = cs_b(r), cs_b(r)^2 - b^2 * sn_b(r)^2 = 1
are used silently. 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 A_b and
V_b increase strictly from 0 to infinity; V_b has an inverse.
The model isoperimetric profile is
I_b(V) = A_b( V_b^(-1)(V) ), V >= 0.
Equivalently, I_b(V) is the perimeter of the round ball of
volume V in the model. I_b(0) = 0, and I_b is continuous and
strictly increasing on [0, infinity).
Perimeter of a rough set. For a measurable E in (M^n, g),
P_g(E) = sup { integral_E div_g X dvol_g :
X smooth compactly supported on M, |X|_g <= 1 }.
This is 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 called sets of
finite perimeter. The theorems are stated for these sets: no
smoothness of E is assumed.
Lemma 1 (distance-Hessian comparison). Fix p in M and put
r(x) = d(p, x). For r > 0 the quadratic form
H := Hess r - (cs_b(r)/sn_b(r)) (g - dr x dr)
is positive semidefinite on T_x M.
Proof. At a point x with r > 0, let gamma be a unit-speed
geodesic from p to x. For a tangent vector X at x, let J be the
Jacobi field along gamma with J(0) = 0 and J(r) = X. The second
variation formula for length gives
Hess r(X, X) = integral_0^r (|D_t J|^2
- ) dt.
Sectional curvature at most -b^2 makes the curvature term at
most -b^2 |J|^2, so
Hess r(X, X) >= integral_0^r (|D_t J|^2 + b^2 |J|^2) dt.
Among all J of this form, the integral is minimized by
J_0(t) = (sn_b(t)/sn_b(r)) X carried by parallel transport:
writing J = J_0 + V with V vanishing at both ends, integration
by parts and J_0'' = b^2 J_0 make the cross term zero, and the
V-energy is nonnegative. The minimum value is, by one more
integration by parts,
= (cs_b(r)/sn_b(r)) |X|^2.
For X perpendicular to gamma' this gives Hess r(X, X) >=
(cs_b(r)/sn_b(r)) |X|^2. For X radial, Hess r(X, X) = 0. For
mixed X both sides vanish. Hence H is positive semidefinite on
every T_x M with r(x) > 0. End of proof.
Corollary 1 (polar comparison). In geodesic polar coordinates
(r, theta) about p, the metric satisfies
g >= dr^2 + sn_b(r)^2 g_{S_p^(n-1)}.
In particular the substitution b = 0 gives Hess D_p >= g, where
D_p(x) = d(p, x)^2/2. That special case is what the equality
theory uses.
Proof. Along a radial geodesic let J be an angular Jacobi
field. Gauss's lemma and commutation of variation fields give
d/dr |J|^2 = 2 Hess r(J, J) >= 2 (cs_b(r)/sn_b(r)) |J|^2.
So |J|^2 / sn_b(r)^2 is nondecreasing, with limit equal to the
initial angular squared norm as r -> 0. Testing on angular
vectors gives the polar bound. For b = 0 the same computation
gives Hess r >= (1/r)(g - dr x dr), and at r = 0 by continuity.
Then
Hess D_p = dr x dr + r Hess r >= dr x dr + (g - dr x dr) = g.
End of proof.
Let S = S(T_p M + R) be the unit sphere in T_p M + R = R^(n+1)
with the round metric of constant curvature 1. For x != p put
r = d(p, x) and theta(x) = exp_p^(-1)(x)/r in the unit sphere of
T_p M. Define, for a > 0,
z(x) = a sn_b(r/2) / cs_b(r/2),
Phi_{p,a}(x) = ( 2z/(1 + z^2) theta(x),
(1 - z^2)/(1 + z^2) ) in T_p M + R,
f_{p,a}(x) = a / ( cs_b(r/2)^2 + a^2 sn_b(r/2)^2 ).
At r = 0 set Phi_{p,a}(p) = (0, 1) and f_{p,a}(p) = a. The map
Phi_{p,a} is inverse stereographic projection composed with the
radial rescaling z. When b = 0 the pair (Phi, f) reduces to the
classical stereographic map with f = a / (1 + a^2 r^2 / 4).
Lemma 2 (comparison maps). Phi_{p,a} is smooth on all of M,
including at p; f_{p,a} is smooth and positive. For every
tangent vector X at every point x of M,
|dPhi_{p,a}(x)(X)|_S <= f_{p,a}(x) |X|_g,
and with u = log f_{p,a},
Hess u <= du x du - (1/2)(b^2 + f_{p,a}^2 + |grad u|^2) g
as quadratic forms. The pair (Phi, f) and its derivatives
through order three depend smoothly on (p, a, x).
Proof. Pullback metric. The round metric on S in stereographic
coordinates is
Phi_{p,a}^* g_S = 4 (dz^2 + z^2 g_{S_p^(n-1)}) / (1 + z^2)^2.
Substitute z(x) = a sn_b(r/2)/cs_b(r/2) and its derivative. The
double-angle identities 1 + z^2 = (cs_b(r/2)^2 + a^2 sn_b(r/2)^2)
/ cs_b(r/2)^2, dz = (a/(2 cs_b(r/2)^2)) dr, and
4 cs_b(r/2)^2 sn_b(r/2)^2 = sn_b(r)^2 simplify the right side
to
Phi_{p,a}^* g_S = f_{p,a}^2 ( dr^2 + sn_b(r)^2 g_{S_p^(n-1)} ).
By Corollary 1 the last factor is at most f_{p,a}^2 g, so
|dPhi_{p,a}(X)| <= f_{p,a} |X| for every tangent X.
Log-conformal Hessian. Write s = sn_b(r), c = cs_b(r),
S = sn_b(r/2), C = cs_b(r/2), A = a^2 + b^2, D = C^2 + a^2 S^2.
Then f = a/D, D' = A s/2, D'' = A c/2. Expanding both sides
of the double-angle identity gives
b^2 D^2 + a^2 + (A^2/4) s^2 = A c D. (2.1)
Differentiate u = log a - log D:
u' = -A s / (2D),
u'' = (u')^2 + u' c/s,
u' c/s = -(1/2)(b^2 + f^2 + (u')^2), (2.2)
the last equivalent to (2.1). Since u is radial, in polar
coordinates
Hess u = u'' dr x dr + u' (Hess r)_ang.
From Lemma 1, (Hess r)_ang = (c/s) g_ang + H with H >= 0, so
Hess u = u'' dr x dr + u' (c/s) g_ang + u' H.
By (2.2) this equals
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
and may be dropped. End of proof.
Given a finite positive measure on M with compact support and
no atoms, a center p and a scale a are needed, so that the
coordinate functions w_1, ..., w_{n+1} of Phi_{p,a} have zero
mu-mean coordinate by coordinate.
Lemma 3 (balancing). Let mu be a finite positive measure on M
with compact support and no atoms. There exist p in M and a > 0
such that the coordinate functions w_1, ..., w_{n+1} of
Phi_{p,a} satisfy
integral w_j dmu = 0 for j = 1, ..., n+1.
The second equation sum_j w_j^2 = 1 is automatic because
Phi_{p,a} lands on the unit sphere.
Idea of proof. Fix p in M and write Phi_{p,a} = (Psi_a, F_2^a)
with Psi_a its T_p M-valued part. Consider
T(xi, a) = Pi_C ( (xi, a) + ( int (Psi_a - xi) dmu,
int F_2^a dmu ) )
on the closed convex cylinder C = { |xi| <= 1 + eps,
delta <= a <= a_1 }, with Pi_C the nearest-point projection.
Brouwer's fixed point theorem gives a fixed point
zeta = (xi, a) of T. At that point the projection inequality
holds:
<= 0 for every y in C, (3.1)
where F(xi, a) is the vector in the parenthetical expression.
Three scalar estimates rule out the boundary.
Bottom face a = delta small: as a -> 0, Psi_a -> 0 and
F_2^a -> 1 uniformly, so F(xi, delta) is dominated by the upward
component mu(M) e_{n+1}. Choosing delta below mu(M)/2 keeps
F pointing strictly upward.
Top face a = a_1 large: pick delta_0 > 0 with mu(B_{delta_0}(p))
< 1/4 (possible because mu has no atoms). Choose a_1 with
a_1 t_b(delta_0) >= sqrt(3), where t_b(s) = sn_b(s/2)/cs_b(s/2).
Then outside B_{delta_0}(p), z(x) >= a_1 t_b(delta_0) >= sqrt(3),
so F_2^a(x) = (1 - z^2)/(1 + z^2) <= -1/2. Inside, F_2^a <= 1.
Therefore
int F_2^a dmu <= (1/4)(1) + (3/4)(-1/2) = -1/8 < 0.
F points strictly downward on the top face.
Outer side face |xi| = 1 + eps: |Psi_a| <= 1 forces
<= mu(M) - (1 + eps) mu(M)
= -eps mu(M) < 0.
F points strictly inward on the outer side.
So zeta lies in the interior of C. Taking both signs of every
small displacement y - zeta in (3.1) forces F(zeta) = 0, which
is the stated balance. Only the existence of (p, a) is used,
not its uniqueness. End of proof.
Let Sigma be an embedded, boundaryless, locally C^{1,1} hyper-
surface of M of dimension m = n - 1 >= 3, with the induced
metric g_Sigma and area measure dsigma. Let N be a choice of
oriented unit normal on Sigma, and let h be the signed normal-
ized mean curvature: the trace of the second fundamental form
with respect to N, divided by m. Reversing N reverses both h
and ; every expression below is invariant.
For a compactly supported Lipschitz function v on Sigma set
Q_Sigma(v) = int_Sigma [ c |grad_Sigma v|^2
+ m (h^2 - b^2) v^2 ] dsigma,
where b >= 0 is the curvature parameter, and
q = 2m/(m-2), c = q - 2 = 4/(m-2),
Y_m = m s_m^(2/m).
The number Y_m is the sharp constant of the Aubin-Talenti
Sobolev inequality in flat R^m:
c int_{R^m} |D phi|^2 dx >= Y_m ( int_{R^m} |phi|^q dx )^(2/q)
for every compactly supported smooth phi, and no smaller
constant works. (The two equivalent sharp forms differ only
in where c sits; see Aubin's book, or Gilbarg-Trudinger,
for the proof.) This is treated as a cited input.
Theorem 3 (Hypersurface Sobolev inequality). Let Sigma be
as above and let b >= 0. For every compactly supported
Lipschitz function v on Sigma,
Q_Sigma(v) >= Y_m ( int_Sigma |v|^q dsigma )^(2/q).
The same holds for the zero extension of every v in H^1_0(U),
where U is any open subset compactly contained in the interior
Sigma^circ and H^1_0(U) is the closure of compactly supported
Lipschitz functions under the induced H^1 norm. No smoothness
of dU is assumed. Sharpness: on a geodesic sphere in the model
space of curvature -b^2, v = 1 attains equality.
Lemma 4 (comparison square). Fix p in M and a > 0. Let
Phi = Phi_{p,a}, f = f_{p,a}, and u = log f be as in Lemma 2,
and put alpha = 2/c = (m-2)/2. For every compactly supported
Lipschitz v on Sigma,
Q_Sigma(v)
>= int_Sigma [ v^2 |dPhi|_Sigma|^2
+ c |grad_Sigma v
- alpha v grad_Sigma u|^2 ] dsigma.
Here |dPhi|_Sigma|^2 = sum_{i=1}^m |dPhi(e_i)|^2 over an
orthonormal frame (e_i) of T_x Sigma, and grad_Sigma u is the
tangential gradient of u along Sigma. The same estimate holds
on H^1_0(U) for U comp contained in Sigma^circ.
Proof. Lemma 2 gives the ambient Hessian bound
Hess u <= du x du - (1/2)(b^2 + f^2 + |grad u|^2) g. (4.1)
Let beta = be the normal derivative. The classical
restriction formula for the Laplacian reads, with tr_{T_Sigma}
the trace over the tangent space of Sigma,
tr_{T_Sigma} Hess u = Delta_Sigma u + m h beta. (4.2)
Substitute (4.2) into the tangential trace of (4.1). The
tangential trace of du x du is |grad_Sigma u|^2; the tangential
trace of g is m (identity); and |grad u|^2 = |grad_Sigma u|^2 +
beta^2. This gives
Delta_Sigma u + m h beta
<= |grad_Sigma u|^2
- (m/2)(b^2 + f^2 + |grad_Sigma u|^2 + beta^2).
Multiply by 2 and rearrange. Using h^2 + 2h beta + beta^2 =
(h + beta)^2, this gives
(m-2)|grad_Sigma u|^2 + 2 Delta_Sigma u
+ m(b^2 + f^2) + m(h + beta)^2 <= m h^2. (4.3)
Now expand the right side of the claimed inequality.
Using c alpha = 2 and c alpha^2 = m - 2,
c |grad_Sigma v - alpha v grad_Sigma u|^2
= c |grad_Sigma v|^2
- 2 c alpha v
+ c alpha^2 v^2 |grad_Sigma u|^2
= c |grad_Sigma v|^2
- 4 v
+ (m - 2) v^2 |grad_Sigma u|^2.
Subtract this and v^2 |dPhi|_Sigma|^2 from Q_Sigma(v):
Q_Sigma(v) - v^2 |dPhi|_Sigma|^2
- c |grad_Sigma v - alpha v grad_Sigma u|^2
= v^2 [m(h^2 - b^2) - |dPhi|_Sigma|^2
- (m-2)|grad_Sigma u|^2]
+ 4 v .
Integrate over Sigma. The v term is an integral of
2 ; since v is compactly
supported, integration by parts gives
int_Sigma 4 v dsigma
= -2 int_Sigma v^2 Delta_Sigma u dsigma.
So
Q_Sigma(v) - |dPhi|^2 term - gradient-square term
= int_Sigma v^2 [ m(h^2 - b^2) - |dPhi|_Sigma|^2
- (m-2)|grad_Sigma u|^2
- 2 Delta_Sigma u ] dsigma.
By (4.3), the bracket is at least
m f^2 + m(h + beta)^2 - |dPhi|_Sigma|^2.
The second term is nonnegative, and Lemma 2 gives
|dPhi|_Sigma|^2 <= m f^2 (sum of m terms, each of norm at most
f^2). So the bracket is >= 0. Therefore Q_Sigma(v) >= integral
of v^2 |dPhi|^2 + c |grad v - alpha v grad u|^2, as claimed.
Approximation extends the estimate to H^1_0(U). End of proof.
Lemma 5 (local Sobolev threshold). Fix U comp contained in
the interior of Sigma and eps > 0. There is C = C(U, eps) <
infinity such that for every v in H^1_0(U),
(Y_m - eps) ( int_Sigma |v|^q dsigma )^(2/q)
<= c int_Sigma |grad_Sigma v|^2 dsigma
+ C int_Sigma |v|^2 dsigma.
Also H^1_0(U) -> L^2(U) embeds compactly.
Proof. Aubin-Talenti, chart by chart. In a C^{1,1} chart of
small radius around a point of U the induced metric is
Lipschitz and a small perturbation of the Euclidean metric:
choosing the chart small enough, (1 - delta) I <= G <=
(1 + delta) I with delta small. For phi compactly supported in
the chart, substituting into Aubin-Talenti and using
dsigma = sqrt(det G) dx gives
c int_Sigma |grad_Sigma phi|^2 dsigma
>= Y_m ((1-delta)/(1+delta))^(m/2)
( int_Sigma |phi|^q dsigma )^(2/q).
(Effect of the metric distortion: gradient norm is at least
(1+delta)^(-1) times Euclidean, area element at least
(1-delta)^(m/2) dx, L^q norm at most (1+delta)^(m/(2q)) times
Euclidean, and 2(m/(2q) + 1/2) = m/q + 1 = (m-2)/2 + 1 = m/2.
Taking the ratio gives the constant above.) Choose delta small
so the factor is >= (Y_m - eps)/Y_m.
Partition of unity. Cover Ubar by finitely many such charts
and choose nonnegative Lipschitz chi_i with chi_i supported
in the i-th chart and sum_i chi_i = 1 on Ubar. Because q >= 2,
int_Sigma |v|^q dsigma
= int_Sigma ( sum_i chi_i^2 |v|^2 )^(q/2) dsigma
<= sum_i int_Sigma |chi_i v|^q dsigma.
By subadditivity of t |-> t^(2/q) (valid for 2/q <= 1),
( int_Sigma |v|^q )^(2/q)
<= sum_i ( int_Sigma |chi_i v|^q )^(2/q).
Apply the chart-local inequality to each chi_i v and sum. The
product rule and sum_i chi_i = 1 give
sum_i |grad_Sigma(chi_i v)|^2
= sum_i |chi_i grad_Sigma v + v grad_Sigma chi_i|^2
= |grad_Sigma v|^2 + v^2 sum_i |grad_Sigma chi_i|^2,
because the cross terms sum to zero (they are 2 v = 0). Therefore
(Y_m - eps) ( int_Sigma |v|^q )^(2/q)
<= c int_Sigma |grad_Sigma v|^2 dsigma
+ c sup_x sum_i |grad_Sigma chi_i(x)|^2
* int_Sigma |v|^2 dsigma.
Set C = c sup_x sum_i |grad chi_i|^2.
Compactness. Let v_j be bounded in H^1_0(U). Extend by zero.
In each chart chi_i v_j is a bounded sequence in H^1(R^m)
supported in a fixed compact set, so by Rellich a subsequence
converges in L^2 in the chart. A diagonal subsequence gives
L^2(U) convergence of v_j. End of proof.
Lemma 6 (attainment below threshold). Let U be an open subset
compactly contained in Sigma^circ, and
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. Take a minimizing sequence v_j. Because Q_Sigma(v_j) is
bounded and Q_Sigma(v_j) >= c ||grad v_j||_2^2 - m b^2
||v_j||_2^2, and because Holder on the bounded set U bounds
||v_j||_2 by sigma(U)^((m-2)/(2m)) ||v_j||_q = sigma(U)^((m-2)/(2m)),
the sequence v_j is bounded in H^1_0(U). A subsequence converges
weakly in H^1_0(U) to some psi. By the compactness part of Lemma
4.3, v_j -> psi strongly in L^2(U) and almost everywhere.
Put t = int |psi|^q dsigma in [0, 1], and z_j = v_j - psi. The
Brezis-Lieb lemma (||f_j||_q^q - ||f_j - f||_q^q -> ||f||_q^q when
f_j -> f almost everywhere with uniformly bounded ||f_j||_q) gives
||z_j||_q^q -> 1 - t.
Weak convergence splits the quadratic form:
Q_Sigma(v_j) = Q_Sigma(psi) + Q_Sigma(z_j) + o(1).
Apply Lemma 5 to z_j:
Q_Sigma(z_j) >= (Y_m - eps) ||z_j||_q^2 - C ||z_j||_2^2.
The L^2 term vanishes because v_j -> psi in L^2. Passing to the
limit j -> infinity and then eps -> 0,
Y = lim Q_Sigma(v_j)
>= Q_Sigma(psi) + Y_m (1 - t)^(2/q)
>= Y t^(2/q) + Y_m (1 - t)^(2/q),
using Q_Sigma(psi) >= Y ||psi||_q^2 and ||psi||_q^2 = t^(2/q).
If t = 0, this gives Y >= Y_m, contradicting Y < Y_m.
If 0 < t < 1, then Y_m > Y gives
Y >= Y t^(2/q) + Y (1 - t)^(2/q)
= Y (t^(2/q) + (1 - t)^(2/q)).
Because 2/q is strictly between 0 and 1, s |-> s^(2/q) is strictly
concave, so t^(2/q) + (1 - t)^(2/q) > 1 for 0 < t < 1, giving
Y > Y, contradiction.
The only remaining possibility is t = 1, which means ||psi||_q =
1 and Q_Sigma(psi) = Y. The infimum is attained. End of proof.
Lemma 7 (balanced variations). Let pi be a normalized
minimizer of Q_Sigma on H^1_0(U) with Y(U) < Y_m. Let
Phi_{p,a}, f_{p,a}, u = log f_{p,a} be the comparison map of
Lemma 2, with the balancing of Lemma 3 applied to the
measure pi^q dsigma. Then
grad_Sigma pi = alpha pi grad_Sigma u dsigma-a.e. on U,
where alpha = 2/c = (m-2)/2.
Proof. First variation. For every bounded Lipschitz zeta
supported in U, differentiate s |-> Q_Sigma(pi (1 + s zeta)) /
||pi (1 + s zeta)||_q^2 at s = 0. For small |s| the factor
1 + s zeta stays positive, and the differentiated integrands
are dominated by a constant times |pi|^q dsigma. The derivative
vanishes because s = 0 is a minimum. This gives
B_Q(pi, pi zeta) = Y int |pi|^q zeta dsigma, (4.8)
where B_Q is the symmetric bilinear form of Q_Sigma.
Apply (4.1) with zeta = w_j. The balancing of Lemma 3 gives
int |pi|^q w_j dsigma = 0, so
B_Q(pi, pi w_j) = 0 for j = 1, ..., n+1. (4.9)
Second variation. Fix one coordinate w = w_j and let B_j =
int |pi|^q w^2 dsigma. Since w has zero |pi|^q-mean, expanding
(1 + s w)^q = 1 + q s w + (q(q-1)/2) s^2 w^2 + O(s^3) and
raising to power 2/q gives
||pi (1 + s w)||_q^2 = 1 + (q-1) B_j s^2 + O(s^3). (4.10)
Expand the numerator using (4.2):
Q_Sigma(pi (1 + s w)) = Y + s^2 Q_Sigma(pi w) + O(s^3). (4.4)
Divide (4.4) by (4.3). The quotient is at least Y because Y is
the infimum; comparing coefficients of s^2 gives
Q_Sigma(pi w) >= (q-1) Y B_j. (4.5)
Product rule. The gradient of pi w is w grad pi + pi grad w.
Multiplying out and integrating,
Q_Sigma(pi w)
= B_Q(pi, pi w^2) + c int |pi|^2 |grad w|^2 dsigma.
(4.6)
The first term is Y B_j by (4.1) with zeta = w^2. Combine
(4.5) and (4.6). Since c = q - 2,
c int |pi|^2 |grad w|^2 dsigma >= (q-2) Y B_j.
Sum over j = 1, ..., n+1:
c int |pi|^2 sum_j |grad w_j|^2 dsigma
>= (q-2) Y int |pi|^q sum_j w_j^2 dsigma = (q-2) Y,
using sum_j w_j^2 = 1 on the sphere. The sum
sum_j |grad_Sigma w_j|^2 equals |dPhi|_Sigma|^2 (the tangential
trace of the pullback metric), and c = q - 2, so
int |pi|^2 |dPhi|_Sigma|^2 dsigma >= Y. (4.7)
Conclusion. Combine (4.7) with the Square Estimate of Lemma 4:
Y = Q_Sigma(pi)
>= int |pi|^2 |dPhi|_Sigma|^2 dsigma
+ c int |grad pi - alpha pi grad u|^2 dsigma
>= Y + c int |grad pi - alpha pi grad u|^2 dsigma.
So the last integral is zero, and grad pi = alpha pi grad u
dsigma-a.e. on U. End of proof.
Contradiction step. Suppose Y(U) < Y_m and take pi as in
Lemma 7. Put F = pi e^(-alpha u). Then
grad F = e^(-alpha u) (grad pi - alpha pi grad u) = 0
dsigma-a.e. on U. Mollify F in coordinate balls; the local
Poincare inequality makes F constant on each ball, and
overlapping balls agree. So F is constant on each connected
component of U. But pi is compactly supported in the interior
of U, so F vanishes near dU. Hence F = 0, so pi = 0, in
contradiction with ||pi||_q = 1.
Therefore Y(U) >= Y_m for every open U compact contained in
Sigma^circ.
Reducing to a general v. Let v be a compactly supported
Lipschitz function on Sigma^circ.
If the support K of v is a proper compact subset of Sigma, pick
a point y in Sigma \ K and a small coordinate ball D around y
disjoint from K. Put U = Sigma \ Dbar. Then U is open, compactly
contained in Sigma^circ, and contains K. So
Q_Sigma(v) >= Y(U) ||v||_q^2 >= Y_m ||v||_q^2.
If Sigma is compact and v does not vanish anywhere, use a
cutoff. Fix a small coordinate ball D and a smooth eta_delta
equal to 0 on the concentric ball D_delta and 1 outside D_{2
delta}, with |grad eta_delta| <= C/delta. The product eta_delta
v is supported in Sigma \ D_delta, which is compactly contained
in Sigma^circ. So Q_Sigma(eta_delta v) >= Y_m ||eta_delta v||_q^2.
Take delta -> 0. The L^q norms converge: eta_delta v -> v in L^q
by dominated convergence. The quadratic form
int |grad(eta_delta v)|^2 dsigma
= int eta_delta^2 |grad v|^2 dsigma
+ 2 int eta_delta v dsigma
+ int v^2 |grad eta_delta|^2 dsigma
has the last term bounded by ||v||_infty^2 C^2 delta^(-2)
sigma(D_{2 delta}), at most C' delta^(m-2), and tends to zero
because m >= 3. The middle term vanishes by dominated convergence
(|grad eta_delta| -> 0 a.e.). The potential term converges by
dominated convergence. So the inequality passes to the limit.
Disconnected Sigma. A compactly supported v meets only finitely
many connected components. Apply the connected result to each
and use subadditivity of t |-> t^(2/q):
Q_Sigma(v) >= Y_m sum_j a_j^(2/q)
>= Y_m (sum_j a_j)^(2/q),
with a_j = int_{Sigma_j} |v|^q dsigma, since 2/q <= 1.
Extending to H^1_0(U). Let v in H^1_0(U). By definition v is
the H^1 limit of compactly supported Lipschitz v_j. Lemma 5
gives v_j -> v in L^q(U); boundedness of h^2 - b^2 and of the
gradient on the compact set Ubar gives Q_Sigma(v_j) ->
Q_Sigma(v). So the inequality passes to the limit.
End of proof of Theorem 3.
Fix an open geodesic ball B = B_{R_B}(o) of finite radius in M.
For each v in (0, |B|) put
I(v) = inf { P_g(E) : |E \ B| = 0, |E| = v }.
The infimum runs over measurable sets E contained in B up to a
null set, of volume v; P_g is full ambient perimeter, including
the part on dB. Full perimeter is what can be controlled without
extra regularity near dB, and it is the natural quantity for a
minimizer that touches dB: pushing the boundary off dB can
change the ambient perimeter unpredictably, but the contact
area counts in full.
Existence. By BV compactness on the compact set Bbar and L^1
continuity of the volume functional, the infimum is attained
for every v in (0, |B|).
Regularity of minimizers (cited input). For each v in (0, |B|)
there is a confined minimizer E whose perimeter support
Gamma = supp |D chi_E| is compact, contained in Bbar, and
decomposes as a disjoint union Gamma = Sigma U Z with:
Sigma: relatively open, boundaryless, embedded, locally
C^{1,1} hypersurface of dimension m = n - 1;
Z: compact singular set with dim_H Z <= n - 8 (empty
when n < 8);
Gamma is C^{1,1} in a collar of dB;
Z is disjoint from a collar of dB;
and the perimeter measure is the induced area on Sigma:
|D chi_E|(A) = sigma(A cap Sigma) for every Borel A in M.
There is a unit normal N on Sigma, outward from E. Moreover
P_g(E; B) > 0 for every 0 < v < |B|: otherwise chi_E would be
constant on the connected ball B, contradicting v in (0, |B|).
Every minimizer thus has a nonempty free smooth patch.
These facts are cited. Kleiner's Theorem 2 gives the boundary
regularity in the confined problem. Ghomi and Spruck supply
the obstacle regularity near dB (their Lemma 7.2(i),(ii), no
total-curvature assumption). Morgan supplies interior
regularity away from dB. None of these inputs uses any
curvature bound beyond Sec_g <= 0. The Kleiner Lemma 7.2(i),
(ii) is the same paper used in dimension three later; it is not
re-derive it here.
Lemma 8. Let v in (0, |B|) be a differentiability point of
I. Let E be a confined minimizer of volume v, let h be its
outward normalized mean curvature on Sigma, and put
h_0 = I'(v)/m. Then
(a) I'(v) > 0;
(b) 0 <= h <= h_0 Sigma-almost everywhere;
(c) h = h_0 on the free part Sigma cap B.
Proof. Free variations. Let Z be a smooth ambient field,
compactly supported in the interior of B, and let Psi^t denote
its flow. For small |t| the sets Psi^t(E) remain in B. Put
P(t) = P_g(Psi^t(E)) and V(t) = |Psi^t(E)|. The volume
Jacobian formula, with the divergence theorem on E (whose
distributional boundary is Sigma), gives
V'(0) = int_E div_g Z dvol_g
= int_{Sigma cap B} dsigma.
The perimeter first variation over the free part reads
P'(0) = int_{Sigma cap B} m h dsigma.
The tangential part of Z contributes a surface divergence, which
integrates to zero on the compact free patch. The contact part
contributes nothing because Z vanishes on dB (it is supported
in the interior of B).
Minimality gives P(t) >= I(V(t)) for small t of both signs,
with equality at t = 0 because P(0) = I(v) = I(V(0)). So
t |-> P(t) - I(V(t)) has a two-sided minimum at t = 0. It is
differentiable there (P and V are smooth in t, I is
differentiable at v by hypothesis), so its derivative vanishes:
P'(0) - I'(v) V'(0) = 0.
Substituting the two first-variation formulas,
int_{Sigma cap B} (m h - I'(v)) dsigma = 0.
This holds for every smooth Z supported in the interior of B.
Writing phi = on the free part, phi is an arbitrary
smooth function on Sigma cap B (extend phi N by a smooth
cutoff to a compactly supported field on the interior of B).
Varying phi gives m h = I'(v) on the free part. This is (c).
Sign of I'(v). Put F(x) = d(o, x)^2 / 2, and let Z = -grad F,
with flow Psi^t(x) = exp_o(e^{-t} exp_o^{-1}(x)). For t >= 0
this flow sends B into itself. Corollary 1 gives Hess F >= g,
hence div_g grad F >= n, so
V'(0) = -int_E Delta F dvol_g <= -n |E| < 0.
For the perimeter derivative, the tangential trace on Sigma
satisfies tr_{T_Sigma} Hess F >= m (trace of an operator that
is >= identity over an m-dimensional space), so
P'(0) = -int_Sigma tr_{T_Sigma} Hess F dsigma
<= -m P_g(E) < 0.
Both P'(0) and V'(0) are strictly negative. From P'(0) >=
I'(v) V'(0) and V'(0) <= 0 (with equality at sign unless the
Hessian bound is sharp), dividing by the negative number V'(0)
flips the inequality:
I'(v) >= P'(0)/V'(0) > 0.
This is (a).
Contact bound. Let C = Sigma cap dB be the contact set, and
N_dB the outward normal of B on dB. Choose a smooth field Z
pointing inward at dB: <= 0 on dB. For t >= 0 the
flow of Z sends B into itself, so the sets Psi^t(E) remain
admissible competitors of volume V(t). Along C the outward
normal N_E of E equals -N_dB, so >= 0 on C.
Differentiate P(t) >= I(V(t)) at t = 0. The free part
contributes the same (m h - I'(v)) integrand, which
vanishes on the free part by (c). Only the contact part
survives:
int_C (m h - I'(v)) dsigma >= 0.
Vary Z on C with <= 0 (and Z compactly supported in a
small collar of C, extended by cutoff to a smooth field). If
m h - I'(v) does not have sign <= 0 almost everywhere on C,
Z can be chosen to make the integral negative, a contradiction.
So m h - I'(v) <= 0 on C, i.e. h <= h_0 on C.
Opposite bound. Let Z be smooth, supported near a point of C,
with > 0 there (pushing the boundary into B on a
small free patch disjoint from the free part of Sigma). For
t >= 0, minimality of E at volume v gives P(t) >= P(0), so
P'(0) = int_C m h dsigma >= 0.
Since > 0, this forces h >= 0 at that point. As the
point was arbitrary on C, h >= 0 on C.
Combining. On the free part h = h_0 > 0 by (a) and (c). On C
0 <= h <= h_0. So 0 <= h <= h_0 everywhere on Sigma.
This is (b). End of proof.
Local control of perimeter near the singular set requires two
elementary estimates.
Lemma 9 (BV deletion). Let E be a set of finite volume and
perimeter in a complete smooth Riemannian manifold M. Let
x in M and 0 < r < inj(x). Then
P_g(E \ B_r(x))
<= P_g(E; M \ Bbar_r(x)) + H^{n-1}(dB_r(x)),
where H^{n-1}(dB_r(x)) is the (n-1)-dimensional area of the
sphere dB_r(x). For all but countably many r in any bounded
interval, P_g(E; dB_r(x)) = 0, so the perimeters of E on the two
open sides add to P_g(E).
Proof. Write chi_{B_r} for the indicator of the open ball B_r =
B_r(x). Since E \ B_r = E cap (M \ B_r), the product rule for
BV functions gives
D chi_{E \ B_r}
= (1 - chi_{B_r}) D chi_E - chi_E D chi_{B_r}.
Taking total variations and using subadditivity,
|D chi_{E \ B_r}|(M)
<= |(1 - chi_{B_r}) D chi_E|(M) + |chi_E D chi_{B_r}|(M).
The first term is |D chi_E|(M \ Bbar_r) because 1 - chi_{B_r}
vanishes on B_r. The second term is set-valued: D chi_{B_r} is
the surface measure on dB_r, oriented outward. Its total
variation is at most H^{n-1}(dB_r).
Summing,
P_g(E \ B_r(x)) <= P_g(E; M \ Bbar_r(x)) + H^{n-1}(dB_r(x)),
as claimed.
For the countable-good-radii statement: the function
R |-> P_g(E \ B_R(x)) is nonincreasing and bounded below by 0,
so it has at most countably many discontinuities. At every
continuity point R, P_g(E; dB_R(x)) = 0. End of proof.
Lemma 10 (area growth). Fix v in (0, |B|) and let E be a
confined minimizing set of volume v. Then
there are C < infinity and r_0 > 0 such that
sigma(Sigma cap B_r(x)) <= C r^m
for every x in the singular set Z and every 0 < r < r_0.
Proof. If Z is empty there is nothing to prove. Otherwise pick
a free smooth boundary point y of E in B (existence by
the confined minimization setup: P_g(E; B) > 0 implies a nonempty free patch), and
a small neighborhood W of y with Wbar contained in B \ Z. Let
Z_field be a nonnegative normal variation of a small patch of
Sigma in W, extended smoothly to a field with compact support in
W. Set
a = int_{Sigma cap W} dsigma > 0.
Let Psi^t be the flow of Z_field. Put G(t) = |Psi^t(E)| - |E|;
G(0) = 0 and, by the volume Jacobian formula, G'(0) = a. Choose
t_0 > 0 so that G'(t) >= a/2 on [0, t_0]. The flow preserves W
and its complement, and any finite-perimeter set F that agrees
with E almost everywhere on W has
|Psi^t(F)| - |F| = G(t) for 0 <= t <= t_0,
|P_g(Psi^t(F)) - P_g(F)| <= C_R t P_g(E; W) for 0 <= t <= t_0,
where C_R is the uniform Lipschitz constant of the tangential
Jacobian of the flow over all tangent m-planes in the compact
support of Z_field. Consequence: every volume deficit
0 <= delta <= a t_0 / 2 can be repaired in flow time t <=
2 delta / a, at perimeter cost at most C_R delta (with C_R
redefined to absorb the constants).
Assume Z is nonempty. Its dimension bound places it in the
interior of B, disjoint from the collar of dB on which Gamma is
C^{1,1}. Decrease rho_*>0 so that B_{2 rho_*}(x) lies in
B \ Wbar for every x in Z. Because the ambient metric is smooth
on the compact set Z and on the balls B_{2 rho_*}(x), there are
constants C_V, C_A with
vol(B_r(x)) <= C_V r^n, H^{n-1}(dB_r(x)) <= C_A r^m
for x in Z and 0 < r < rho_*. Decrease rho_* further so that
C_V rho_*^n <= a t_0 / 2.
Fix x in Z. For almost every r in (0, rho_*) the sphere
dB_r(x) carries no perimeter of E (Lemma 9). At such r, put
F = E \ B_r(x).
Then F agrees with E on M \ B_r(x) superset W. Lemma 9 gives
P_g(F) <= P_g(E; M \ Bbar_r(x)) + C_A r^m.
The volume deficit is
delta = |E \ F| = |E cap B_r(x)| <= vol(B_r(x)) <= C_V r^n.
Repair by the flow G: find F' of volume v, in B, with
P_g(F') <= P_g(F) + C_R delta
<= P_g(E; M \ Bbar_r(x)) + C_A r^m + C_R C_V r^n.
Minimality of E at volume v gives P_g(E) <= P_g(F'). Since
P_g(E) = P_g(E; B_r(x)) + P_g(E; M \ Bbar_r(x)) and the second
term cancels, this gives
P_g(E; B_r(x)) <= C_A r^m + C_R C_V r^n <= C_0 r^m,
with C_0 = C_A + C_R C_V rho_* (using r <= rho_*).
Perimeter-area identity. In the interior of B, the perimeter
measure of E is the induced area on Sigma:
P_g(E; B_r(x)) = sigma(Sigma cap B_r(x)).
For an arbitrary 0 < r < rho_*/2, choose a good radius rho in
(r, 2r) and use monotonicity of the perimeter measure:
sigma(Sigma cap B_r(x)) <= sigma(Sigma cap B_rho(x))
<= 2^m C_0 r^m.
So C = 2^m C_0 and r_0 = rho_*/2 work. End of proof.
The singular set has vanishing (m-2)-capacity, so it can be
removed from any Sobolev test at small Dirichlet cost.
Lemma 11 (capacity cutoffs). Let Gamma be a compact subset of a
smooth Riemannian manifold, and let S subset Gamma be closed.
Suppose 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, 0 < r < r_0.
Then there exist 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. Assume S nonempty (otherwise eta_j = 1). For each j
construct 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}.
Hausdorff nullity of S in dimension m-2 gives a countable open
cover by balls with sum of (m-2)-radii arbitrarily small.
Recentering at points of S with a fixed enlargement preserves
the sum up to a constant. Compactness of S gives a finite
subcover without increasing the radii. This yields the cover.
Let 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_i eta_{ji}(x)|_Sigma.
Each 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
(overlaps only help: the min is bounded above by each term).
Applying area growth at the doubled radii,
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 of the cover form a neighborhood of S on which
eta_j vanishes; its support is therefore contained in a
compact subset of Gamma \ S, hence compact in Sigma (embedding
charts identify subspace and manifold topologies). For each
fixed x in Sigma, closedness of S gives d(x, S) > 0; as radial
sizes tend to zero, the doubled balls eventually miss x, so
eta_j(x) -> 1. End of proof.
Everything now assembles into a single inequality for each
confined minimizer.
Lemma 12 (constant test). Let v in (0, |B|) be a
differentiability point of I, and let E be a confined
minimizer at volume v with perimeter support Sigma U Z. Then
m int_Sigma (h^2 - b^2) dsigma >= Y_m I(v)^((m-2)/m).
Proof. Lemma 8 gives 0 <= h <= h_0 = I'(v)/m < infinity
almost everywhere on Sigma. If Z is nonempty, its dimension
bound dim_H Z <= n - 8 = m - 7 implies H^{m-2}(Z) = 0 for
m >= 3, and area growth (Lemma 10) supplies the doubling
estimate of Lemma 11. That lemma gives cutoffs eta_j; if Z is
empty take eta_j = 1.
Apply the Hypersurface Sobolev theorem (Theorem 3) to
eta_j on Sigma with the curvature parameter b:
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) = sigma(Sigma) is finite. On Sigma,
|h^2 - b^2| <= h_0^2 + b^2, a bounded function. As j -> infinity,
eta_j -> 1 pointwise by Lemma 11, and dominated convergence
applied to (h^2 - b^2) eta_j^2 gives
m int_Sigma (h^2 - b^2) eta_j^2 dsigma
-> m int_Sigma (h^2 - b^2) dsigma.
Similarly int_Sigma eta_j^q dsigma -> int_Sigma 1 dsigma = I(v),
and int_Sigma |grad_Sigma eta_j|^2 dsigma -> 0 by Lemma 11.
Taking the limit j -> infinity with 2/q = (m-2)/m,
m int_Sigma (h^2 - b^2) dsigma >= Y_m I(v)^((m-2)/m).
This covers every required dimension: when 4 <= n <= 7 the
singular set is empty (dim_H Z <= n - 8 < 0), and when n = 8 it
is at most zero-dimensional and Lemma 11's coverings still
apply. Dimensions n = 2 and n = 3 use separate arguments
(dimensions two and three below). End of proof.
Proposition 1. For almost every v in (0, |B|),
I'(v) >= m sqrt( b^2 + (s_m / I(v))^(2/m) ).
Proof. The profile I is locally Lipschitz on (0, |B|): for
nearby volumes v, w pick minimizers and use the volume-repair
flow of Lemma 10 to convert one to the other at O(|v - w|)
perimeter cost. Being Lipschitz, I is differentiable almost
everywhere. At a differentiability volume v, choose a minimizer
E and invoke Lemma 8 (h_0 = I'(v)/m), then Lemma 12:
m (h_0^2 - b^2) I(v) >= Y_m I(v)^((m-2)/m).
Both I(v) > 0 and h_0 > 0 (by Lemma 8(a)), so dividing by
m I(v):
h_0^2 - b^2 >= (Y_m/m) I(v)^((m-2)/m) = s_m^(2/m) I(v)^(-2/m).
(Here Y_m = m s_m^(2/m).) Therefore
h_0^2 >= b^2 + (s_m / I(v))^(2/m),
and taking the positive square root and using h_0 = I'(v)/m,
I'(v) >= m sqrt( b^2 + (s_m/I(v))^(2/m) ).
End of proof.
Define, for A > 0,
W_b(A) = int_0^A da / (m sqrt(b^2 + (s_m/a)^(2/m))).
W_b is smooth and strictly increasing on (0, infinity), with
W_b(A) -> 0 as A -> 0. Its derivative is
W_b'(A) = 1 / (m sqrt(b^2 + (s_m/A)^(2/m))) > 0.
The model identity. Substitute a = A_b(s), the model sphere
area function:
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, the square root becomes
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),
using cs_b(s)^2 - b^2 sn_b(s)^2 = 1. Also
A_b'(s) = m s_m sn_b(s)^(m-1) cs_b(s). The integrand therefore
simplifies to
A_b'(s) / (m cs_b(s)/sn_b(s)) = s_m sn_b(s)^m = V_b'(s).
So W_b(A_b(r)) = V_b(r) - V_b(0) = V_b(r).
The integrated profile inequality. Multiply Proposition 1 by
the positive factor 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.
So (W_b o I)'(v) >= 1 almost everywhere on (0, |B|). Since I is
locally Lipschitz and W_b is C^1 on its range, W_b o I is
absolutely continuous on compact subintervals of (0, |B|), and
the fundamental theorem of calculus gives, for 0 < v_0 < v < |B|,
W_b(I(v)) - W_b(I(v_0)) >= v - v_0.
Take v_0 -> 0+. The inequality I(v_0) -> 0 for v_0 -> 0 (a
small geodesic ball of volume v_0 has perimeter of order
v_0^((m-1)/m) -> 0, so the infimum does too). So W_b(I(v_0)) ->
W_b(0) = 0, and
W_b(I(v)) >= v for every 0 < v < |B|.
Conclusion in dimension n >= 4. Let E subset M be bounded, of
finite volume V = |E| and finite perimeter A = P_g(E). Choose
an open geodesic ball B with E subset B; then V < |B|. Let r > 0
solve V_b(r) = V, so r = V_b^(-1)(V). From
W_b(I(V)) >= V = V_b(r) = W_b(A_b(r)) and strict monotonicity
of W_b,
I(V) >= A_b(r) = A_b(V_b^(-1)(V)) = I_b(V).
Minimality of E in the confined problem (E is admissible at
volume V because E subset B) gives P_g(E) >= I(V). Therefore
P_g(E) >= I_b(V) = I_b(vol_g(E)).
This proves Theorem 1 for bounded sets of finite perimeter in
every dimension n >= 4 and for every kappa <= 0.
Dimensions n = 2 and n = 3, and the unbounded-set case, are
handled in the next two sections.
The Hypersurface Sobolev theorem requires m = n - 1 >= 3, i.e.
n >= 4. Dimensions two and three use their own inputs, but the
logic is identical: prove the comparison for smooth domains,
then pass to finite-perimeter sets.
Smooth approximation. Every bounded finite-perimeter set E in a
Riemannian manifold can be approximated in L^1 by bounded smooth
open sets E_j with |E_j Delta E| -> 0, |E_j| -> |E|, and
P(E_j) -> P(E). This is the standard strict BV approximation
(Guneysu-Pallara, Theorem 2.12). Consequently, if for every
bounded smooth domain D of volume V and boundary area A one has
A >= I_b(V), then the same holds for every bounded
finite-perimeter E, by taking a strict approximation and using
continuity of I_b.
Let (M^2, g) be a Cartan-Hadamard surface with Gaussian
curvature K <= kappa <= 0. The classical Weil-Bol inequality, in
the smooth-disk formulation of Chavel and Feldman, states that a
smooth disk D of area V and boundary length L under K <= kappa
satisfies
L^2 >= 4 pi V - kappa V^2. (7.1)
Direct evaluation. In dimension two, the model functions are
V_b(r) = 2 pi integral_0^r sn_b(t) dt and A_b(r) = 2 pi
sn_b(r). Solving V_b(r) = V for r and substituting into A_b(r)
gives the classical formula I_b(V) = sqrt(4 pi V - kappa V^2),
matching the right side of (7.1). So for smooth disks, (7.1) and
L >= I_b(V) are the same inequality.
Reduction to smooth domains. Let D be a smooth bounded domain,
with connected components D_1, ..., D_k. In each component,
fill the holes (replace the included region by the disk bounded
by the outer boundary). This increases area and decreases
boundary length. The resulting smooth disks Delta_i satisfy
sum_i |Delta_i| >= |D|, sum_i length(dDelta_i) <= length(dD).
Apply (7.1) to each disk and use subadditivity of s(V) =
sqrt(4 pi V - kappa V^2) in V (shown below):
length(dD)^2 >= sum_i length(dDelta_i)^2
>= sum_i (4 pi |Delta_i| - kappa |Delta_i|^2)
>= 4 pi |D| - kappa |D|^2.
Taking the positive square root, length(dD) >= I_b(|D|).
Subadditivity of s. For V_1, V_2 >= 0 one has
s(V_1 + V_2)^2 - s(V_1)^2 - s(V_2)^2 = -2 kappa V_1 V_2 >= 0,
since kappa <= 0. Squaring again (both sides nonnegative),
s(V_1 + V_2)^2 <= (s(V_1) + s(V_2))^2,
because the cross term 2 s(V_1) s(V_2) is >= 0. So
s(V_1 + V_2) <= s(V_1) + s(V_2). By induction this extends to
any finite number of summands.
Conclusion in dimension two. For every bounded smooth domain D
and then, by strict approximation, for every bounded
finite-perimeter E of finite volume, P_g(E) >= I_b(vol_g(E)).
Let (M^3, g) be a Cartan-Hadamard three-manifold with sectional
curvature at most kappa. Kleiner's Theorem 2 states that every
smooth compact domain D subset M of volume V and boundary area A
satisfies
A >= I_b(V),
where b = sqrt(-kappa). Kleiner's theorem is stated for
arbitrary kappa <= 0 and every complete simply connected
three-manifold. It is proved by a separate, dimension-specific
argument that is not reproduced.
Passing to finite-perimeter sets. Let E be bounded with finite
volume V and finite perimeter A. Take smooth E_j with
|E_j Delta E| -> 0, |E_j| -> V, P(E_j) -> A. Then
A = lim P(E_j) >= lim I_b(|E_j|) = I_b(V).
So the comparison holds for every bounded finite-perimeter E in
dimension three.
Combining Sections 6, 7.1, 7.2, the isoperimetric comparison
P_g(E) >= I_b(vol_g(E))
holds for every bounded measurable E of finite volume and finite
perimeter, in every dimension n >= 2 and for every kappa <= 0.
Sections 6 and 7 give P_g(E) >= I_b(vol_g(E)) for every bounded
finite-perimeter E, in every dimension n >= 2. Removing the
boundedness assumption needs one cutoff and one BV coarea
identity. Neither uses any curvature hypothesis, so the estimate
is stated once and reused below for any increasing continuous
perimeter lower bound.
Lemma 13 (Passage to finite volume). Let M be a complete
connected smooth Riemannian manifold without boundary. Let
J: [0, infinity) -> [0, infinity) be continuous and
nondecreasing, with J(0) = 0. Suppose P_g(F) >= J(|F|) for
every bounded finite-perimeter F subset M. Then the same holds
for every measurable E with finite volume and finite ambient
perimeter.
Proof. Two BV identities. For compactly supported Lipschitz phi
and bounded u in BV(M), the product rule reads
D(phi u) = phi Du + u grad phi dvol_g. (8.1)
It follows from the distributional rule for smooth multipliers
by mollifying phi locally in charts: uniform convergence handles
the measure term, L^1 convergence handles the gradient, and
boundedness of u handles the second term. For 0 <= u <= 1,
coarea gives
|Du|(M) = integral_0^1 P_g({u > t}) dt. (8.2)
The integrand is measurable because the defining perimeter
supremum can be taken over a countable dense family of
compactly supported test fields. Both identities extend from
smooth functions to BV by strict approximation.
Fix o in M and, for R > 0, define
phi_R(x) = min{1, max{0, R + 1 - d(o, x)}}.
Completeness makes the support compact. The function equals one
on B_R(o), vanishes outside B_{R+1}(o), and is 1-Lipschitz.
The product rule and coarea give
integral_0^1 P_g({phi_R chi_E > t}) dt
= |D(phi_R chi_E)|(M)
<= integral_M phi_R d|D chi_E|
+ integral_E |grad phi_R| dvol_g
<= P_g(E) + |E \ B_R(o)|. (8.3)
Almost every level 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 therefore bound the left
side of (8.3) below by J(v_R). Since v_R -> |E| and
|E \ B_R(o)| -> 0 as R -> infinity, continuity of J gives
P_g(E) >= J(|E|). End of proof.
Proof of Theorem 1 (main theorem). Null sets have zero perimeter
and I_b(0) = 0. The comparison for bounded sets of finite
perimeter and Lemma 13 applied with J = I_b prove the
assertion for all n >= 2 and every normalized b in {0, 1}.
For an arbitrary negative bound, put b = sqrt(-kappa) > 0 and
g~ = b^2 g. Then
Sec_{g~} = b^{-2} Sec_g <= -1,
|E|_{g~} = b^n |E|_g,
P_{g~}(E) = b^{n-1} P_g(E).
The last identity holds for ambient BV perimeter: a constant
metric rescaling leaves divergence unchanged, while
dvol_{g~} = b^n dvol_g and |X|_{g~} = b |X|_g. Under the
substitution Y = b X in the dual supremum, multiplication is
by b^{n-1}. The model functions have the matching relations
V_b(r) = b^{-n} V_1(b r),
A_b(r) = b^{-(n-1)} A_1(b r),
I_b(V) = b^{-(n-1)} I_1(b^n V).
Applying the b = 1 comparison to g~ and dividing by b^{n-1}
gives the claimed inequality for g and kappa. The case kappa = 0
was proved directly, without a curvature limit. End of proof.
Turn to the equality case (Theorem 2). Let (M^n, g) be a
complete simply connected smooth Riemannian manifold with
Sec_g <= 0 and n >= 2. Let E subset M be bounded measurable,
with V = |E| in (0, infinity) and finite perimeter A, such that
A = n omega_n^(1/n) V^((n-1)/n). (9.1)
Put m = n - 1, R = (V/omega_n)^(1/n), and H_0 = m/R. The
plan has three stages. First, (9.1) makes E a local perimeter
almost-minimizer. Standard regularity then gives a
C^{1,alpha} regular boundary, a compact singular set of
Hausdorff dimension at most n - 8, and area growth near the
singular set. The boundary carries a constant outward trace
mean curvature H_0. Second, the equality Sobolev test and
radial flux identities below put the perimeter support
on a round sphere of radius R. Third, BV phase constancy and
metric comparison recover both the set and its induced metric,
which is Euclidean throughout.
The present section proves the first stage. This is the only
place where regularity of perimeter minimizers enters.
Proposition 2 (Regularity of the boundary). Under the
hypotheses of (9.1), the perimeter support
Gamma = supp |D chi_E| is compact and decomposes as
Gamma = Sigma U Z with:
Sigma is a relatively open, boundaryless, embedded
C^{1,1}_loc hypersurface;
Z is compact with Hausdorff dimension at most n - 8,
in particular Z is empty when n < 8;
the occupied phase of E defines an outward unit normal N
on Sigma; the perimeter measure equals induced area on
Sigma; sigma(Sigma) = s_m R^m; and the closure of Sigma is
Gamma;
there are C < infinity and r_0 > 0 such that
sigma(Sigma cap B_r(x)) <= C r^m for x in Gamma and
0 < r < r_0;
the outward trace mean curvature H = div_{T Sigma} N
equals H_0 almost everywhere on Sigma.
The assertions concern the perimeter support and the measurable
class of E, not the topological boundary of an arbitrary
representative.
Write theta = (n-1)/n and Lambda = A/V, and put z = |F|/V for
a bounded finite-perimeter competitor F. Theorem 1 gives
P(F) >= I_b(|F|) = A z^theta, so
P(E) - P(F) <= A (1 - z^theta). (9.4)
The map z -> z^theta is concave with theta < 1, so
1 - z^theta <= max{0, 1 - z}. Bounding the nonnegative branch
A (1 - z^theta) <= Lambda |E| max{0, 1 - |F|/|E|}
<= Lambda |E delta F|. (9.5)
The last inequality is ||E| - |F|| <= |E delta F|. Suppose
E delta F lies, up to null sets, in a compact subset of a
bounded open set O. Locality of the perimeter gives
P(E; M \ O) = P(F; M \ O), so subtracting the common part
yields
P(E; O) <= P(F; O) + Lambda |E delta F|. (9.6)
Only competitors with P(F; O) < infinity matter: agreement
with E near the complement of O then makes P(F) finite and
equal to P(F; O) + P(E; M \ bar O).
Local smooth-manifold regularity. Apply Corollary 1.6 of
Antonelli, Pasqualetto and Pozzetta to the localized
inequality (9.6). Define, on measurable subsets of O,
G(F) = Lambda |F delta (E cap O)|.
Then G(empty) < infinity and
|G(F_1) - G(F_2)| <= Lambda |F_1 delta F_2|, so the continuity
exponent required by their Condition (1.3) is 1 > 1 - 1/n.
Under (9.6), (E, O) is a (Lambda, G)-minimizer in the sense of
that paper. The remaining hypothesis P(E; O) > 0 is automatic
for a ball O centered on Gamma. Their theorem, which holds in
every n >= 2 with no lower Ricci bound, provides a
C^{1,alpha}_loc regular part of full perimeter, a relatively
closed singular part of Hausdorff dimension at most n - 8, and
local upper area growth.
Set Sigma to be the regular part, using the density-one
representative whose boundary equals the perimeter support.
Then Gamma = boundary of E is compact by boundedness of E and
completeness of M. Taking a finite cover of Gamma by the local
regularity neighborhoods makes the local area estimates
uniform: there are C < infinity and r_0 > 0 such that
sigma(Sigma cap B_r(x)) <= C r^m for x in Gamma,
0 < r < r_0. (9.7)
The support identity. Every neighborhood of a support point
meets Sigma, since the singular part has Hausdorff codimension
at least two. So closure(Sigma) = Gamma. Fix a regular point p;
locally the perimeter support is a graph dividing a coordinate
ball U into two connected sides. The distributional derivative
D chi_E vanishes on each side by mollification in smaller
balls (metrics coefficients are Lipschitz, and overlapping
balls agree). So chi_E is locally constant almost everywhere on
each side. The two side values cannot be equal: the graph has
zero ambient measure, and equality would make chi_E locally
constant on U, contradicting P(E; U) > 0. The occupied side
therefore defines a consistent outward unit normal N on Sigma.
The divergence formula on a graph identifies the perimeter
measure with induced dsigma in U, and the finitely many chart
pieces patch to
|D chi_E| = dsigma|Sigma on every Borel set. (9.8)
In particular sigma(Sigma) = A = s_m R^m. The perimeter
support now carries the perimeter measure. The graph equation identifies its mean curvature with the constant H_0.
Choose graph coordinates (y, z) in R^m x R so that Sigma is
locally z = u(y) and E lies below the graph. Let G(y, z) be
the ambient metric matrix in these coordinates and let
W(y, z) = sqrt(det G(y, z)). The graph area integrand is
L_g(y, z, p) = W(y, z)
[(-p, 1)^T G(y, z)^{-1} (-p, 1)]^{1/2}. (9.9)
For a smooth compactly supported zeta, the graph z = u(y) +
t zeta(y) gives admissible finite-perimeter competitors for
both signs of small t. The deficit
P(F) - n omega_n^(1/n) |F|^theta is nonnegative by Theorem 1
and vanishes at F = E. Its volume derivative at F = E has
coefficient
n omega_n^(1/n) * theta * V^{theta - 1} = m / R = H_0. (9.10)
Define
B(y, z, p) = d_z L_g(y, z, p) - H_0 W(y, z). (9.11)
Then the first variation reads
integral [ d_p L_g(y, u, Du) . D zeta
+ B(y, u, Du) zeta ] dy = 0 (9.12)
for every zeta in C_c^infty. The matrix d_pp L_g is positive
definite: its quadratic form is the Hessian of a positive
definite norm in directions transverse to the radial kernel of
(-p, 1), and any nonzero slope variation therefore contributes
a strictly positive second-order term. The equation (9.12) is
uniformly elliptic on every compact argument set.
Local graph regularity. Lemma 14 below upgrades the
C^{1,alpha}_loc solution u of (9.12) to C^{1,1}_loc, without
an antecedent second-derivative bound. Transversally, Sigma is
a C^{1,1}_loc graph. On a C^{1,1} graph the induced metric a_ij
is locally Lipschitz, so for any tangential field
X = X^i F_i in a parametrization F, almost everywhere and
weakly,
div_Sigma X = d_i X^i + (1/2) X^i a^{jk} d_i a_{jk}
= (1/sqrt(det a))
d_i (sqrt(det a) X^i). (9.13)
Compactly supported tangential Lipschitz fields therefore have
zero integral divergence. Splitting a general ambient field Y
into tangential and normal parts gives
div_{T Sigma} Y = div_Sigma(Y^T) + H , (9.14)
with H = div_{T Sigma} N.
Now take Y = zeta d_z in the graph equation. The area
derivative is integral div_{T Sigma} Y dsigma; the volume
derivative is the outward flux dsigma. Substituting
(9.14) into (9.12) gives
integral_Sigma (H - H_0) zeta dsigma = 0. (9.15)
The factor is strictly positive, because E occupies
the lower side. Equivalently its product with the graph area
density is W. Every smooth compactly supported zeta therefore
gives
H = H_0 almost everywhere on Sigma. (9.16)
The coefficient H_0 came from the same global deficit on every
patch, so (9.16) is the same signed outward trace curvature on
every component. Proposition 2 is now proved except that the
local graph regularity has only been cited. It is the content
of the next subsection.
Lemma 14 (Stationary graphs are C^{1,1}). Let Omega subset R^m
be open and let u in C^{1,alpha}_loc(Omega) with
0 < alpha < 1. Let G(y, z) be the metric matrix of a smooth
Riemannian metric on a neighborhood of the graph of u in
R^m x R, put 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_z L_g(y, z, p) - H_0 W(y, z).
If
integral_Omega [ d_p L_g(y, u, Du) . D zeta
+ B(y, u, Du) zeta ] dy = 0 (9.19)
for every zeta in C_c^infty(Omega), then u in C^{1,1}_loc(Omega).
Proof. The positivity of d_pp L_g was verified after (9.12).
Shrink the graph chart so that every argument of G, W and L_g
along the graph stays inside a single compact ellipticity
region. Fix a Euclidean ball B_{4 rho} compactly contained in
the parameter domain.
Difference quotients and energy. For a coordinate unit vector
e and 0 < delta < rho, set
v(y) = (u(y + delta e) - u(y)) / delta, y in B_{3 rho}.
Each v is C^1, though no uniform bound on Dv is yet known. The
identity
v(y) = integral_0^1 d_e u(y + s delta e) ds
does give a uniform C^{0,alpha} bound for v.
Subtract the translated equations and divide by delta. To
describe the coefficients explicitly, write
T_s(y) = (1 - s)(y, u(y), Du(y))
+ s (y + delta e, u(y + delta e), Du(y + delta e)).
The resulting weak equation is
integral [ (a Dv + f) . D zeta
+ (b . Dv + d) zeta ] dy = 0, (9.20)
where
a = integral_0^1 d_pp L_g(T_s) ds,
b = integral_0^1 d_p B(T_s) ds,
f = integral_0^1 (d_e d_p L_g(T_s)
+ d_z d_p L_g(T_s) v) ds,
d = integral_0^1 (d_e B(T_s)
+ d_z B(T_s) v) ds.
Here d_e differentiates only the explicit y argument. The known
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, together with the bound for v, makes a and f
uniformly C^{0,alpha} and b and d uniformly bounded. The
matrices a are symmetric and uniformly elliptic.
Test (9.20) with chi^2 v, where chi is supported in B_{3 rho}
and equals one on B_{2 rho}. Such tests are justified by smooth
approximation. Ellipticity produces a positive multiple of
integral chi^2 |Dv|^2 on the left. Every other term is bounded
by Young's inequality and the uniform bounds on v, f, b and d,
absorbing the linear terms in Dv into that positive multiple.
The first-order term b . Dv is absorbed this way. The remaining
bound is a constant times integral (chi^2 + |Dchi|^2), so
integral_{B_{2 rho}} |Dv|^2 dy <= C, (9.21)
uniformly in the difference step. No second derivative of u has
been used. The gradient bound comes next.
Harmonic 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_g(a(x) Dw) = 0, (9.22)
where a from (9.20) is frozen at the center x. Existence and
uniqueness follow by minimizing the positive quadratic
Dirichlet energy in v + H^1_0(B_r(x)). Put z = v - w.
Substituting (9.20) and testing against z, noting that the
constant vector f(x) integrates to zero against Dz, and using
ellipticity together with the coefficient bounds, gives
lambda ||Dz||_2^2
<= C r^alpha (||Dv||_2 + |B_r|^{1/2}) ||Dz||_2
+ C (||Dv||_2 + |B_r|^{1/2}) ||z||_2. (9.23)
Here all norms are over B_r(x); the first term is the
pointwise coefficient error a(y) - a(x) of size r^alpha, and
the second term is the first-order and zeroth-order part.
Zero-boundary Poincare gives ||z||_2 <= C r ||Dz||_2, and
r <= r^alpha for r <= 1. Dividing through,
(fint_{B_r(x)} |Dv - Dw|^2)^{1/2}
<= C r^alpha
(1 + (fint_{B_r(x)} |Dv|^2)^{1/2}). (9.24)
Energy minimization 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}, (9.25)
after subtracting the affine function p . y from both v and w.
Harmonic oscillation. For 0 < gamma <= 1/4 and any constant
vector p,
(fint_{B_{gamma r}(x)}
|Dw - (Dw)_{B_{gamma r}(x)}|^2)^{1/2}
<= C gamma
(fint_{B_r(x)} |Dw - p|^2)^{1/2}, (9.26)
where subscripts denote averages. To justify this at the weak
regularity at hand, mollify Dw - p in an interior ball. Its
components solve the same constant-coefficient equation. A
linear change of coordinates, with distortion controlled by
ellipticity, identifies them as harmonic functions. The mean
value formula reproduces each such function by convolution
with a smooth radial kernel of radius comparable to r.
Differentiating the kernel and applying Cauchy-Schwarz bounds
the gradient on B_{r/2}(x) by C r^{-1} times the L^2 mean of
|Dw - p|. Letting the mollification scale tend to zero
upgrades the estimate to every representative.
Iteration. Define
p_r = fint_{B_r(x)} Dv,
J(r) = (fint_{B_r(x)} |Dv - p_r|^2)^{1/2}.
Take p = p_r in (9.25) and (9.26), and compare averages across
the two balls. Then
J(gamma r)
<= C gamma J(r)
+ C gamma^{-m/2} r^alpha
(1 + |p_r| + J(r)), (9.27)
|p_{gamma r} - p_r| <= gamma^{-m/2} J(r). (9.28)
Fix gamma small enough that C gamma <= 1/2, and fix 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). Start from one
fixed radius r_* <= min{rho, 1}. Its starting average p_{r_*}
is bounded uniformly in delta and e by (9.21). Along the
geometric scales r_j = gamma^j r_*, the product of the factors
1 + C' r_j^alpha is finite because sum r_j^alpha < infinity.
So |p_{r_j}| stays bounded uniformly. Each fixed-step
difference quotient is C^1, hence p_{r_j} -> Dv(x) as
j -> infinity. Therefore
|Dv(x)| <= C'' uniformly in x in B_rho, e and delta.
Finally, Dv(y) = (Du(y + delta e) - Du(y)) / delta holds
classically. Uniform control of |Dv| bounds increments of Du
along every coordinate direction independently of the step
delta. On a smaller coordinate cube, join two points by
coordinate segments and add the increment bounds; the total
length is at most sqrt(m) times the Euclidean distance.
Therefore every first derivative of u has a locally uniform
modulus of continuity, and u is in C^{1,1}_loc(Omega). End of
proof.
The regular part Sigma of the perimeter support is not assumed
compact or complete. The singular set has zero (m-2)-capacity,
so the same cutoffs as in Lemma 11 extend both the first
variation formula and the hypersurface Sobolev inequality to
smooth tests supported on all of Gamma.
Lemma 15 (Cutoffs for an equality boundary). Under the
hypotheses of Proposition 2, there are eta_j in Lip_c(Sigma)
with 0 <= eta_j <= 1 everywhere on Sigma, eta_j(x) -> 1 for
every x in Sigma, and
integral_Sigma |grad_Sigma eta_j|^2 dsigma -> 0. (9.29)
Proof. If Z is empty, take eta_j = 1. Otherwise n >= 8,
m >= 7, and dim_H Z <= m - 7 < m - 2, so H^{m-2}(Z) = 0. The
compactness of Z, the finiteness of sigma(Sigma), the local
embedding, and the area growth (9.7) verify exactly the
hypotheses of Lemma 11 (capacity cutoffs). That lemma supplies
the eta_j. End of proof.
Global first variation. For every smooth ambient field Y
defined near Gamma, the cutoffs and (9.14) imply
integral_Sigma div_{T Sigma} Y dsigma
= H_0 integral_Sigma dsigma. (9.30)
Indeed, apply the compact-support identity to eta_j Y and
expand:
integral_Sigma eta_j div_{T Sigma} Y dsigma
+ integral_Sigma dsigma
= H_0 integral_Sigma eta_j dsigma.
The middle integral is bounded in absolute value by
||Y||_infty A^{1/2} ||grad_Sigma eta_j||_2, which tends to
zero. All other terms converge by dominated convergence, since
the ambient data are bounded near the compact set Gamma. The
same identity holds on any compact boundaryless component of
Sigma directly, by integrating (9.14) on that component.
Extended Sobolev test. When n >= 4, the same cutoffs extend the
hypersurface Sobolev inequality (Theorem 3) to every smooth
ambient function restricted to Sigma. The normalized mean
curvature is h = H/m = 1/R by (9.16), and the ambient curvature
bound is b = 0. With q = 2m/(m-2) and c = 4/(m-2), the result
reads
c integral_Sigma |grad_Sigma v|^2 dsigma
+ (m / R^2) integral_Sigma v^2 dsigma
>= m s_m^(2/m)
(integral_Sigma |v|^q dsigma)^{2/q}. (9.31)
To justify passage from eta_j v to v: the boundedness of v and
its gradient near Gamma gives convergence in L^2 and L^q,
while
grad_Sigma(eta_j v) - grad_Sigma v
= (eta_j - 1) grad_Sigma v + v grad_Sigma eta_j
converges to zero in L^2. The first term goes by dominated
convergence, the second by Lemma 15. The whole Dirichlet
energy, including the cross term, therefore converges.
Constant functions and their ambient perturbations are now
legitimate tests; no completeness of the regular locus is used.
This completes the proof of Proposition 2 and supplies the
tool used below to locate the perimeter support.
Retain the notation above. Put m = n - 1,
V = |E|, R = (V/omega_n)^(1/n), A = s_m R^m. Proposition 2
gives the decomposition Gamma = Sigma U Z, with all the
perimeter carried by Sigma and outward trace curvature
H = H_0 = m/R. The remaining problem is to show that Gamma is
contained in the exponential image of a Euclidean sphere of
radius R. The argument is split by dimension: at least four,
two, and three. A single final step then recovers both E and
its induced metric.
Two elementary observations. First, for any compactly
supported finite measure mu on M, the function
F(p) = integral_M D_p(x) dmu(x), D_p(x) = d(p, x)^2 / 2,
is continuous and proper. If the support of 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 p*.
Squared distance is smooth on M x M, and differentiating at a
minimum gives
integral_M log_{p*} x dmu(x) = 0. (10.1)
Uniqueness of the minimizer is not used.
Second, let T = Sigma or a compact boundaryless component of
Sigma, and let a = area(T) < infinity. The global first
variation (9.30) applies on T. The distance comparison
Hess D_p >= g from Corollary 1 gives, for every p in M,
m a
<= integral_T div_{T Sigma} grad D_p dsigma
= (m / R) integral_T dsigma
<= (m / R)
(a integral_T d(p, x)^2 dsigma(x))^{1/2}. (10.2)
The middle equality is (9.30) with Y = grad D_p, and the last
inequality is Cauchy-Schwarz with the pointwise bound
|grad D_p| = d(p, x). All fields may be multiplied by an
ambient cutoff equal to one near Gamma, so the noncompactness
of grad D_p on M causes no difficulty. In particular,
integral_T d(p, x)^2 dsigma(x) >= R^2 a. (10.3)
If the opposite inequality also holds, every inequality in
(10.2) is an equality. The last of them forces grad D_p
parallel to N almost everywhere on T, and since |N| = 1,
grad D_p = R N almost everywhere on T. (10.4)
For example, the squared L^2 norm of the difference is
integral_T d(p,x)^2 dsigma + R^2 a
- 2 R integral_T dsigma = 0.
Continuity upgrades (10.4) to every point of T: a nonempty
open regular patch has positive area. Since |N| = 1, (10.4)
puts all of T on the distance sphere r_p = R. This is the
property each of the three dimension regimes will establish
for T = Sigma, hence for its closure Gamma.
Here m >= 3 and n >= 4. the argument above has established (9.31) for
every smooth ambient function restricted to Sigma. The
constant function v = 1 attains equality in (9.31), since
A = s_m R^m gives LHS = (m/R^2) s_m R^m = m s_m R^{m-2} and
RHS = m s_m^(2/m) (s_m R^m)^{(m-2)/m} = m s_m R^{m-2}.
Let phi be such a restriction with int_Sigma phi dsigma = 0.
Boundedness permits the Taylor expansion of v = 1 + tau phi,
(integral_Sigma |1 + tau phi|^q dsigma)^{2/q}
= A^{2/q} [1 + (q-1) (tau^2/A)
integral_Sigma phi^2 dsigma + o(tau^2)]. (10.10)
The constant terms in (9.31) agree, and the linear terms
vanish because phi has zero mean. Since
m s_m^(2/m) A^{2/q - 1} = m/R^2 and c = q - 2, comparing the
coefficients of tau^2 gives
integral_Sigma |grad_Sigma phi|^2 dsigma
>= (m / R^2) integral_Sigma phi^2 dsigma. (10.11)
The cutoff limit was taken before the expansion; no mean-zero
condition on cutoff approximations is asserted.
Choose a barycenter p of the area measure dsigma on Sigma and
an orthonormal basis e_1, ..., e_n of T_p M. The smooth
functions phi_i(x) = have zero mean by (10.1).
Because d log_p is contracting (Corollary 1),
sum_{i=1}^n |grad_Sigma phi_i|^2
= sum_{j=1}^m |d log_p(v_j)|^2 <= m (10.12)
for any orthonormal frame v_1, ..., v_m of T_x Sigma.
Summing (10.11) over i gives
sum_i integral_Sigma |grad_Sigma phi_i|^2 dsigma
<= m A,
(m / R^2) sum_i integral_Sigma phi_i^2 dsigma
= (m / R^2) integral_Sigma r_p^2 dsigma,
so
integral_Sigma r_p^2 dsigma <= R^2 A. (10.13)
Together with (10.3), this yields (10.4) on all of Sigma. Its
density in Gamma, via closure(Sigma) = Gamma, therefore gives
Gamma subset {x in M : d(p, x) = R}. (10.14)
Neither connectedness nor completeness of the regular part
was used.
Now m = 1 and n = 2. Then Gamma = Sigma is a compact
embedded C^{1,1} curve without boundary, and A = 2 pi R,
because s_1 = 2 pi. Choose a connected component T of Gamma
and let L = length(T) > 0.
A connected compact one-dimensional manifold without boundary
admits a periodic arclength parametrization of period L. To
see this here, continue an oriented arclength chart: a
traverse of arbitrarily large length cannot stay injective in
a component of finite length, and local uniqueness produces a
least positive period. The image of one least period is open
and closed in the component, so it exhausts the component.
The periodic Wirtinger inequality states that
int_T |grad_T phi|^2 dsigma
>= (2 pi / L)^2 int_T phi^2 dsigma,
whenever int_T phi dsigma = 0. (10.18)
For completeness: in arclength coordinates the assertion
follows by expanding a trigonometric polynomial in Fourier
modes, the zero mode being absent. Periodic convolution
approximates a C^1 function and its derivative uniformly,
preserving the zero mean, and the same inequality holds for
every test used below.
Take a barycenter p of the area measure on T. Apply (10.18)
to phi_i(x) = , whose zero mean follows from
(10.1), and use the contraction of d log_p exactly as in
(10.12). Then
int_T r_p^2 dsigma <= L^3 / (2 pi)^2. (10.19)
On the other hand, (10.3) gives
int_T r_p^2 dsigma >= R^2 L. Combining,
R^2 L <= L^3 / (2 pi)^2, hence L >= 2 pi R = A. (10.20)
Since L <= A by definition, L = A. This component carries
all the length, and every inequality used is an equality.
Another component would carry a regular open arc of positive
length, impossible because the total is A. Therefore
T = Gamma, and (10.4) gives (10.14) with the same barycenter
p.
Now m = 2 and n = 3. Then Sigma = Gamma is a compact
embedded C^{1,1} surface, A = 4 pi R^2, and h = 1/R (so the
trace curvature is 2 h and the normalized mean curvature is
h). The spectral argument for m >= 3 does not apply for
m = 2; a punctured radial flux calculation replaces it.
Fix p in Sigma. Away from p, define
r = r_p, Y = r^{-2} grad D_p = grad r / r,
t = . (10.21)
Since Hess D_p >= g and |grad r| = 1 (off p),
div_{T Sigma} Y
= r^{-2} tr_{T Sigma} Hess D_p
- 2 r^{-2} |grad_Sigma r|^2
>= 2 r^{-2} (1 - |grad_Sigma r|^2) = 2 t^2, (10.22)
the last identity following from |grad r|^2 = |grad_Sigma r|^2
+ t^2 = 1.
Choose a smooth nondecreasing chi : [0, infinity) -> [0, 1]
that vanishes on [0, 1] and equals one on [2, infinity). The
field chi(r/eps) Y extends smoothly by zero near p. Applying
the global first variation (9.30) to this field and
differentiating the cutoff gives
int_Sigma chi(r/eps) (2 h t - 2 t^2) dsigma >= J_eps,
J_eps = int_Sigma (chi'(r/eps) / (eps r))
|grad_Sigma r|^2 dsigma. (10.23)
The left integrand is 2 h t - 2 t^2 = h^2/2 - 2 (t - h/2)^2,
and the chi' term enters with a positive coefficient that is
not subtracted.
Annular flux. The limit
lim_{eps -> 0} J_eps = 2 pi (10.24)
holds. Use exponential coordinates at p, with the first two
coordinate axes in T_p Sigma. In a small neighborhood of p
the whole support is a single graph (y, b(y)), where
b(0) = 0 and Db(0) = 0. For small eps this graph contains
the whole annulus contributing to J_eps, with no other sheet
or component. Set y = eps z. For fixed z != 0,
r / eps -> |z|, dsigma / eps^2 -> dz,
|grad_Sigma r| -> 1,
because r = |(y, b(y))| exactly. On the support of chi' one
has 1 <= r/eps <= 2, so z stays in a bounded disk and, using
the vanishing slope of b, away from zero. The area density
is uniformly bounded and |grad_Sigma r| <= 1. Dominated
convergence after extension by zero to the fixed disk yields
lim J_eps = int_{R^2} chi'(|z|)/|z| dz
= 2 pi int_1^2 chi'(s) ds = 2 pi.
Since h^2 A/2 = (1/R^2)(4 pi R^2)/2 = 2 pi, (10.23) gives
0 <= int_Sigma chi(r/eps) (t - h/2)^2 dsigma
<= (h^2/4) int_Sigma chi(r/eps) dsigma
- (1/2) J_eps (10.25)
with a right side that tends to zero. Fatou's lemma then
gives t = h/2 almost everywhere off p, without any
integrability assumption on t^2. Continuity upgrades this to
every point of Sigma \ {p}.
Off-diagonal form. The off-diagonal identity t = h/2 is
proved for every fixed p. Fix one x_0 in Sigma and evaluate
at x_0 for every p != x_0; no intersection of uncountably
many exceptional null sets is needed. Using
grad D_p(x_0) = -log_{x_0}(p) and Y = grad D_p / r^2,
- / r_p(x_0)^2 = 1 / (2 R). (10.26)
Completing the square,
|log_{x_0}(p)|^2 + 2 R = 0,
|log_{x_0}(p) + R N(x_0)|^2 = R^2,
so
Gamma subset exp_{x_0}
{v in T_{x_0} M : |v + R N(x_0)| = R}. (10.27)
The Euclidean sphere in (10.27) contains the origin of
T_{x_0} M; its center is -R N(x_0).
All three dimension regimes give the same conclusion: for some
o in M, the perimeter support Gamma is contained in exp_o(S),
where S is a Euclidean sphere of radius R in T_o M. The
sphere S need not be centered at the tangent-space origin.
Let U_0 be the open ball in T_o M bounded by S, and put
U = exp_o(U_0). The global exponential diffeomorphism maps
the two connected components of T_o M \ S onto the two
connected components of M \ exp_o(S). The bounded interior
component is U (its closure is a compact image); the exterior
component is unbounded because vectors escaping to infinity in
the Euclidean exterior have images exp_o(v) at distance
d(o, exp_o(v)) = |v| escaping to infinity. Exterior
connectedness uses n >= 2. The smooth hypersurface exp_o(S)
has zero ambient volume.
Phase recovery. The distributional derivative of chi_E is
supported on Gamma, hence vanishes on each of M \ U and
M \ bar U (bar U = exp_o(bar U_0)). A function whose weak
coordinate derivatives all vanish is locally constant almost
everywhere, as can be seen by mollifying in coordinate balls
and then chaining overlapping balls. Thus chi_E is almost
everywhere constant on each of the two connected components.
Its constants lie in {0, 1}. Boundedness of E forces the
exterior constant to be zero, because the open exterior
contains balls of positive volume at arbitrary distance from
E. Since V > 0 and the separating sphere exp_o(S) has zero
volume, the interior constant is one. Hence
E = U up to a set of volume zero. (10.28)
Only the containment of the perimeter support in exp_o(S) has
been used.
Metric recovery. Corollary 1 gives g~ = (exp_o)^* g >= g_o,
the constant Euclidean metric of T_o M, on all of T_o M. This
holds in particular on U_0. There,
vol_{g~}(U_0) = V = omega_n R^n = vol_{g_o}(U_0). (10.29)
The smooth relative density sqrt(det_{g_o} g~) is at least
one. Equality of its integral with vol_{g_o}(U_0) and
continuity force it to be identically one on U_0. All
eigenvalues of g~ relative to g_o are at least one and their
product is one, so each eigenvalue equals one. Therefore
g~ = g_o on U_0, (10.30)
and exp_o : U_0 -> U is a Riemannian isometry. This is the
necessity direction of Theorem 2.
Sufficiency. Let F : B_R(0) subset R^n -> U subset M be a
Riemannian isometry onto an open region with its induced
metric. 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). (10.31)
Consequently U = B_R(p). The formula (10.31) extends smoothly
to the closed Euclidean ball through the ambient exponential
map, and its pullback metric is Euclidean in the interior,
therefore on the boundary by continuity. The boundary is a
smooth sphere of area s_{n-1} R^{n-1}, which equals its
ambient perimeter by the divergence formula. Its volume is
omega_n R^n, so A = n omega_n^(1/n) V^((n-1)/n) holds in
(9.1). Perimeter and volume are unchanged by null
modifications. This completes the sufficiency in Theorem 2.
No flatness outside U is implied.
Theorems 1 and 2 reduce the isoperimetric problem to a sharp
perimeter bound. That bound is the exact input needed for the
classical rearrangement theorems comparing the first Dirichlet
eigenvalue and the torsional rigidity with those of the model
ball. The proofs are short once the isoperimetric comparison
is available.
For a smooth relatively compact domain D of a Riemannian
manifold (M, g) and 1 < p < infinity, let W^{1,p}_0(D) be the
closure of C_c^infty(D) in the Sobolev norm, and define
lambda_{1,p}(D) = inf_{0 != v in W^{1,p}_0(D)}
(int_D |grad_g v|_g^p dvol_g) /
(int_D |v|^p dvol_g), (11.1)
T_p(D) = sup_{0 != v in W^{1,p}_0(D)}
(int_D |v| dvol_g)^p /
(int_D |grad_g v|_g^p dvol_g). (11.2)
The first is the Dirichlet eigenvalue of the unnormalized
p-Laplacian -Delta_{p,g} u = div_g(|grad_g u|_g^{p-2}
grad_g u). The second is the p-torsional rigidity in the
quotient normalization of Briani, Buttazzo and Prinari.
Corollary 2 (Model-ball Faber-Krahn and Saint-Venant).
Let n >= 2, kappa <= 0, and 1 < p < infinity. Let (M^n, g)
be complete, simply connected, smooth and without boundary,
with Sec_g <= kappa. For a nonempty smooth domain D via
subset M, let B_kappa(|D|) be a geodesic ball of volume |D|
in the simply connected n-dimensional space form of curvature
kappa. Then
lambda_{1,p}(D) >= lambda_{1,p}(B_kappa(|D|)),
T_p(D) <= T_p(B_kappa(|D|)). (11.3)
Proof. Let v in C_c^infty(D). Extend u = |v| by zero on M;
this is a compactly supported Lipschitz function of finite
support. Put b = sqrt(-kappa) and take the interval density
a(r) = A_b(r) on (0, infinity), whose cumulative mass is
F_a(r) = V_b(r). So a(F_a^{-1}(s)) = I_b(s).
By Theorem 1, the support-neighborhood bound required by
Polya-Szego holds at every finite volume, and trivially also
for sets of infinite perimeter. The positive superlevels of u
have finite volume, and the relaxed metric p-energy is at most
the Riemannian p-energy. The Polya-Szego theorem of Nobili
and Violo, together with their Lemma 3, gives a decreasing
equimeasurable rearrangement u^* for a(r) dr that is bounded
and locally absolutely continuous, with weighted derivative
energy at most
int_D |grad_g v|_g^p dvol_g. (11.4)
Write B = B_kappa(|D|), with center o. Since supp u is
compactly contained in D, the support of u has volume strictly
below |D|. So u^* vanishes on some interval before
R = V_b^{-1}(|D|). (11.5)
Use the radial construction in Section 7.1 of Nobili and
Violo. The polar density is
a(r) = n omega_n r^{n-1} w_b(r), (11.6)
where w_b(r) = (sn_b(r) / r)^{n-1} extends smoothly and
positively at zero and is log-convex. Define
u~(x) = u^*(d_kappa(o, x)). (11.7)
In normal coordinates u~ is bounded and locally W^{1,p} away
from the origin with finite gradient p-energy. Its gradient
extends across the origin: integrating by parts outside the
radius-eps ball, the inner boundary term is O(eps^{n-1})
because u^* is bounded, and eps -> 0 since n >= 2. The weak
gradient therefore extends across the origin. Equivalence of
Sobolev norms in smooth coordinates and vanishing near R now
give u~ in W^{1,p}_0(B).
The model radial gradient is the radial derivative, and
int_B |grad_kappa u~|_kappa^p dvol_kappa
<= int_D |grad_g v|_g^p dvol_g,
int_B |u~|^q dvol_kappa
= int_D |v|^q dvol_g, q = 1, p. (11.8)
Taking the Rayleigh infimum and the torsion-quotient supremum
gives both comparisons in (11.3). The defining density of
C_c^infty(D) and the finite volume of D allow both variational
values to be computed on the tests used above. End of proof.