Comparison of Exit Moment Spectra for Extrinsic Metric Balls

We prove explicit upper and lower bounds for the L1-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds Pm in ambient Riemannian spaces Nn. We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds Nn themselves.


Introduction
We consider a complete Riemannian manifold (M n , g) and the induced Brownian motion X t defined on M. The L p -moments of the exit time of X t from smooth precompact domains D in the manifold are given by the following integrals (see [7,10,13,14,23]): where the functions u k are defined inductively as the sequence of solutions to the following hierarchy of boundary value problems and for k 2, Here denotes the Laplace-Beltrami operator on (M n , g) . The first solution u 1 (x) is the mean time of first exit from D for the Brownian motion starting at the point x in D, see [7,15].
The quantity A 1,1 (D) is known as the torsional rigidity of D. This name stems from the fact that if D ⊆ Ê 2 , then A 1,1 (D) represents the torque required per unit angle of twist and per unit beam length when twisting an elastic beam of uniform cross section D, see [1] and [28]. The torsional rigidity plays a role in the exit moment spectrum similar to the role played by the first positive Dirichlet eigenvalue in the Dirichlet spectrum. See also [3,4] and [2,32]. Perhaps the most relevant example and token of interest in these problems is given by the St. Venant torsion problem. It is a precise analog of the Rayleigh conjecture about the fundamental tone of a membrane. In 1856 Saint-Venant conjectured that among all cross sections with a given area, the circular disk has maximum torsional rigidity. The first proof of this conjecture was given by G. Polya in 1948, see [29] and [28].
In view of the isoperimetric inequality for domains in Ê 2 and in view of the domain monotonicity of A 1,1 (D) it thence follows, that among all cross sections with a given circumference, the circular disk has maximum torsional rigidity. In other words, in Since we shall similarly only be concerned with p = 1, and since our results for the higher moments in the exit time moment spectrum are also in this sense isoperimetric type inequalities we define: Definition 1. 1 The isoperimetric exit moment spectrum of D is defined by If we formally define u 0 (x) = 1 for all x ∈ D, then all the solutions u k -including u 1 (x)-are uniformly generated by induction from Eq. 1.3. With this natural extension of the u k sequence we thence have from Definition 1.1: , (1.5) which is precisely the isoperimetric quotient for D.
We will henceforth refer to the list , · · · } as the extended isoperimetric exit moment spectrum of D.
Here we restrict our study to be concerned with the exit moment spectra of a specific kind of domains, the so-called extrinsic R-balls D R defined in submanifolds P m which are properly immersed into ambient Riemannian manifolds N n with controlled sectional curvatures.
Suppose p is a pole in N, see [30]. An extrinsic p-centered R-ball D R of the submanifold P is then, roughly speaking, the intersection between the submanifold and the ambient metric R-ball centered at p in the ambient space N.
The isoperimetric relations satisfied by these extrinsic balls have been studied and applied in a number of contexts, see e.g. [11,18,19,21,22,27]. In these works we use R-balls and R-spheres in tailor made rotationally symmetric (warped product) model spaces M m w as comparison objects. The simplest settings considered are given by the minimal submanifolds P m in real space forms Ã n (b) of constant sectional curvature b 0. In these specific cases we have the following isoperimetric inequalities, see [5,15,16,18,27]: denote, respectively, the geodesic R-ball and the geodesic R-sphere in the real space form Ã n (b).
With the notation introduced above we may state this result as follows: In passing we note that when equality is attained in Eq. 1.7 for some fixed radius R, and when the ambient space N n is the hyperbolic space À n (b), b < 0, then the minimal submanifold itself is a totally geodesic hyperbolic subspace À m (b) of À n (b), see [27]. Thus, in analogy with the St. Venant torsion problem-and in analogy with the classical isoperimetric problem itself-we also obtain strong rigidity conclusions from equalities in these isoperimetric estimates.

A First Glimpse of the Main Results
In the present paper we extend the inequalities 1.7 and prove isoperimetric inequalities of this type for every element A k (D R ), k 0, in the extended isoperimetric exit moment spectrum for extrinsic metric balls. Before stating this extension for minimal submanifolds in constant curvature ambient spaces below we note, that this is but a shadow of our main results, Theorem 4.1 and Theorem 4.2 in Section 4, where we prove both upper and lower bounds for the isoperimetric exit moment spectrum under more relaxed curvature conditions. The main condition for the lower bounds is a lower bound on the sectional curvatures of the ambient space and the upper bounds for the spectrum stem similarly from an upper bound on the ambient sectional curvatures. Moreover, in our general results the submanifolds are not assumed beforehand to be minimal.
. When the ambient space is hyperbolic space À n (b), b < 0, then equality in Eq. 1.8 for some radius R and for some value of k 0 implies that D R -and in fact all of P m -is totally geodesic in À n (b ), so that equality is attained for all k and for every smaller p-centered extrinsic ball in P m .
In order to illustrate our use of the upper and lower bounds on the ambient space sectional curvatures in the more general setting alluded to above-and since we believe that the following result is also in itself of independent interest-we extract here a purely intrinsic consequence from the proofs of Theorems 4.1 and 4.2. The notion of radial sectional curvatures and the geometric analytic notions associated with the model spaces are defined precisely in Section 2 below.

Theorem 1.3 Let B N
R be a geodesic ball of a complete Riemannian manifold N n with a pole p and suppose that the p-radial sectional curvatures of N n are bounded from below (respectively from above) by the p w -radial sectional curvatures of a w-warped model space M n w . Then the extended isoperimetric exit moment spectrum of B N R satisf ies for all k 0 the following respective inequalities: where B w R is the geodesic ball in the model space M n w . Equality in Eq. 1.9 for some k 0 implies that B N R is isometric to the warped product model ball B w R and hence again that equality is attained for all k 0 and for every smaller p-centered extrinsic ball in P m .
The proofs of these results, Theorems 1.2 and 1.3 are given in Section 5 at the end of this paper.

Preliminaries and Comparison Setting
We first consider a few conditions and concepts that will be instrumental for establishing our results.

Extrinsic Metric Balls
We consider a properly immersed m-dimensional submanifold P m in a complete Riemannian manifold N n . Let p denote a point in P and assume that p is a pole of the ambient manifold N. We denote the distance function from p in N n by Since p is a pole there is-by definition-a unique geodesic from x to p which realizes the distance r(x). We also denote by r the restriction r| P : P −→ Ê + ∪ {0}. This restriction is then called the extrinsic distance function from p in P m . The corresponding extrinsic metric balls of (sufficiently large) radius R and center p are denoted by D R ( p) ⊆ P and defined as any connected component which contains p of the set: Since P m is assumed to be unbounded and properly immersed into N, we have for every R that B R ( p) ∩ P = P.

The Curvature Bounds
We now present the curvature restrictions which constitute the geometric framework of our investigations.

Definition 2.1 Let p be a point in a Riemannian manifold M and let
The sectional curvature K M (σ x ) of the two-plane σ x ∈ T x M is then called a p-radial sectional curvature of M at x if σ x contains the tangent vector to a minimal geodesic from p to x. We denote these curvatures by K p,M (σ x ).
In order to control the mean curvatures H P (x) of P m at distance r from p in N n we introduce the following definition: The p-radial mean curvature function for P in N is defined in terms of the inner product of H P with the N-gradient of the distance function r(x) as follows: In the following definition, we are going to generalize the notion of radial mean convexity condition introduced in [11,22]. Definition 2.3 (see [22]) We say that the submanifold P satisfies a radial mean convexity condition from below controlled by a smooth radial function h 1 (r) (respectively, from above controlled by a smooth radial function h 2 (r)) from the point p ∈ P such that The radial bounding functions h 1 (r) and h 2 (r) are related to the global extrinsic geometry of the submanifold. For example, it is obvious that minimal submanifolds satisfy a radial mean convexity condition from above and from below, with bounding functions h 2 = 0 and h 1 = 0. On the other hand, it can be proved, see the works [6,22,26,31], that when the submanifold is a convex hypersurface, then the constant function h 1 (r) = 0 is a radial bounding function from below.
The final notion needed to describe our comparison setting is the idea of radial tangency. If we denote by ∇r and ∇ P r the gradients of r in N and P respectively, then we have the following basic relation: When the submanifold P is totally geodesic, then ∇r = ∇ P r in all points, and, hence, ∇ P r = 1. On the other hand, and given the starting point p ∈ P, from which we are measuring the distance r, we know that ∇r( p) = ∇ P r( p), so ∇ P r( p) = 1. Therefore, the difference 1 − ∇ P r quantifies the radial detour of the submanifold with respect the ambient manifold as seen from the pole p. To control this detour locally, we apply the following Definition 2. 4 We say that the submanifold P satisfies a radial tangency condition at p ∈ P when we have a smooth positive function g(r) so that Remark 2.5 Of course, we always have Remark 2. 6 We observe, that the assumption ∇ P r(x) > 0 implies that the properly immersed extrinsic ball D R in P can have only trivial topology. It follows directly from Theorem 3.1 in [24], since r(x) is a smooth function on P − {p} without critical points, that D R is diffeomorphic to the standard unit ball in R m .

Model Spaces
As mentioned previously, the model spaces M m w serve first and foremost as comparison controller objects for the radial sectional curvatures of N n . Definition 2.7 (See [8,9]) A w−model M m w is a smooth warped product with base (2.5) Note that for b > 0 the function Q b (r) admits a smooth extension to r = π/ √ b . For b 0 any center point is a pole.
In the papers [8,9,20,21,25], we have a complete description of these model spaces and their key properties. In particular the sectional curvatures K pw,Mw in the radial directions from the center point p w are determined by the radial function and the mean curvature of the distance sphere of radius r from the center point is

The Isoperimetric Comparison Spaces
Given the bounding functions g(r), h(r) (when in the following no specific index is given, then h represents any one of the bounding functions h 1 (r) or h 2 (r)), and the ambient curvature controller function w(r) described is Sections 2.2 and 2.3, as in [11,22] we construct new model spaces C m w,g,h . For completeness, we recall their construction: Definition 2.9 Given a smooth positive function g(r(x)) > 0 satisfying g(0) = 1 and g(r(x)) 1 for all x ∈ P, a stretching function s is defined as follows dt . where the auxiliary function (r) satisfies the following differential equation: Note that, when g(r) = 1 for all r and h(r) = 0 for all r, then the stretching function s(r) = r and W(s(r)) = w(r) for all r. In this case C m w,g,h simply reduces to the w warped model space M m w . The spaces M m W = C m w,g,h will be applied as those spaces, where our bounds on the exit moment spectrum are attained.

Balance Conditions
In the paper [11] we considered and applied a balance condition on the general model spaces M m W , that we shall also need in the sequel: Here q W (s) is the isoperimetric quotient function . (2.13)

Remark 2.12
In particular the w-balance condition for M m W = C m w,g,h implies that η w (r) − h(r) > 0 (2.14) wherever g(r) > 0.

Remark 2.13
The above definition of a (strict) w−balance condition for M m W is clearly an extension of the balance condition (from below) as defined in [21,Definition 2.12]. The condition in that paper is obtained precisely when g(r) = 1 and h(r) = 0 for all r ∈ [0, R] so that r(s) = s, W(s) = w(r), and q w (r)η w (r) 1/m .

Comparison Constellations
We now present the precise settings where our main results take place, introducing the notion of comparison constellations as they were previously defined in [11]. For that purpose we shall bound the previously introduced notions of radial curvature and tangency by the corresponding quantities attained in the special model spaces, the isoperimetric comparison spaces defined above.
Definition 2.14 Let N n denote a complete Riemannian manifold with a pole p and distance function r = r(x) = dist N ( p, x). Let P m denote an unbounded complete and properly immersed submanifold in N n . Suppose p ∈ P m and suppose that the following conditions are satisfied for all x ∈ P m with r(x) ∈ [0, R]: 1. The p-radial sectional curvatures of N are bounded from below by the p w -radial sectional curvatures of the w−model space M m w : ) .
2. The p-radial mean curvature of P is bounded from below by a smooth radial function h 1 (r): 1 (r(x)) .
3. The submanifold P satisfies a radial tangency condition at p ∈ P, with smooth positive radial function g(r) such that A constellation bounded from above is given by the following dual setting defining the special W-model spaces C m w,1,h2 with the uniform choice g = 1:

Definition 2.15
Let N n denote a Riemannian manifold with a pole p and distance function r = r(x) = dist N ( p, x). Let P m denote an unbounded complete and properly immersed submanifold in N n . Suppose the following conditions are satisfied for all x ∈ P m with r(x) ∈ [0, R]: 1. The p-radial sectional curvatures of N are bounded from above by the p w -radial sectional curvatures of the w−model space M m w : (r(x)) .
2. The p-radial mean curvature of P is bounded from above by a smooth radial function h 2 (r): Let C m w,1,h2 denote the W-model with the specific warping function W : π(C m w,1,h2 ) → Ê + constructed in Definition 2.10 via w, g = 1, and h = h 2 . Then the triple {N n , P m , C m w,1,h2 } is called an isoperimetric comparison constellation bounded from above on the interval [0, R].

Theorem 2.16 Let N n be a manifold with a pole p, let M m
w denote a w−model space with center p w . Let us consider a smooth function f : Ê + → Ê and the restricted distance function from the pole r : P → Ê.
Then we have the following dual Laplacian inequalities for the modif ied distance functions (i) Suppose that every p-radial sectional curvature at x ∈ N − {p} is bounded by the p w -radial sectional curvatures in M m w as follows: Then we have for every smooth function f (r) with f (r) 0 for all r, (respectively f (r) 0 for all r):

19)
where H P denotes the mean curvature vector of P in N.
(ii) Suppose that every p-radial sectional curvature at x ∈ N − {p} is bounded by the p w -radial sectional curvatures in M m w as follows: (2.20) Then we have for every smooth function f (r) with f (r) 0 for all r, (respectively f (r) 0 for all r): where H P denotes the mean curvature vector of P in N.

Exit Moment Spectra of R-balls in Model Spaces
We have the following result concerning the exit moment spectrum of a geodesic R-ball B w R ⊆ M m w :

Proposition 3.1 Letũ k be the solution of the boundary value problems 1.3, def ined on the geodesic R-ball B w R in a warped model space M m w . Thenũ
Therefore, Proof A straightforward computation gives the boundary conditionũ k (R) = 0 is satisfied and as a consequence of the Maximum Principle for elliptic operators, the functionsũ k are the only solutions to the boundary value problems defined on B w R and given by Eq. 1.3.
Therefore, applying the Divergence Theorem, we obtain where S w R is the geodesic R-sphere in M m w , and the claim is proved.

A Key Lemma
Let us consider now an isoperimetric comparison model space M m W and letũ W k be the radial functions given by Eq. 3.2, which are the solutions of the problems 1.3 defined on the geodesic ball B W s(R) . We define the functions f k : where s is the stretching function given by Eq. 2.8.
Then we have the following lemma, which will be of instrumental importance for the proofs of the main results below:

Lemma 3.2 Let M m W be an isoperimetric comparison model space that is w-balanced in the sense of Def inition 2.11 with h
If k 2 or if M m W is strictly balanced, then the inequality is in fact a strict inequality: Proof By Eq. 2.8, f k (r) =ũ W k (s(r))(s (r)) 2 +ũ W k (s(r))s (r) Since the functionsũ W k are the solution of the problems 1.3 on B W s(R) , using Eq. 3.4, Taking into account the explicit construction of M m W , i.e. Eqs. 2.9 and 2.10, a straightforward computation shows that and consequently, + (η w (r)g(r) + g (r))ũ W k (s(r)).
Replacing the expression ofũ W" k (s(r)) in Eq. 3.6 we obtain that and Since f k (r) =ũ W k (s(r))/g(r) < 0, the functions f k are strictly decreasing in ] 0, R ] for all k 1 and consequently by Eq. 3.2 , (3.9) where the last equality is obtained using Eq. 2.13. Note that we can assume that u 0 ≡ 1 and therefore f 0 ≡ 1 too, so that only in the case k = 1 can we have equality in Eq. 3.9. Finally, combining the above inequality with Eq. 3.7 we get: by the balance condition 2.12-respectively the strict balance condition-and the fact that g and f k−1 are positive functions.

Lower and Upper Bounds for the Isoperimetric Exit Moments
We are now ready to prove the first of our main results.
Theorem 4.1 Let {N n , P m , C m w,g,h1 } denote a comparison constellation bounded from below in the sense of Def inition 2.14. Assume that M m W = C m w,g,h1 is w-balanced in the sense of Def inition 2.11. Let D R be an extrinsic R-ball in P m , with center at a point p ∈ P which also serves as a pole in N. According to Remark 2.6, our assumption g(r(x)) > 0 implies trivial topology of the extrinsic ball D R . For all k 0, i.e. for the extended exit moment spectrum, we also have: Proof Consider the functions f k =ũ W k • s of Lemma 3.2. Let r denote the smooth distance to the pole p on M. We define v k : D R → Ê by v k (q) = f k (r(q)).
Using Theorem 2.16, Lemma 3.2, Eq. 3.7 and the fact that f k (r) 0, we have that Now, we are going to prove inductively that if we denote by u k the solutions of the hierarchy of boundary value problems on D R given by Eq. 1.3, then v k u k on D R .
For k = 1, since f 0 is assumed to be identically 1, inequality 4.2 gives us that Applying the Maximum Principle we conclude that v 1 u 1 on D R . Suppose now that v k u k on D R , then as a consequence of inequality 4.2 we get and (v k+1 − u k+1 ) = 0 on ∂ D R , so applying again the Maximum Principle we have v k+1 u k+1 . Summarizing we have so far: v k u k and P v k P u k on D R for all k 1. Taking these inequalities into account and applying Divergence theorem we then get Since f k+1 (R) =ũ W k+1 (s(R))/g(R) 0 and ∇ P r g(r), we conclude that

4)
where B W R is the geodesic ball in C m w,1,h2 . If M m W is strictly balanced then equality in Eq. 4.4 for some f ixed radius R and some f ixed k 0 implies that D R is a geodesic cone in N and that the equality is in fact attained for all k 0 and for every smaller p-centered extrinsic ball in P m .
Proof The proof of this theorem follows closely the lines of the proof of Theorem 4.1. Since there are, however, some crucial and obvious differences we take this space to point them out explicitly. In the present case we have s(r) = r because g(r) ≡ 1 (see Eq. 2.8). Therefore f k+1 =ũ W k+1 so that v k+1 =ũ W k+1 • r. Thence v k+1 is the solution of the boundary value problems 1.3 on B W R transplanted to D R .
The new geometric setting given by the comparison constellation bounded from above gives us now: Again we prove inductively that if u k denotes the family of solutions of the hierarchy of boundary value problems on D R given by Eq. 1.3, then v k u k on D R . For k = 1, since f 0 is still assumed to be identically 1, inequalities 4.6 and 4.5 give us that Suppose now that v k u k on D R , then again as a consequence of inequalities 4.5 and 4.6 we get P v k+1 −(k + 1) v k −(k + 1) u k = P u k+1 , and (v k+1 − u k+1 ) = 0 on ∂ D R , so applying again the Maximum Principle we have v k+1 u k+1 .
We have: v k u k and P v k P u k on D R for all k 1. The Divergence theorem gives the claim in Eq. 4.4: Suppose that M m W is strictly balanced and that we have equality in Eq. 4.4. Then we must have equalities in Eqs. 4.8, 4.7, and 4.6 as well. In particular the last mentioned equality gives ∇ P r ≡ 1 because we have from Eq. 3.2 that ( f k (r) − f k (r)η w (r)) > 0. Therefore ∇ P r = ∇ N r and D R is a geodesic cone swept out by the radial geodesics from p.

Intrinsic and Constant Curvature Results
In this short section we finally show how to obtain the results stated in the introduction from Theorems 4.1 and 4.2.
Proof of Theorem 1.2 This theorem follows immediately from Theorem 4.2 once we show that the comparison space M m W is strictly w-balanced. But we have g = 1 and h 2 = 0 so that M m W is M m Remark 2.13. The equality case gives even more significant rigidity: Since D R is here a minimal geodesic cone, then by analytic continuation D R and in fact all of P m is totally geodesic in the hyperbolic space À n (b), see [15].
Proof of Theorem 1. 3 We consider the intrinsic versions of (the proofs of) Theorem 4.1 and Theorem 4.2 assuming that P m = N n . In this case, the extrinsic distance to the pole p becomes the intrinsic distance in N, so, the extrinsic domains D R become the geodesic balls B N R of the ambient manifold N and for all x ∈ P we have: ∇ P r(x) = ∇r(x), As a consequence, ∇ P r ≡ 1, so g(r(x)) = 1 and C(x) = h 1 (r(x)) = h 2 (r(x)) = 0. The stretching function becomes the identity s(r) = r, W(s(r)) = w(r), and the isoperimetric comparison spaces C m w,g,h1 and C m w,1,h2 reduce to the same auxiliary model space M m w . Since ∇r ≡ 1, we do not need to control the sign of ( f k (r) − f k (r)η w (r)) in Eqs. 2.19 and 2.21. For this reason it is not necessary to assume any w-balance conditions in these cases. The theorem and the two-sided bounds in Eq. 1.9 then follow directly from the inequalities in Theorem 4.1 and Theorem 4.2. If equality is satisfied, then B N R has all its radial curvatures equal to the radial curvatures of M m w , hence they are isometric, see [21].