Comparison theorems for exit times Almut Burchard and
Michael Schmuckenschl¨agery
February 9, 2000
Abstract We compare the exit time of Brownian motion from a subset of a smooth complete Riemannian manifold with the exit time from a geodesic disc in a manifold of constant curvature.
1 Introduction Let M be smooth complete Riemannian manifold, and denote Brownian motion on M by Xt (for a description of Brownian motion on manifolds we refer to [22]). The exit time from a subset A M is defined by
TA = inf ft > 0 : Xt 62 Ag ;
(1.1)
it is a stopping time provided that A is Borel measurable. The subject of this paper are bounds on the exit time of Brownian motion from a A, in terms of geometric quantities such as the volume of A and the curvature of M . More precisely, we compare the distribution function of the exit time from A
uA(t; x) = Px(TA > t)
(1.2)
with the distribution function of the exit time from a related geodesic disc in a space of constant curvature. We prove two such comparison results, one giving upper and lower bounds for small geodesic discs in terms of the radius and of curvature bounds, and the second giving upper bounds for subsets of manifolds of constant curvature in terms of their volume. Both can be read as statements about the Dirichlet heat kernel on A, and analogous inequalities hold for certain spectral invariants. Our third result, which plays an auxiliary role here, generalizes Riesz’ rearrangement inequality to multiple integrals. Finally, we show that the comparison theorems discussed here contain classical isoperimetric inequalities as limiting cases.
Department of Mathematics, University of Virginia, Charlottesville, VA 22903.
[email protected] y Institut f¨ur Analysis und Numerik, Johannes Kepler Universit¨at Linz, 4040 Linz, Austria.
[email protected]
1
1.1 Comparison results To formulate the first theorem, let M be a smooth complete Riemannian manifold. The dimension m of the manifold is fixed throughout the paper. We generally assume both upper and lower bounds on the curvature of M . The distance between two points x; y 2 M is denoted by d(x; y ). We write dx for integration with respect to the natural Riemannian volume on M , and V (A) for the volume of a Borel set A. The symbol P t refers to the heat semigroup on A, and p(t; x; y ) to the corresponding integral kernel. For 2 R , denote by M the sphere ( > 0), Euclidean space ( = 0), or hyperbolic space ( < 0) of constant curvature . Let D be a disc of radius R in M . By symmetry, the distribution function of the exit time from D can be written as
uD (t; x) = h(t; d(x; x)) ;
(1.3)
where x is the center of D, and h (t; R) = 0. The function h (t; r) is strictly decreasing in r for 0 < r < R, t > 0 (see Corollary 4.3). Theorem 1 (Exit time from geodesic discs) Let A be a geodesic disc of radius R and center xo in a smooth m-dimensional Riemannian manifold M . Assume that R is small enough so that A does not meet the cut locus of xo . 1. If the Ricci curvature of M is bounded below by (m ? 1), then
uA(t; x) h(t; d(x; xo )); where h (t; r) is defined by the distribution function of the exit time from a disc D the same radius R, as in (1.3).
(1.4)
M of
2. If the the sectional curvature of M is bounded above by , then the reverse inequality holds:
uA(t; x) h(t; d(x; xo )): If equality occurs in either (1.4) or (1.5) for some isometric to D .
(1.5)
t > 0 and x in the interior of A, then A is
The exit time from a sufficiently regular open set (and in particular from a small disc as described above) solves the heat equation
@t u = u
x 2 A; t > 0
(1.6)
x 2 @A; t > 0
(1.7)
with Dirichlet boundary conditions
uA(t; x) = 0 2
and initial values
x2A:
uA(0; x) = 1
(1.8)
(Note that our convention differs from [22] by a factor of 2.) Theorem 1 says that if is a bound for the curvature of M , then the solution of the Dirichlet problem (1.6)-(1.8) on a small disc is bounded by the solution of the corresponding problem on a disc of the same radius in M . Analogous comparison are known for the heat kernel p(t; x; y ) (see [20], VIII.3). The next theorem applies only to subsets of M . Theorem 2 (Rearrangement for exit times on M ) Let A M be an open set of finite volume, and let A be the open geodesic disc of equal volume as A centered at x . Then, for all t > 0, the exit time from A is dominated by the exit time from A in the sense that Z
F (uA(t; x)) dx
Z
F (uA (t; x)) dx
(1.9)
for every convex increasing function F with F (0) = 0. In particular,
max u (t; x) uA (t; x ) : x2A A Equality in either (1.9) when F (uA (t; x )) a set of zero capacity.
(1.10)
> 0 or in (1.10) implies that A differs from a disc by
The majorization statement in equation (1.9) admits many equivalent reformulations, some of which are discussed in Proposition 5.1 (see [2] for a general discussion). It implies in particular that, unless A is a disc, all Lp -norms of uA (with p 1) are strictly smaller than the corresponding norms of uA . Another implication is that for A M , the moments
Mp;n(A) := kEx[TAn]kp
(1.11)
are maximized, among sets of a given volume, by geodesic balls (see Corollary 6.1). Such moment inequalities were established by McDonald [37] who showed furthermore that discs are the only critical points of the moment functionals. We will prove (1.9) in the form Z
B
uA(t; x) dx
Z
B
uA (t; x) dx
(1.12)
for any Borel set B A. The special case in (1.10) asserts the intuitively obvious fact that Brownian motion starting at some point x 2 A tends to leave A earlier than Brownian motion starting at x would leave A . Inequalities for exit times analogous to those in Theorem 2 were proven in R n by Bandle [6], and on Gauss space by Borell [14]. The one-dimensional case was proven by Friedberg and Luttinger in [24]. Closely related are geometric inequalities for the eigenvalues of the Laplacian, for 3
solutions of elliptic boundary value problems [7], for subharmonic functions [5], and for harmonic measure [27]. Thus, the main contribution of Theorem 2 is the characterization of the cases of equality, which was not attempted in the above sources. In an independent development, Morpurgo has recently proved a more general statement of the inequalities in Theorem 2, including the cases of equality when F is strictly convex [39]. We believe that our result remains interesting, as our proof yields a lower bound for the difference between the two sides of (1.10) and (1.12), expressed as the Wiener measure of a set of paths which we characterize geometrically. Our proofs show that inequalities (1.9), (1.10) and (1.12) actually hold for any Borel set A M . We did not succeed in characterizing the cases of equality in this more general setting, but are convinced that, up to sets of zero volume, they are only the obvious ones. Theorems 1 and 2 will be proven by completely different methods. To obtain inequalities (1.4) and (1.5), we write uA (t; ) in geodesic coordinates about xo and compare it directly with the solution of the corresponding problem on M . The inequalities and their proofs are local, as they can be settled by computations in a single coordinate patch. Theorem 2, on the other hand, is a global result, as it requires no assumptions on the size or shape of A. For the proof, we rely on rearrangement techniques that exploit the symmetries of the spaces M . The key observation is that the so-called two-point rearrangements, which are simple geometric manipulations that push A closer to A , can only increase the exit time from a set A. In this context, we call A the spherical symmetrization of A.
1.2 Inequalities for multiple integrals For technical reasons, we do not apply rearrangement arguments directly to the proof of Theorem 2 (see Section 5). Instead, we obtain (1.12), via a Trotter product formula, from a limit as n ! 1 of rearrangement inequalities for multiple integrals
J (f1 ; : : : ; fn) =
Z
Z
Y 1in
fi(xi )
Y 1i 0. Theorem 3 was proved independently by Morpurgo [39] (see also the announcement in [38]), with essentially the same characterization of the cases of equality, but a slightly different combinatorial proof. The special case of (1.14) with n = 2 Z
f (x)k(x; y)g(y) dxdy
Z
f (x)k(x; y)g(y) dxdy
(1.16)
is known as Riesz’ rearrangement inequality [42]. Here, f and g are again nonnegative measurable functions on M that vanish at infinity, and k is a nonincreasing function of the distance. Inequality (1.14) on R m is contained in an inequality of Brascamp, Lieb and Luttinger which does not require the kernels on the left hand side to be decreasing functions of the distance but rather allows them to be rearranged along with the functions [15]. The cases of equality are fully understood only if the multiple integral is in the form of a convolution [36, 18]. The basic idea of approximating the spherical symmetrization with simpler rearrangements was proposed by Steiner for a proof of the isoperimetric inequality [44]. Following Hardy, Ahlfors replaced Steiner symmetrization with the even simpler two-point rearrangement in the study of convolution kernels on the unit circle [1]. Friedberg and Luttinger used two-point rearrangements to prove Theorem 2 in one dimension [24]. They appear, as compressions, in connection with discrete isoperimetric inequalities [13, 23]. More about two-point rearrangements can be found in [28, 5, 8, 10, 9, 4, 16]. Our proof of Theorem 3 is modeled after the proof of Riesz’ rearrangement inequality on spheres by Baernstein and Taylor [5]. In particular we use the fact that the spherical symmetrization can be approximated in Lp -spaces by repeated two-point rearrangements, which was first proved there.
1.3 Relation with isoperimetric inequalities Our final topic is the relation of the rearrangement inequalities in Theorems 2 and 3 with isoperimetric inequalities. It is well known that inequalities of the type (1.9) can be obtained via the co-area formula from isoperimetric inequalities (see [7]). We are interested in the converse direction, namely recovering isoperimetric inequalities from estimates for heat kernels. Our approach is motivated by the work of Ledoux on isoperimetric inequalities in Gauss space [35]. 5
There are many definitions of the perimeter of a set, all of which coincide for open sets with smooth boundary. We will prove that for sufficiently regular sets A M , r
Z 1 ? uA(t; x) dx : Per(A) = tlim (1.17) !0+ 4t A The isoperimetic inequality appears in the t ! 0+ limit of Theorem 2 in the following way: For a subset A of finite volume in M , inequality (1.9) with F (z ) = z is equivalent to Z
A
1 ? uA(t; x) dx
Z
A
1 ? uA (t; x) dx ;
(1.18)
(note that V (A) = V (A ) by definition of the spherical symmetrization). Plugging both sides of (1.18) into (1.17) shows that r
Z 1 ? uA(t; x) dx Per(A) = tlim (1.19) !0+ 4t A r Z 1 ? u (t; x) dx tlim (1.20) A + !0 4t A = Per(A) ; (1.21) at least for sufficiently regular sets A. The isoperimetric inequality for more general Borel sets
then follows from the semicontinuity properties of the perimeter (see (3.2)). However, information about the cases of equality is lost in the limit t ! 0+ of (1.20). For large times, the exit time from A behaves asymptotically as Const:e?1 (A)t , where 1 (A) is the lowest eigenvalue of the (negative) Dirichlet Laplacian on A. Hence inequality (1.12) implies that
1(A) 1(A ) ;
(1.22)
which is a statement of the Faber-Krahn inequality. We note without proof that our lower bound for the difference between the two sides of (1.12) decays sufficiently slowly with t to yield a positive lower bound for the difference 1 (A) ? 1 (A ) unless A is already a disc. In summary, Theorem 2 contains the isoperimetric inequality and the Faber-Krahn inequality as limiting cases. One may wonder in what form Theorem 2 could extend to general manifolds. Can one find, for a given set A M , a suitable disc A M whose exit time dominates the exit time from A? Is there any hope for rearrangements that can symmetrize simultaneously the manifold and the functions living on it? We offer some speculations along these lines in Section 7. Note that there can be no general lower bound on the exit time in terms of the curvature of M and the volume of A alone, since it is always possible to make the exit time small by removing a subset of small volume from A. Some intriguing inequalities for Dirichlet eigenvalues in two dimensions point to possible lower bounds for the exit time in terms of the conformal class of A [40, 33]. The key to our proof of (1.17) is the following formula. 6
Theorem 4 (The heat flow over the boundary as a measure for the perimeter) Let A M be a Borel set of finite volume, and let P t denote the heat semigroup on M at time t 0. Then r
Z P tI dx ; t Ac A
Per(A) tlim (1.23) !0+ where Ac is the complement of A. In particular, the right hand side is finite for every set of finite perimeter. If the boundary of A is a twice continuously differentiable submanifold, then (1.23) holds with equality (and the limit on the right hand side exists). Theorem 4 was proved for Gauss space by Ledoux in connection with the isoperimetric inequality [35]. An argument similar to (1.18)-(1.20) exhibits the isoperimetric inequality as a t ! 0+ limit of Theorem 3: By Riesz’ rearrangement inequality, Z
Ac
P tI
A
Z
(A )c
P t IA :
(1.24)
Applying (1.23) on both sides of (1.24) yields the isoperimetric inequality (see [35]). The relation between (1.17) and Theorem 4 relies on the approximation uA(t; x) IAP t(IA ? IAc ) : (1.25) This is exact if A is a half-space in M , since then both sides of (1.25) solve the same Dirichlet problem. We will show that, for any A M with twice continuously differentiable boundary, Z
1 ? uA(t; x) dx = 2
Z
as t ! 0 : (1.26) A Ac Recalling that P t is self-adjoint and P t IA + P t IAc = 1, we recognize (1.26) as an integrated version of (1.25). Thus, Theorem 4 implies formula (1.17).
P t IAc dx + o(t1=2 )
It is beyond the scope of this paper to fully discuss how the the right hand sides of (1.17) and (1.23) relate to the various definitions of the perimeter of a Borel set A (see [46, 17] for precise definitions of the perimeter). While the two formulas make sense for general Borel sets (where they may take the value +1), we do not know whether they have desirable semicontinuity properties analogous to (3.2).
1.4 Outline of the paper The paper is organized as follows. In Section 2, we recall some tools from differential geometry and prove the local comparison results of Theorem 1. Section 3 contains the proof of Theorem 4. The main issue here is to estimate the small-time behavior of the heat kernel. In the same section, we justify the approximation (1.25) and prove formula (1.17). Section 4 is devoted to the development of rearrangement tools and the proof of Theorem 3. We also show there that the exit time generally increases under two-point rearrangements, and give an estimate by how much. In Section 5, we prove Theorem 2 as a limit of Theorem 3. This is followed, in Section 6, by some examples how the statement and proof of Theorem 2 yield corresponding inequalities for eigenvalues and exit time moments. We conclude in Section 7 with two conjectures regarding extensions of Theorems 2 and 3 to manifolds of non-constant curvature. 7
2 Exit time from geodesic discs Our goal in this section is the proof of Theorem 1. We need to recall a few facts from Differential Geometry (see [20, 25, 12] for details). Every point in M can be represented as x = expxo (rw), where w is a unit tangent vector at xo , expxo is the exponential map from the tangent space of M at xo into M , and r = d(x; xo ) > 0. This representation is unique in some sufficiently small neighborhood of xo , and we refer to (r; w) as geodesic coordinates about xo . If we identify the unit sphere in the tangent space of M at xo with Sm?1, we may express the volume form on M in terms of Lebesgue measure on R + and the standard surface measure on Sm?1 as dx = (r; w) dr dw . The gradient and Laplacian of (x) = d(x; xo ) at a point x = expxo (rw) can be written as
r(x) = @r expxo (rw) ; (x) = @r log (r; w) :
(2.1)
On any m-dimensional manifold, we have
(r; w) = rm?1 (1 ? 61 Ric (w; w)r2 + o(r2))
(r ! 0):
(2.2)
For the manifolds of constant curvature M , one can explicitly compute (r) = Sk (r)m?1 , where
Sk (r) :=
8 < :
p p
sin(r )= pr p sinh(r ?)= ?
if > 0 if = 0 if < 0
(2.3)
(see Theorem II.5.1 of [20], or Theorem 4.21 of [25]). Fix , and choose geodesic coordinates simultaneously about xo in M and about x in M . Let be a diffeomorphism from a neighborhood of xo in M to a neighborhood of x in M with (xo ) = (x ) which preserves geodesic coordinates. Two classical inequalities relate the volumes of geodesic balls about xo 2 M and x 2 M . Bishop’s comparison theorem says that if the Ricci curvature of M is bounded from below by (m ? 1), then
@r (r;(rw)) 0 ;
(2.4)
(x) ( (x)) ;
(2.5)
or, equivalently,
where is the Laplacian on M [12], Theorem 11.15 (see [20], III.3). One implication is that V (A) V (D). G¨unther’s comparison theorem asserts the reversed inequalities if the sectional curvature is bounded above by : @r (r;(rw)) 0 ; (2.6) 8
that is,
(x) ( (x)) ;
(2.7)
and consequently V (A) V (D) [26] (see [20], III.2).
P ROOF OF T HEOREM 1 Let uA(t; x) be the exit time from a geodesic disc A of radius R and center xo in M , as in the statement of the theorem. As discussed in the introduction, uA solves the Dirichlet problem (1.6)-(1.8). Let D be the disc of the same radius R in M . We want to compare uA with h (t; (x)), where (x) is the distance form x to xo in M , and h(t; r) is determined by (1.3). Using that r(x) is a unit vector, we calculate for x 2 A, t > 0,
(@t ? )h(t; (x)) = @t h(t; (x)) ? @r2 h(t; (x)) ? (x) @r h(t; (x)) : (2.8) For the first claim of Theorem 1, assume that (m ? 1) is a lower bound for the Ricci curvature of M . Since h (t; r) decreases with r, Bishop’s inequality (2.5) implies that (@t ? )h(t; (x)) @t h(t; (x)) ? @r2 h(t; (x)) ? ( (x))@r h(t; (x)) (2.9) = (@t ? )u(t; (x)) (2.10) = 0 (2.11) where : A ! D is the diffeomorphism introduced in (2.5). On the other hand, (@t ? )uA(t; x) = 0 : (2.12) Since h (t; (x)) satisfies the same initial and boundary conditions as the desired solution uA (t; x), it follows by the maximum principle that h (t; (x)) uA (t; x), as claimed. To see the second claim, assume that is an upper bound for the sectional curvature. It is possible (when > 0) that R is greater than half the diameter of M . In that case, however, h(t; x) 1, and the claim is trivially satisfied. Otherwise, we argue as above and conclude with (2.7) that
(@t ? )h (t; (x)) 0 (@t ? )uA(t; x) ;
(2.13)
and the claim follows again by the maximum principle. Suppose that either (1.4) or (1.5) holds with equality for some (t1 ; x1 ) with t1 > 0 and x1 in the interior of A. By the strong maximum principle for parabolic equations (see [41], 3.2 Theorem 2), equality holds for all 0 t t1 and all x 2 A, and by analyticity, for all t 0. Since h (t; r) decreases strictly with r, it follows that
( (x)) = (x)
that is,
@r (r;(rw)) = 0
(x 2 A) ;
(0 r < R; w 2 Sm?1) :
We conclude that (r; ! ) (r), and that maps A isometrically onto D. 9
(2.14)
(2.15)
3 Perimeter and heat flow In this section we prove Theorem 4 and inequality (1.17). Since the right hand side of (1.23), which involves only a single integral over the heat kernel, is more manageable than the right hand side of (1.17), we prove Theorem 4 first. We define the perimeter of a set A as the total variation of its characteristic function, that is,
Per(A) := sup
Z
Y : jY j1 A
div Y (x) dx ;
(3.1)
where the supremum is over all smooth compactly supported vector fields with jY j 1, and div Y denotes the divergence of Y (see [46, 17]). In general, the inequality Z
Per(A) lim sup gn(x) div Y (x) dx = lim krgnk1 n!1 Y
n!1
(3.2)
holds for any sequence of smooth functions gn converging to IA in L1loc . In particular, the perimeter is lower semicontinuous with respect to convergence in measure. If A has finite perimeter, one can choose the approximating sequence gn so that
Per(A) = nlim krgnk1 ; !1
and consequently
Z
(3.3)
Z
h (x); Y (x)i dS := nlim h?rgn(x); Y (x)i dx (3.4) !1 @A exists for every smooth bounded vector field Y on M . If A has smooth boundary, then (3.1) coincides with the definition of the perimeter as a surface integral. Furthermore, the right hand side of (3.4) coincides with the usual formula for the flux of Y across the boundary of A, where (x) is the outward unit normal to the boundary of A at a boundary point x, the pairing h ; i denotes the inner product on the tangent space of M at x, and dS is integration with respect to (m ? 1)-dimensional surface measure. 3.1 A simple perimeter estimate on spheres The following inequality will not be needed below. We present it here to introduce, in a simpler setting, the idea for the proof of Theorem 4. Proposition 3.1 Let M be a sphere of constant curvature > 0, and let Pt be the heat semigroup on M . For every A M ,
1 Z P t I dx 1 Z P t I dx ; Per(A) Ac A Per(H ) H c H
where H is a hemisphere.
10
(3.5)
Remark Passing to the limit t ! 1 in (3.5) yields that V (A)V (Ac)
Per(A)
V (H )2 Per(H ) ;
(3.6)
which is related with Cheeger’s inequality (see [20]). The right hand side of (3.6) is maximized for V (A) = V (H ) = V (M )=2, when the statement becomes a special case of the classical isoperimetric inequality with the best constant. P ROOF : By symmetry, the heat kernel on M can be written in the form
p(t; x; y) = q(t; d(x; y)) ; (3.7) where q (t; r) is a strictly decreasing function of of r for each fixed t > 0. For any pair of smooth nonnegative functions f and g on M , we compute Z
f (Pt g ? g) dx
Z tZ
=
0
= ? = ?
M
f (Psg) dxds
Z tZ 0
M
0
M
Z tZ
(3.8)
hrg; r(Psf )i dxds Z
M
hrg(x); rxq(s; d(x; y))i f (y)dxdyds
(3.9) (3.10)
where h; i denotes the inner product on the tangent space of M at x. We have used the heat equation in the (3.8), integrated by parts in (3.9), and spelled out the heat kernel in (3.10). Setting f = IAc and switching the order of integration yields Z Z tZ Z t hrg(x); rxq(s; d(x; y))i dydxds : Pg ? g dx = ? (3.11) Ac 0 M Ac Fix s > 0 and integrand,
?
x2
Z
Ac
M . The inner integral can be estimated by taking the negative part of the
hrg(x); rxq(s; d(x; y))i dy
Z
Z
M
hrg(x); rxq(s; d(x; y))i? dy
(3.12)
Z 1
hrg(x); wi? dw (3.13) @r q(s; r) (r) dr : S m?1 0 We have switched to geodesic coordinates y = expx (r; w), and used the fact that does not depend on the direction vector w to separate the integrals. Clearly, the right hand side of (3.13) is a constant multiple of jrg (x)j, where the constant depends only on s, the dimension m, and the curvature . Performing the integrations over x and s, we obtain =
Z
Ac
Pt g ? g dx C (t; m; ) krgk1 : 11
(3.14)
Using an approximation as in (3.3) we conclude that Z
Ac
Pt IA dx C (t; m; )Per(A) :
(3.15)
We claim that for a hemisphere H , inequality (3.15) is an identity. Since H has a smooth compact boundary, there exists a sequence of functions gn approximating IH in the sense of (3.3). Inserting this sequence into (3.11), taking limits, and using (3.4), we see that Z
H
Pt IH dx c
=
Z tZ 0
Z
@H H c
h (x); rx q(s; d(x; y))i dydxds
(3.16)
where (x) is the outward normal to @H at x. The key observation is that hemispheres have the property that for x 2 @H , the outward normal (x) forms an acute angle with the geodesic from x to y, if and only if y 2 H c. Since q(s; d(x; y)) increases along that geodesic, it follows that for x 2 @H , Z
Ac
h (x); rxq(s; d(x; y))i dy =
Z
M
h (x); rxq(s; d(x; y))i+ dy :
Continuing the computation as in (3.12) and (3.13), we see that This completes the proof.
(3.17)
H satisfies (3.15) with equality.
3.2 Proof of Theorem 4 From the proof of Proposition 3.1, we see that the special case of inequality (1.23) in R m can be proven by computing the constant C (t; m; 0) in (3.14) and studying its asymptotics as t ! 0. On a general manifold, such a computation has to be replaced by estimates, as explicit formulas for the heat kernel are not available. We approximate the heat kernel p(t; x; y ) on M by q (t; d(x; y )), where
q(t; r) = (4t1)m=2 e?r2 =(4t)
(3.18)
is the function defining the heat kernel on R m . We remark that the integral operators Qt given by the kernels q (t; d(x; y )) can be used to construct the heat kernel on M exactly (see, for example, [11, 43]). Here, we mainly need that each Qt is a bounded linear operator on Lp and that Qt f ! f as t ! 0 pointwise at least for smooth functions f with compact support. Another useful observation is that for R > 0 and t sufficiently small, q (t; d(x; y )) is concentrated on the diagonal in the sense that the contribution of Z Z
d(x;y)R
f (x)q(t; d(x; y))g(y) dxdy kf k1 kgk1 q(t; R)
(3.19)
is exponentially small in t?1 as t ! 0, uniformly for R bounded away from zero. The next lemma shows that Qt f almost solves the heat equation. 12
Lemma 3.2 Assume that the Ricci curvature of M is bounded below, and the sectional curvature is bounded above. Then there exists a constant C (depending only on M ) such that for t 1:
(@t
? )Qt f 1 C kf k1
(3.20)
P ROOF : Let (x) = d(x; y ). Using geodesic coordinates as in the proof of Theorem 1, we calculate
(@t ? )q(t; (x)) = (x) (x) ? (m ? 1) q(t; (x)) : 2t (x)
(3.21)
It is easy to see from the explicit formulas in (2.3) that on M , (x)
? m(?x)1 C () (x) :
(3.22)
By our assumptions on the curvature and the inequalities (2.4) and (2.6) of Bishop and G¨unther, a statement of the form (3.22) holds also on M , and we conclude that
j(@t ? )q(t; (x))j C (M ) (xt ) q(t; (x)) 2
(3.23)
in any patch where geodesic coordinates are valid. It follows that (@t
? )Qt f (x)
d(x; y )2 1 2t x d(x; y ) M Z Z R(y;w) 2
kf k
Z
? dm(x;?y1) dy
C (M ) kf k1
Sm?1 0
(3.24)
r q(t; r)(r; w) drdw ; t
(3.25)
where R(y; w) denotes the maximal length of a distance-minimizing geodesic starting at y in direction w. In the last step we have used the fact that in a complete manifold, the cut locus has zero volume. We claim that the last integral is finite. Indeed, using Bishop’s inequality (2.4) again, we see that for y 2 M and w 2 Sm?1, Z R(y;w) 2 r 0
t q(t; r)(r; w) dr =
Z 1 2 r Z
0 0
t q(t; r)(r)pdr 1 2 (r t) q(1; r) rt(m+1) =2 dr ;
(3.26) (3.27)
where is determined by the lower bound on the Ricci curvature of M . Since the right hand side of (3.27) is an increasing function of t and takes a finite value for t = 1, it is uniformly bounded for t 1. Inequality (3.20) implies a corresponding bound for the difference between heat kernel P t : 13
Qt and the true
Lemma 3.3 Under the assumptions of Lemma 3.2, we have for t 1
t
(Q ? P t )f C kf k t;
1
1
where C is the constant of Lemma 3.2.
(3.28)
P ROOF : By Lemma 3.2, the function
v(t; x) = (P t ? Qt)f (x) ? C kf k1 t
(3.29)
satisfies
(@t ? )v(t; x) 0 (t 1): Since v (0; x) = 0, we conclude by the maximum principle that v (t; x) 0.
(3.30) Repeating the argu-
ment for
v~(t; x) = ?(P t ? Qt )f (x) ? C kf k1 t gives the claim. Inequality (4) holds with P t replaced by Qt :
Lemma 3.4 If A M is a bounded set with finite perimeter, then
V (@A) tlim !0+
r
Z Qt I dx t Ac A
(3.31)
P ROOF : Lemma 3.2 implies that for any two smooth nonnegative functions f and g with compact support, Z Z tZ t f (Q g ? g) dx = (3.32) f @sQsg dxds 0 M Z tZ = (3.33) f (Qsg) dxds + kf k1 kgk1 O(t) : 0 M Hence we may repeat the computation in (3.8)-(3.11) to obtain Z Z tZ Z t (Q g ? g) dx = ? hrg(x); rxq(s; d(x; y))i dxdyds + kgk1 O(t) : (3.34) Ac 0 M Ac The inner integral on the right hand side can be estimated as in the proof of Proposition 3.1
?
Z
Ac
hrg(x); rxq(s; d(x; y))i dy =
Z
Z
M
hrg(x); rxq(s; d(x; y))i? dy
S m?1
hrg(x); wi? 14
Z R(x;w) 0
@r q(s; r) (r; w) drdw:
(3.35) (3.36)
In the first step, we have taken the positive part of the integrand. In the second step, we have switched to geodesic coordinates y = expx (r; w); here R(x; w) is the maximal length of a distanceminimizing geodesic starting at x in direction w. Since g is compactly supported, we have
Ro = x2inf inf R(x; w) > 0 : suppg w2Sm?1
(3.37)
For given " > 0, there exists a positive number R" 2 (0; Ro ) such that j(r; w) ? r m?1 j " for all r < R"; ! 2 S m?1. On the other hand, using Bishop’s inequality (2.4) again, we see that Z
[R" ;R(x;r)]
Z 1
r q(s; r) (r) dr R" 2s Z 1 1 2ps p q(1; r)r?jj(r) dr ;
@r q(s; r) (r; w) dr
R" = s
(3.38) (3.39)
which is exponentially small in s?1 as s ! 0. It follows that Z R(x;w) 0
r
Z 1 ? @r q(s; r)(r; w) dr = s (4s)?(m+1)=2 e?r2=4s rm dr 1 + o(1) 0 r = s ((m + 1)!m+1)?1 (1 + o(1))
uniformly for x in the support of g and w hand side of (3.36) is easily computed as Z Sm?1
(3.40) (3.41)
2 S m?1 . The value of the spherical integral on the right
hrg(x); wi? dw = !m?1 jrg(x)j
(3.42)
where !m?1 is the volume of the unit ball in dimension m ? 1. Combining equations (3.41) and (3.42) with (3.36) yields
?
Z
A
hrg(x); rxq(s; d(x; y))i dy c
r
!m?1 jrg(x)j (1 + o(1)) : s (m + 1)!m+1
(3.43)
We perform the integrations over x and s in (3.34) to obtain r
Z Z Z t ! 1 1 ds (3.44) m?1 t p p Q g ( x ) dx jr g j dx lim lim + + t!0 t!0 t Ac (m + 1)!m+1 M t 0 s = krgk1 (3.45) The claim follows by approximating IA with a suitable sequence of smooth functions and applying
(3.4). Lemma 3.5 If the boundary of A is twice continuously differentiable, then the limit in (3.31) exists, and the inequality holds with equality. 15
P ROOF : Approximating IA with a sequence of smooth compactly supported functions gn as in (3.3) and the limits n ! 1 in (3.32-(3.34) yields Z
Ac
Qt I
A dx
=
Z tZ
Z
@A Ac
0
h (x); rxq(s; d(x; y))i dxdyds + kgk1 O(t) ;
(3.46)
where is the outward unit normal to A at the boundary point x. Similarly, (3.35) becomes Z
Ac
h (x); rx q(s; d(x; y))i dy
Z
M
h (x); rx q(s; d(x; y))i+ dy :
(3.47)
We need to estimate the difference between the two sides of (3.47). Consider first, for given x 2 @A, points y 2 Ac where the shortest geodesic from x to y forms an obtuse angle with (x). The smoothness and compactness of @A imply that for x 2 @A, the (m ? 1)-dimensional surface measure of the set
S?(x; r) = fw 2 Sm?1 j expx(rw) 2 Ac; hw; (x)i < 0g
(3.48)
is of the order r 2 uniformly in x 2 @A, which shows that
Z r q(s; r)(r) hw; (x)i? dwdr hrxq(s; d(x; y)); (x)i? dy S? (x;r) Ac 0 2s Z 1 3 Const: 2rs q(s; r)(r)dr p 0 = O( s) :
Z
Z 1
Similar considerations show that Z
A
p hrxq(s; d(x; y)); (x)i+ dy = O( s)
uniformly for x 2 @A. Thus, the difference between the two sides of (3.47) is of order The claim follows by integrating over x 2 @A and s 2 [0; t] . P ROOF
OF
(3.49)
p
O( s).
T HEOREM 4 By Lemma 3.3, Z
Ac
(P t ? Qt )I
A dx
=
Z
A
(P t ? Qt )IAc C V (A) t
(3.50)
for 0 t 1, with a constant C which depends only on M . Hence it is sufficient to prove the claims for Qt in place of P t , which was done in Lemmas 3.4 and 3.5.
16
3.3 Proof of formula (1.17) As discussed in the introduction, (1.17) is related to Theorem 4 via the the approximation
uA(t; x) P t(IA ? IAc )(x) = 1 ? 2P tIA(x)
(3.51)
(recall that P t IA + P t IAc = 1). It remains to prove that (3.51) is justified for sufficiently regular sets. By a theorem of Varadhan [47], the heat kernel on M satisfies 2 lim ? 4 t log p ( t ; x; y ) = d ( x; y ) + t!0
(3.52)
(for more precise results on small-time asymptotics see [30]). This suggests that both sides of (3.51) should be exponentially small in dist (x; Ac )2 =4t as t ! 0, which would justify (3.51) in the interior of A. Lemma 3.7 proved below contains a weaker statement which suffices for our purposes. We first consider the important special case of a half-space in M . Lemma 3.6 Let H
M be a half-space. Then uH (t; x) = IH P t(IH ? IH c ) ;
(3.53)
1 ? uA(t; x) C1 e?dist (x;Ac))2 =4t + C2t
(3.54)
and
holds uniformly for 0 t 1, x 2 H . P ROOF : The two sides of (3.53) agree, since they solve the same Dirichlet problem on H . For the second claim, we write
1 ? uH (t; x0 ) = 2P tIH c (x) = 2Qt IH cZ(x) + O(t) 1 2!m?1 q(t; r)(r) dr + O(t)
(3.55) (3.56) (3.57)
R
where we have used Lemma 3.3 in the second line, and replaced in the third. The last integral is estimated by Z 1
R
q(t; r)(r) dr
Z 1
p
H by a ball of radius R about x
pjjt !m?1
?r q(1; r) re p
t p C (m; )q(1; R= t) ; R= t
dr
(3.58) (3.59)
where we have used (2.3) in the first line, and used that the integrand increases with t in the second. Combining (3.57) and (3.59) gives the claim.
17
Lemma 3.7 Assume that the Ricci curvature of M is bounded below by (m ? 1) (where m is the dimension of M ), and let A M be a Borel set. Then for every fixed Ro > 0, there exist constants C1; C2; C3 such that
P tIAc (x) 1 ? uA(t; x) C1 e?(C2 dist (x;Ac))2 =4t + C3t
(3.60)
holds for all x with dist (x; Ac ) Ro , and all t > 0. P ROOF : The first inequality in (3.60) is just the fact that
uA(t; x) P tIA(x) 1 :
(3.61)
To prove the second inequality, set R = dist (x; Ac ). Clearly, replacing R about x only decreases the exit time. By the first part of Theorem 1,
uA(t; x) uD (t; x0)
A by a disc of radius (3.62)
where D M is a disc of radius R centered at x0 , and is determined by the lower bound on the Ricci curvature. Next, we choose a collection of m +1 half-spaces Hi M such that their intersection contains x0 and is contained in D. Then
1 ? uD (t; x0 )
X
i
(1 ? uHi )(t; x0) :
(3.63)
It can be arranged that dist (x0 ; Hic ) CR, where the constant depends on , m, and (in case < 0) on Ro. We complete the proof by combining inequalities (3.62) and (3.63) with Lemma 3.6 and adjusting the constants. The next step is an estimate near the boundary. Lemma 3.8 Assume that the boundary of A tiable. Then there exists a constant C such that
M
is compact and twice continuously differen
p
sup P t(IA ? IAc ) ? u(t; x) C t
(3.64)
v(t; x) = P t(IA ? IAc ) ? u(t; x)
(3.65)
x2A
for all t > 0. P ROOF : The restriction of
satisfies the heat equation, with zero initial values, and boundary conditions
v(t; x) = P t(IA ? IAc )(x) x 2 @A : 18
(3.66)
By the maximum principle, it suffices to prove that v satisfies the claimed bound on the boundary of A. By Lemma 3.3 we may replace P t with Qt in (3.66). We compute
Qt (I
A ? IAc )
=
Z 1 0
q(t; r)
Z
Sm?1
IA(expx(r!)) ? IAc (expx(r!)) (r; !) d!dr :
(3.67)
The inner integral on the right hand side is the difference between the areas occupied by A and Ac, respectively in the geodesic sphere of radius r about x. Since the boundary of A is twice differentiable, p the fraction of both areas is 1=2 + O(r). It is easy to compute from there that v(t; x) = O( t) as t ! 0 uniformly for t 1 and x 2 @A. Proposition 3.9 If A M is a bounded set with twice continuously differentiable boundary, then
Per(A) =
r
Z 1 ? u (t; x) dx : A 4t A
(3.68)
P ROOF : By Lemma 3.8, the function r
?1 ? u(t; x) ? 2P tI c (3.69) A 4t is bounded uniformly for 0 < t 1 and x 2 A. By Lemma 3.7, it converges to zero pointwise almost everywhere as t ! 0. Lebesgue’s dominated convergence theorem implies that r Z r Z P tI c dx : lim (3.70) 1 ? u ( t; x ) dx = lim A A t!0+ 4t A t!0+ t A Due to the regularity assumptions on A, the limit on the right hand side exists and equals the perimeter of A.
4 Rearrangements on M We now introduce the rearrangements needed in the proof of Theorem 3.
4.1 Definitions Recall that we denote by M the m-dimensional sphere, Euclidean space, or hyperbolic space of constant curvature , and by x a distinguished point of M . The spherical symmetrization A of a set A M of finite volume is the geodesic disc centered at x that has the same volume as A. The spherical symmetrization f of a nonnegative measurable function f on M is the decreasing function of d(x; x ) which is equimeasurable with f and lower semicontinuous. This definition makes sense if f vanishes at infinity in the sense that all its positive level sets have finite volume. As explained in the introduction, we will approximate the spherical rearrangement by sequences of two-point rearrangements. We define a reflection on a metric space M to be an isometry with the following three properties: 19
1. 2.
2x = x for all x 2 M ; M is the disjoint union of the set of fixed points H o, and two half-spaces H ? and H + which are exchanged by , that is, x = x x 2 H o ; H + = H ? ;
3.
(4.1) (4.2)
d(x; y) < d(x; y) for all (x; y) 2 H +.
It is convenient to write
A+ = A \ H + ; A? = A \ H ? ;
(4.3)
as well as
x = +
x 2 H+ ; x 2 H?
x x
and x?
=
x x
x 2 H+ : x 2 H?
(4.4)
The two-point rearrangement of a function f with respect to a reflection is given by
f (x)
=
8 < :
maxff (x); f (x)g ; minff (x); f (x)g ; f (x) ;
x 2 H+ x 2 H? x 2 Ho :
(4.5)
Since level sets of f are related with level sets of f by
A \ H + = (A [ A) \ H + A \ H ? = (A \ A) \ H ? ;
(4.6)
this defines an equimeasurable rearrangement. Note that the definition of f in (4.5) does not require any assumptions on f . Measurability and decay at infinity are needed to show that the spherical symmetrization can be approximated by repeated two-point rearrangements. The spaces M are characterized by large families of reflections. The reflections on a sphere, seen as embedded in R m+1 , are the Euclidean reflections at hyperplanes passing through the center of the sphere. The reflections on a hyperbolic space, seen in the Poincar´e model as the unit disc in R m with the hyperbolic metric, are the conformal inversions at spheres that intersect the bounding unit sphere at right angles. In either case, there exists for each pair of points x; y 2 M a unique reflection with (x) = y , and the reflections generate the entire group of isometries.
20
4.2 Reflections of Brownian paths We next show that two-point rearrangements increase exit times. Consider an open A of finite volume in M . Denote conditional Wiener measure on paths in M joining x with y on the interval t . Let [0; t] by Wx;y
pA(t; x; y) =
Z
t inf IA(Xs) dWx;y
0st
(4.7)
be the conditional Wiener measure of all paths Xs parametrized over [0; t] which connect x with y in A. In other words, pA(t; x; ) is the probability density at time t of a Brownian path starting at x. We remark that PAt has the semigroup property, and is one of the possible definitions of the Dirichlet heat kernel on A. Let be a reflection on M (with a choice of positive half-space H + ), and let A be the twopoint rearrangement of A under . With the above definitions,
uA(t; x) = PAt IA (x) ; and
Z
uA(t; x) dx = =
Z Z Z
(4.8)
pA(t; x; y) dydx
Z
X
H+ H+
pA(t; x; y) dydx ;
where x denotes x+ or x? , and the sum is over the four possible choices of (x ; y ). The following proposition says that the integrand can only increase under two-point rearrangement. Proposition 4.1 Fix a reflection on M . Let A be a Borel measurable subset of M , and let and g be arbitrary functions on M . Then, for any pair of points x; y 2 H + , we have X
f (x)pA(t; x ; y)g(y)
X
f (x )pA (t; x ; y)g (y) ;
f
(4.9)
and similarly
f (x+)pA(t; x+; x+ ) + f (x?)pA(t; x?; x?) f (x+ )pA (t; x+ ; x+) + f (x?)pA (t; x? ; x?) :
(4.10)
Suppose that A is open and A is connected. Then equality in (4.9) implies that either (i) both sides of (4.9) vanish, or (ii) (iii)
A = A, f (x+) f (x?), g(y+) g(y?), or A = A, f (x+) f (x?), g(y+) g(y?) . 21
Equality in (4.10) implies that either (iv) both sides of (4.10) vanish, or (vi) (vi)
A = A and f (x+ ) f (x?), or A = A and f (x+ ) f (x? ).
All equations between sets are understood up to sets of zero capacity.
K
+
H
A
+
H K
σA
-
H
o
-
Figure 1: A set A and its reflection A. The shaded region is the rearranged set
A . To every path assign a path in A by replacing Xs with its Xs on certain subintervals where the path hits K? .
Xs in A we
Proof Let us consider only inequality (4.9), as the proof of (4.10) is similar. We may assume without loss of generality that f and g are characteristic functions of sets B and C , respectively. By definition, the left hand side of (4.9) equals the conditional Wiener measure of sets of paths which start in B , end in C , and join x with y within A, while the right hand side equals the corresponding quantity for A , B , and C . We will construct a map which assigns to every path in A a corresponding path in A , using multiple reflections at H o . It will be clear that the map preserves Wiener measure and is one-to-one, but in general not onto. To be a bit more precise, set
K = A n A ; K + = K \ H + ;
and K ?
= K \ H?
(4.11)
(see Figure 1, and define a sequence of stopping times Tj as follows. If 8
0jXs 2 K ?g; x+ 2 B n B : T1 = 0; T2 = inf fs > 0jXs 2 K +g; : x 2B: T1 = inf fs > 0jXs 2 K g; 22
(4.12)
62 B , both sides of inequality (4.9) vanish, and there is nothing to show. Recursively, if 8 XTj 2 K + : terminate if Xs does not hit K ? until time t, > > < otherwise Tj +1 = inf fs > Tj jXs 2 K ? g; (4.13) ? : terminate if Xs does not hit K + before t, X 2 K > T j > : otherwise Tj +1 = inf fs > Tj jXs 2 K + g: Finally, if Tj is the last of these stopping times and either XTj 2 C n C and Xt 2 C n C , or XTj 2 K ? and Xt 2 C n C , set Tj+1 = t. Let N the number of these stopping times. By Lemma 4.2, which is proved below, the value of N is almost surely finite. Let Sn be the set of sample paths in A with N = n. If n > 1, then, by continuity, every sample path Xs in Sn must if x
hit H o at least once between Tj ?1 and Tj (j
= 2; : : : ; n). Let tj = supft 2 (Tj?1; Tj ) j Xt 2 H og (j = 2; : : : ; n) (4.14) be the last time before Tj that Xt hits H o . Note that tj is not a stopping time (since it depends on Xs for s > tj ) but nevertheless measurable with respect to the filtration associated with Brownian motion up to time t. For each path Xs in Sn , the times tj cut the interval [0; t] into n subintervals, where Xs hits K + and K ? on alternating subintervals. Consider the transformation L on Sn defined by
LXs =
Xs Xs
for s 2 [tj ?1 ; tj ] if Xso else
2 K ? for some so 2 [tj?1; tj ] :
(4.15)
By construction, the image LSn of Sn consists of continuous paths in A starting either at x or at x and ending at y or y . Claim (4.9) follows, since L is one-to-one and preserves Wiener measure. For the statement about the cases of equality, we estimate the difference between the two sides of inequality (4.9). Let E t be the event that a sample path Xs remains in A during the interval [0; t], and that moreover [0; t] contains a subinterval in which Xs remains in A \ H + and hits both K + and K ?. By construction, LSn \ E t = ; ; (4.16) and hence X f (x )pA (t; x ; y)g (y)
? f (x)pA(t; x ; y)g(y)
X
Wxt ;y (E t ) :
(4.17)
The right hand side is positive, since A, and hence A and A \ H + are open and connected. Lemma 4.2 Let be a reflection on M , with a given choice of a positive half-space H + . Let A M be a Borel set, and define the sets K , K +, K ? an the stopping times Ti as in the proof of Proposition 4.1. Then
N =
the number of stopping times Tj
0
0, there corresponds a collection of 2n sets of paths which are clearly pairwise disjoint, and equimeasurable with the original set of paths. Namely, each of the first n intervals (tj ?1 ; tj ) can be reflected independently — since the original paths avoid K , this defines 2n disjoint collections. In summary, X
Wxt ;y ([jnSj ) 2?n
X
p(t; x ; y) ;
(4.19)
which shows the stronger statement that the expected value of N is finite. The following corollary was pointed out to us by Laugesen [34]. Corollary 4.3 Let D be a disc in M , centered at x , with D 6= M . Then the functions pD (t; x; x) and uD (t; x) are strictly decreasing functions of d(x; x ) for t > 0, x 2 D. P ROOF : By symmetry, both pD (t; x; x) and uD (t; x) depend only on t and d(x; x ). Let x and y be two points in D with d(x; x ) < d(y; x). We want to show that
pD (t; x; x) > pD (t; y; y) ; and (4.20) uD (t; x) > uD (t; y) : (4.21) Assume without loss of generality that x lies on the shortest geodesic connecting y with x . Let be the reflection mapping x to y, let H o be the hypersurface of fixed points, and let H + be the half-space containing x and x. Clearly, D = D. To see the first inequality, choose A = D and f = g = If y g. Then f = g = Ifxg , and Proposition 4.1 implies (4.20). Similarly, setting A = D, f = Ifyg , and g = 1 in Proposition 4.1 implies (4.21). 4.3 Reflections on the discrete cube We next prove a discrete version of Theorem 3. Consider the two-point space f+; ?g, with the metric defined by d(+; ?) = 1. The map that exchanges + and ? is a reflection, with no fixed points, and H + = f+g and H ? = f?g as the positive and negative half-spaces. For any function on f+; ?g, let be the corresponding two-point rearrangement of f :
(+) = max f (+); (?)g ; (?) = min f (+); (?)g :
(4.22)
Lemma 4.4 Let 1 ; : : : ; n be nonnegative functions on the set f+; ?g, with the metric, the reflection , and the two-point rearrangement of a function as explained above. For each pair ij , let kij ("; "0) be a decreasing function of the distance between " and "0, i.e.,
kij ("; "0) = aij + bij I"="0 24
with aij ; bij
0. Then the value of J ( 1; : : : ; n) :=
X Y
1in
i("i)
Y 1i 0. Assume additionally that ?o is connected. Then either (i) i = i for i = 1; : : : ; n, or (ii) i = i for i = 1; : : : ; n. P ROOF : We write
X
J ( 1; : : : ; n) = where ? runs over all graphs on the vertices of ?, and
K? =
Y
ij 62?
aij
Y
ij 2?
bij ;
?
C ? C =
4
Y
i2C : i (0) i (1)
i(0) ?
C =
C?
Y
C
i2C
i(0) +
3 2
Y
i2C : i (0) i (1)
which is clearly nonnegative. Hence rearrangement. Moreover,
Y
(4.25)
i = 1; : : : ; n, C runs over the connected components
Defining C accordingly, we may factor 2
K?
i(1)5 4
Y
i2C
i(1) :
Y
i2C : i (0) 0 sufficiently small), we see that f () ? f (1 ) = f () ? f ( ? ) has a definite sign for ?=2 < < =2 and 0 < < =2 ? , which implies that f is monotone on [0; ]. This completes the proof in dimension m = 1. For m > 1, assume that we already know that a function f on a m-dimensional sphere is symmetric under j ? 1 commuting reflections 1 ; : : : ; j ?1 , for some 1 j m. We argue as for m = 1 that the average of ( ) over all reflections whose normals lie in the intersection of the hyperplanes H o (l ) (l = 1; : : : ; j ? 1 vanishes, and by the continuity of there exists j with (j ) = 0; we conclude that f is symmetric under j . By induction over j , we find m perpendicular hyperplanes of symmetry for f . We may assume that these are the hyperplanes x1 = 0; : : : ; xm = 0, which all pass through the north pole (0; : : : ; 0; 1). Since f is symmetric under (x1 ; : : : ; xm; xm+1 ) ! (?x1 ; : : : ; ?xn ; xm+1 , we have () = 0 for any hyperplane containing the xm+1 -axis. It follows that f is symmetric under rotation about the xm+1 -axis. Applying the one-dimensional result to the restriction of f to the x1 ? xm -hyperplane we see that f is spherically decreasing. Case 2: M is a hyperbolic space ( < 0). Some minor adjustments are needed to deal with the non-compactness of the space. For m = 1, each point on the line segment (?1; 1) determines a (hyperbolic) reflection having that point as a fixed point. We specify R the interval to the right as the positive half-space. The integrability of f implies that ( ) ! f as the fixed point approaches 1. By the intermediate value theorem, there exists a reflection 1 with (1 ) = 0. The argument is completed as in the case of the unit circle. For m > 1, we view hyperbolic space as the unit disc in R m with the hyperbolic metric. A reflection is given by a sphere which intersects the boundary of the disc at a right angle, together with a choice of a positive half-space. Consider the family of such spheres with centers on the x1 coordinate axis. Again, the intermediate value theorem implies that the family contains a reflection 1 with (1) = 0. Using the symmetries of hyperbolic space, we may assume that 1 is the Euclidean reflection at the hyperplane x1 = 0. Repeating the argument with families of spheres centered on the other coordinate axes yields a collection of d commuting reflections 1 ; : : : ; m , which we may take to be the reflections at the coordinate hyperplanes. It follows as above that f is rotationally symmetric. The proof is completed by using the one-dimensional case to show that the restriction of f to a line segment passing through the center of the unit disc depends monotonically on the distance from the center.
5 Proof of Theorem 2 5.1 Restatement of inequality (1.9) The following Proposition 5.1 leads to a useful reformulation of Theorem 2. It due to Alvino, Trombetti, and Lions (see [2], Proposition 2.1). 28
Proposition 5.1 Let f and g be nonnegative measurable functions on M which vanish at infinity, and suppose that g is symmetrically decreasing about a point x of M . The following statements are equivalent. 1. For every Borel set B :
Z
B
Z
f (x) dx
B
g(x) dx ;
(5.1)
with equality only if B has zero volume. 2. For every nonnegative convex function F with F (0) = 0, Z
with equality only if F
F (f (x)) dx
Z
F (g(x)) dx ;
(5.2)
g 0.
P ROOF : To see that (5.1) implies (5.2), consider first the case where
F (y) = (y ? a)+ = max(y ? a; 0) with some a 0. Clearly, assumptions, and choose
F
is nonnegative, convex, and
F (0) = 0.
(5.3) Let
B = fx 2 M j f (x) > ag Then
Z
(f (x) ? a)+ dx =
=
Z ZB
f (x) dx ? a V (B )
ZB
g(x) dx ? a V (B )
(g(x) ? a)+ dx ;
f
and
g be as in the (5.4)
(5.5) (5.6) (5.7)
where we have used (5.1) in the second step. Equality can only occur when B is a set of zero volume, that is, when a g (x ). The proof is completed by noting that any nonnegative convex function F with F (0) = 0 can be written as
F (y) =
Z 1 0
(y ? a)+ d(a)
with some nonnegative Borel measure .
29
(5.8)
To see the converse direction, let B be given, and let a be the number so that g (x) a when x 2 B , and g(x) a when x 62 B . Then Z
B
f (x) dx
Z
(f (x) ? a)+ dx + aV (B ) (Zg(x) ? a)+ dx + aV (B ) = g(x) dx ; B
where we have used (5.2) in the second step. Clearly, equality can occur only when zero volume.
(5.9) (5.10) (5.11)
B is a set of
Remark It is clear from the proof that inequalities (5.1) and (5.2) also imply the seemingly stronger statement that Z
Z
F (f (x))h(x) dx
F (g(x))h(x) dx
(5.12)
for F as above, and any nonnegative measurable function h which vanishes at infinity, with equality only if (F g ) h 0.
5.2 Trotter products It is tempting to obtain Theorem 2 directly from Proposition 4.1 by approximating the spherical symmetrization with a sequence of two-point rearrangements. The difficulty is that the exit time has does not depend continuously on A with respect to symmetric difference. We avoid this issue by using rearrangement methods only to prove Theorem 3 (for which the convergence of A1 ;:::;n to A in symmetric difference was sufficient). We turn to the relation between Theorems 2 and 3. We mentioned in the introduction that for a sufficiently regular open set A, the distribution function of the exit time uA(t; x) solves the heat equation on A with Dirichlet boundary conditions. Similar relations hold between exit times and heat kernels on measurable sets. There are, however, several meaningful definitions of “exit time” and “heat equation with Dirichlet boundary conditions” which all coincide for sufficiently regular open sets, but may differ for more general measurable sets. We already defined TA , the exit time of Brownian motion from a Borel set A, and the corresponding heat kernel p(t; x; y ) as the conditional Wiener measure of the set of all paths on [0; t] connecting x with y in A. The heat semigroup PAt is both positivity preserving and positive definite on Ho1 (A) (the closure of Co1 (A) in H 1 ), and vanishes on its complement. Following Stroock [45], we define the first penetration time of Xt into the complement of A by
T~A = inf t > 0 :
Z t
30
0
IA(Xs) < t :
(5.13)
Clearly, T~A is a stopping time, and TA
T~A. Hence, the corresponding distribution function u~A(t; x) = Px(T~A > t) : (5.14)
satisfies for all t, x,
uA(t; x) u~A(t; x) :
(5.15)
If (5.15) holds with equality for all x, then the complement of A is called Kac-regular. There are several reasons for considering u ~ in place of u; see [45, 29] for a detailed discussion. We mention here that u ~A(t; x) is upper semicontinuous in x for any t > 0 and any Borel set A. The function u ~A(t; x) solves the heat equation (1.6) in the sense that
u~A(t; x) =
Z
p~A(t; x; y)IA(y) dy ;
(5.16)
where p~A (t; x; y ) is the kernel of the Trotter product
P~At = nlim (I P t)n !1 A
(5.17)
(see, e.g. [29]). Here, P t is the heat operator on M at time t, and IA is the indicator function of A. The existence of the limit in the strong L2-sense is guaranteed by a theorem of Kato [31], since both multiplication with IA and the heat semigroup P t define contraction semigroups on L2 . The semigroup P~At defined by (5.17) is positivity preserving and positive definite on the intersection of H 1(M ) with the set of L2-functions supported on A, and vanishes on its orthogonal complement. We note that P~At is another candidate for the Dirichlet heat kernel on A. In contrast with uA (t; x) and pA (t; z; y ), the functions u ~A(t; x) and p~A(t; x; y) are not affected ~ if A is changed by a set of zero volume. Let A be the set consisting of all points of density one in A. By a theorem of Stroock [45], any path that hits the complement of A~ almost surely spends positive time outside A~, so that
u~A = u~A~ = uA~ : We will show that
Z
B
Z
u~A(t; x) dx
B
u~A (t; x) dx
(5.18)
(5.19)
for any pair of Borel sets A and B , which shows (by Proposition 5.1) that Z
F (~uA(t; x)) dx
Z
F (~uA (t; x)) dx :
(5.20)
~A but uA = u~A . This is a stronger statement than inequality (1.9), since, in general, uA u t ~ Inequalities (4.9) and (4.10) of Proposition 4.1 also hold for PA . This immediately follows by combining (5.18) with Proposition 4.1, and noting that every point of A~ has density one:
u~A = uA~ uA~ uAf = u~A : 31
(5.21)
An alternate proof for this fact, which we will not present here, uses Varadhan’s inequality (3.52) and Lemma 4.5 to estimate the contributions of various sets of paths to Trotter’s formula. That proof results in a description of the difference between the two sides of (5.21) as the Wiener ~ t. The event E~ t occurs for a path Xs with T~A > t, if the exists some measure of an event E subinterval [t1 ; t2 ] [0; t] during which Xt does not leave H + and spends positive time in both A n A and A n A. We believe that equality in (5.20) should (except for the trivial cases), occur only when A is a ball up to a set of zero volume, but were not able to prove this assertion. This question could be t (E~ t ) > 0 for at least some x; y 2 A under resolved, if one could guarantee the positivity of Wx;y reasonable conditions on A. The following conjecture (or the corresponding statement for p~) would do the trick: Conjecture 5.2 Let A M be a Borel set of finite volume which is symmetric under some reflection . Assume that A consists exactly of its points of density one. Let H + and H ? be the half-spaces corresponding to , and let A+ = A \ H + , and A? = A \ H ? . Then, for x; y 2 H + ,
pA(t; x; y) > 0 =) pA+ (t; x; y) > 0 :
(5.22)
We remark that the conjecture clearly holds in two dimensions (since Brownian motion tends to encircle any point it hits), and for open sets (since A \ H + is connected). P ROOF Z
B
OF
T HEOREM 2 Let A M be an open set, and B an arbitrary Borel set. Then
u~A(t; x) dx =
Z
IB (x)~pA(t; x; y)IA(y) dydx
Z
Z
Z
Z
= nlim IB (x1 )p(t=n; x1; x2 )IA(x2 ) : : : p(t=n; xn?1 ; xn)IA (xn) dx1 : : : dxn !1 = nlim IB (x1 )p(t=n; x1; x2 )IA (x2) : : : p(t=n; xn?1 ; xn)IA (xn) dx1 : : : dxn !1 =
Z
Z
IB (x)~pA (t; x; y)IA (y) dydx
u~A (t; x) dx ; B where we have used (5.16) in the first and last lines, the definition of p~A (t; x; y ) in the second and fourth lines, and Theorem 3 in the third line. We conclude that =
Z
B
uA(t; x) dx
Z
B
uA (t; x) dx :)
By Proposition 5.1, this proves (1.9). Inequality (1.10) follows by setting F (y ) obtain
kuA(t; )kp kuA (t; )kp (p 1) 32
(5.23)
= yp in (1.9) to (5.24)
and then letting p ! 1. To discuss the cases of equality in (5.23), let us assume for the moment that A is connected. If A is not a disc (up to a set of zero volume), then, by Lemma 4.6 proved below, there exists a reflection so that A differs from both A and A by a set of positive volume. Since A is open and connected, the intersection A \ H + is again open and connected. It follows that the difference term of Proposition 4.1 (with f = IB and g = IA ) is positive for every x+ 2 B and y + 2 A , and hence that Z
B
Z
uA(t; x)
g(t)] dt =
(6.4)
Z 1 0
u(g(t); x) dt
(6.5)
where g (t) is the inverse function of G and u(t; x) is the distribution function of the exit time. With Fubini’s theorem we see that Z
B
Ex[G(TA )] dx =
Z 1Z 0
B
u(g(t); x) dxdt :
(6.6)
Applying Theorem 2 to the inner integral completes the proof. The next inequality is a by-product of the proof, rather than the statement of Theorem 2. Corollary 6.2 Let A M be an open set of finite volume. Then Z
f (x) PAt g(x) dx
Z
f (x) PAt g(x) dx
(6.7)
for any pair of nonnegative measurable functions f and g which vanish at infinity. If neither f nor g vanish almost everywhere, then equality occurs only when A is a disc (up to a set of zero capacity), and the restrictions of f and g to A are nondecreasing functions of the distance to the center. 34
P ROOF : For characteristic functions f = IB , g = IC the claim is obtained by replacing u(A(t; x) with PAt IC (and correspondingly u ~A with P~ tIAc ) in the proof of Theorem 2. The general claim (6.7) follows by writing f and g with the layer-cake principle as
f (x) =
Z 1 0
If (x)>s ds ; g(x) =
Z 1 0
Ig(x)>s ds :
(6.8)
Corollary 6.2 immediately implies the Faber-Krahn inequality: Let 1 (A) and 1 (A ) be the first Dirichlet eigenvalues of A and A , respectively, and let A and A be the corresponding eigenfunctions. Then
e?1 (A)
=
Z
A(PAt A) dx
Z
( A) PAt ( A) dx e?1 (A ) ;
(6.9)
where the first equation is the definition of A , the second is (6.7), and the third is the variational characterization of e?1 (A )t . Taking logarithms on both sides yields the Faber-Krahn inequality 1(A) 1 (A). Equality occurs (for open sets A) only when A is a disc. Using (4.10) instead of (4.9 in the proof of Theorem 2 yields the corresponding inequality for the trace of the heat kernel: Theorem 2’ Under the assumptions of Theorem 2, trace PAt trace PAt
;
(6.10)
with equality only when A differs from a disc by a set of zero capacity .
As in the case of Theorem 2, inequality (6.10) holds for any Borel set A M . It also holds with the heat operator PAt (corresponding to the exit time from A) replaced by P~At (corresponding to the first penetration time into the complement of A). Again, we were not able to analyze the cases of equality in this more general setup. Theorem 2’ translates to bounds on the eigenvalues of the Dirichlet Laplacian: Corollary 6.3 (Bounds for eigenvalues) Let A; A M be as in Theorem 2 or Theorem 2’. Denote by i (A) the sequence of eigenvalues of ?A in increasing order, repeated according to multiplicity, and by i (A ) the corresponding sequence for A . Then, for every t > 0, X X e?i(A)t e?i(A )t : (6.11) i0 i0 In particular, the zeta functions of A and A are related by
1 X 1 s s i0 i (A ) i0 i (A)
X
(6.12)
for every s > 0 for which the sums converge. In either case, equality occurs only when A differs from a disc by a set of zero volume. 35
Inequality (6.12) is complementary to the lower bounds for the sums of the reciprocals of the eigenvalues proved by P´olya and Schiffer for two-dimensional domains under conformal mapping normalization [40]. In analogy with the lower bounds of Laugesen and Morpurgo in [33], we suspect that n n X X 1 1 F (A) F (A ) ; (6.13) i i i=1 i=1 for every convex increasing function F with F (0) = 0.
Rearrangement inequalities analogous to the ones in Theorems 2 and 2’ hold for solutions of equations of the form
@t u ? u + V (x)u = 0 ;
u(0; x) = 1
(6.14)
where V is a potential which is bounded below and increases at infinity. We define the symmetric increasing rearrangement of V by ?
V = log e?V
and similarly for V . Let us assume for convenience that (6.14) is given by
V
;
(6.15)
is continuous and nonnegative. Then the solution of
uV (t; x) =
Z
pV (t; x; y) dy ;
(6.16)
where the heat kernel pV can be represented by the Feynman-Kac formula as
pV (t; x; y) =
Z
e?
Rt
t : Wx;y
0 V (x(s)) ds d
(6.17)
Due to the continuity assumption on V , the Trotter product formula ? t=n ?V t=n n PVt = nlim P e !1
(6.18)
results in the same kernel (i.e., we need not distinguish between uV and u ~V ). Using Lemma 4.5 ? V t=n (with ki;i+1 (x; y ) = p(t=n; x; y ) and fi = e for all i) implies that Z
uV (t; x) dx
Z
uV (t; x) dx
(6.19)
for every reflection . The difference between the two sides can be estimated by the limit of the difference terms given in the proof of Lemma 4.5: Z
uV (t; x) dx ?
Z
Z
H+ H+
Z
Z
uV (t; x) dx
Rt e? 0 V (x) ds
(6.20)
Rt 1 ? e? 0 [V (x)?V (x)]+ ds
36
Rt 1 ? e? 0 [V (x)?V (x)]? ds
t dydx ; dWx;y
where the integration is over all continuous paths x(s) in the positive half-space H + . This difference term is positive unless V = V . We conclude that Z
F (uV (t; x)) dx
Z
F (uV (t; x)) dx
for all convex nonnegative functions with F (0) = 0, with equality only when identically, or V is already spherically increasing.
(6.21)
F uV vanishes
7 Concluding remarks We close by formulating two plausible generalizations of Theorems 2 and 3 to manifolds of nonconstant curvature. The first extends a long-standing conjecture due to Aubin [3] that on simply connected manifolds of non-positive curvature, the isoperimetric inequality should hold with the sharp Euclidean constants, with equality only for flat discs. While significant progress has been made in recent years [21, 32, 19], the conjecture is open for smooth manifolds of dimension larger than four. Conjecture 7.1 Suppose that M is a simply connected, complete smooth manifold of non-positive curvature. For a given A M of finite volume, let A be the Euclidean disc centered at x with V (A) = V (A). Then Z
A
F (uA(t; x)) dx
Z
A
F (uA (t; x)) dx
(t 0)
(7.1)
for all nonnegative convex functions F with F (0) = 0, and all t 0. Furthermore, Z
A
P tIA
Z
A
Pt IA
(t 0):
(7.2)
By formula (1.17) and Theorem 4, either (7.1) or (7.2) would imply
Per(A) Per(A ) ;
(7.3)
which is the conjectured isoperimetric inequality described above. The assumption that M is simply connected is crucial, since otherwise M may be compact, in which case A = M , uM 1, P tIM 1, and Per(A) = 0 would contradict (7.1), (7.2), and (7.3). Note that (7.1) does not conflict with the lower bound on uA in (1.4), since the disc D defined there generally has smaller volume than A. Let now M be a compact smooth manifold whose Ricci curvature is bounded below by (m ? 1), where > 0, and set = VV((MM )) 1 : (7.4) 37
For A M , let A
M be a disc with V (A) = V (A ) :
(7.5)
The Gromov-Levy isoperimetric inequality says that
Per(A) Per(A)
(7.6)
at least for sufficiently regular sets (see [20], IV.2, Remark 2). Conjecture 7.2 Let A Then Z
A
and A M be as in the assumptions of the Gromov-Levy inequality. Z
F (uA(t; x)) dx
A
F (uA (t; x)) dx
(t 0)
(7.7)
for all nonnegative convex functions with F (0) = 0, and Z
A
P tI
A
Z
t
PIA A
(t 0) :
(7.8)
Inequality (7.6) is contained in the t ! 0 asymptotics of either (7.7) or (7.8). Note that (7.7) and (7.8) are saturated for t = 0 and in the limit t ! 1 by definition of . For M = M , we have = 1, so that (7.7) reduces to (1.9) of Theorem 2, and (7.8) reduces to (1.24). If M is not isometric to M , then the Gromov-Levy inequality (7.6) is strict. In that case, formula (1.17) and Theorem 4 imply that Conjecture 7.2 holds at least for small values of t.
Acknowledgments We wish to thank A. Baernstein, C. Kerce, M. Loss, E. H. Lieb, R. S. Laugesen, P. McDonald, C. Morpurgo, and L. E. Thomas for many useful discussions, and R. Howard and M. Ledoux for valuable references. Special thanks go to M. Loss for initiating this collaboration by posing the motivating question and establishing the first contact between the authors. We also gratefully acknowledge the hospitality of MSRI during the Spring 1996 program on Convex Geometry and Geometric Functional Analysis. A.B.’s work was supported in part by National Science Foundation grant number DMS-9971493, and by a Sloan Fellowship.
References [1] L. V. Ahlfors. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill Series in Higher Mathematics, 1973. 38
[2] A. Alvino, G. Trombetti, and P.-L. Lions. On optimization problems with prescribed rearrangements. Nonlinear Anal., 13:185–220, 1989. [3] T. Aubin. Probl`emes isop´erim´etriques et espaces de Sobolev. J. Differential Geometry, 15:573–598, 1976. [4] A. Baernstein. A unified approach to symmetrization. Sympos. Math., XXXV:47–91, 1995. [5] A. Baernstein and B. A. Taylor. Spherical rearrangements, subharmonic functions, and *functions in n-space. Duke Math. J., 43:245–268, 1976. [6] C. Bandle. On symmetrization in parabolic equations. J. d’Analyse Math´emathique, 30:98– 112, 1976. [7] C. Bandle. Isoperimetric Inequalities and Applications. Pitman, Boston, 1986. [8] W. Beckner. Sobolev inequalities, the Poisson semigroup, and analysis on the sphere S n . Proc. Natl. Acad. Sci. USA, 89:4816–4819, 1985. [9] W. Beckner. Geometric inequalities in Fourier analysis. Essays on Fourier analysis in honor of Eli M. Stein (Princeton, NJ 1991), 36–68. Princeton Unversity Press. [10] Y. Benyamini. Two-point symmetrization, the isoperimetric inequality on the sphere and some applications. In Texas Functional Analysis Seminar 1983/84, Austin, Texas, pages 53– 76. Longhorn Notes, University of Texas Press, Austin, TX, 1984. [11] M. Berger, P. Gauduchon, and E. Mazet. Le spectre d’une vari´et´e riemannienne. Springer LNM 194, 1971. [12] R. L. Bishop and R. J. Crittenden. Geometry of manifolds. Academic Press, 1964. [13] B. Bollob´as and I. B. Leader. Compressions and isoperimetric inequalities. J. Combin. Theory Ser. A, 56:47–62, 1991. [14] C. Borell. Geometric bounds on the Ornstein-Uhlenbeck velocity process. Z. Wahrsch. Verw. Gebiete, 70:227–237, 1985. [15] H. J. Brascamp, E. H. Lieb, and J. M. Luttinger. A general rearrangement inequality for multiple integrals. J. Funct. Anal., 17:227–237, 1974. [16] F. Brock and A. Yu. Solynin. An approach to symmetrization via polarization. Trans. Amer. Math. Soc., to appear. [17] Yu. D. Burago and V.A. Zalgaller. Geometric inequalities (English translation of the Russian original). Springer Grundlehren 285, 1980. [18] A. Burchard. Cases of equality in the Riesz rearrangement inequality. Annals of Mathematics, 143:499–527, 1996. 39
[19] J. Cao. An isoperimetric comparison theorem for surfaces of variable curvature. In Proceedings of the Third Summer School on Differential Geometry, Partial Differential Equations and Numerical Analysis (in Spanish). Colec. Mem., 7, Acad. Colombiana Cienc. Exact. Fis. Natur., pages 1–7, Bogota, 1996. [20] I. Chavel. Eigenvalues in Riemannian Geometry. Academic Press, 1984. [21] C. B. Croke. A sharp four-dimensional isoperimetric inequality. Comment. Math. Helvet., 59:187–192, 1984. [22] M. Emery. Stochastic calculus in manifolds. Springer, 1990. [23] P. Frankl and Z. F¨uredi. A short proof for a theorem of Harper about Hamming spheres. Discrete Math., 34:311–313, 1981. [24] R. Friedberg and J. M. Luttinger. Rearrangement inequalities for periodic functions. Arch. Rat. Mech. Anal., 61:35–44, 1976. [25] S. Gallot, D. Hulin, and J. Lafontaine. Riemannian Geometry. Springer-Verlag, 1990. [26] P. G¨unther. Einige S¨atze u¨ ber das Volumenelement eines Riemannschen Raumes (german). Publ. Math. Debrecen, 7:78–93, 1960. [27] K. Haliste. Estimates of harmonic measures. Ark. Math, 6:1–31, 1965. [28] G. E. Hardy, J. E. Littlewood, and G. P´olya. Inequalities. Cambridge University Press, London and New York, 1952. [29] I. Herbst and Z. Zhao. Sobolev spaces, Kac-regularity, and the Feynman-Kac formula. In Seminar on Stochastic Processes, 1987 (Princeton, NJ 1987), pages 171–191. Birkh¨auser Boston, Boston, MA, 1988. [30] Y. Kannai. Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Commun. Partial Differential Equations, 2:781–830, 1977. [31] T. Kato. Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. In Topics in Functional Analysis (I. Gohberg and M. Kac, editors), pages 185–195, Academic Press, New York, 1978. [32] B. Kleiner. An isoperimetric comparison theorem. Invent. Math., 108:37–47, 1992. [33] R. Laugesen and C. Morpurgo. Extremals for eigenvalues of Laplacians under conformal mapping. J. Funct. Anal., 155:64–108, 1998. [34] R. Laugesen . Personal communication, 1996. [35] M. Ledoux. Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space. Bull. Sci. Math., 118:485–510, 1994. 40
[36] E. H. Lieb. Existence and uniqueness of the minimizing solution to Chocquard’s nonlinear equation. Studies in Applied Mathematics, 57:93–105, 1977. [37] P. McDonald. Isoperimetric conditions, Poisson problems, and diffusion in Riemannian manifolds. preprint, 1999. [38] C. Morpurgo. Sharp trace inequalities for intertwining operators on S n and R n . Intern. Math. Research Notices, 20:3565-3572, 1999. [39] C. Morpurgo. Conformally invariant Sobolev inequalities and the spectrum of intertwining operators. preprint. [40] F. P´olya and M. Schiffer. Convexity of functionals by transplantation. J. Analyse Math., 3:245–345, 1953-54. [41] M. H. Protter and H. F. Weinberger. Maximum principles in differential equations. Springer, 1967. [42] F. Riesz. Sur une in´egalit´e int´egrale. J. London Math. Soc., 5:162–168, 1930. [43] T. Sakai. Riemannian Geometry. AMS Translations of Mathematical Monographs, 149, 1996. [44] J. Steiner. Einfacher Beweis der isoperimetrischen Haupts¨atze. Crelle J. Reine Angew. Math., 18:281-296, 1838, and Ges. Werke, 2:77-91, Berlin 1882 (in German). [45] D. W. Stroock. The Kac approach to potential theory, Part I. J. Math. Mech., 16:829–852, 1967. [46] G. Talenti. The standard isoperimetric theorem. Handbook of convex geometry, A:73–123, 1993. [47] S.R.S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math., 20:431–455, 1967.
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