Volume growth of submanifolds and the Cheeger Isoperimetric Constant

We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above .


INTRODUCTION
The Cheeger isoperimetric constant I ∞ (M ) (see [5]) of a non-compact Riemannian manifold of dimension n ≥ 2 is defined as: where Ω ranges over open submanifolds of M possessing compact closure and smooth boundary, Vol(∂Ω) denotes the (n − 1)-dimensional volume of the boundary ∂Ω, and Vol(Ω) denotes the n-dimensional volume of Ω, (concerning this definition, see also [3] and [4]). This paper focuses on obtaining sharp upper and lower bounds for the Cheeger isoperimetric constant I ∞ (P ) of a complete submanifold P with controlled mean curvature and properly immersed in an ambient manifold N with sectional curvatures bounded from above and which possess at least one pole.
As a consequence of these upper and lower bounds, and as a preliminary view of our main theorems (Theorems 3.2 and 3.3 in section §.3), we present the following results, which constitute a particular case of them when a complete, non-compact and minimal submanifold properly immersed in a Cartan-Hadamard manifold is considered. In contrast, if we focus on compact and minimal submanifolds of a Riemannian manifold satisfying other geometric restrictions, we refer to the work [12], where certain isoperimetric inequalities involving these submanifolds have been proven.
Theorem A. Let P m be a complete non-compact and minimal submanifold properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above as K N ≤ b ≤ 0, and suppose that Sup t>0 ( Then Theorem B. Let P m be a complete non-compact and minimal submanifold properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above The lower bounds for I ∞ (P ) in Theorem B come from direct application of the divergence theorem to the Laplacian of the extrinsic distance defined on the submanifold using the distance in the ambient manifold, following the arguments of Proposition 3 in [21] and of Theorem 6.4 in [4].
On the other hand, the upper bounds in Theorem A were obtained by assuming that the (extrinsic) volume growth of the submanifold is bounded from above by a finite quantity. As we shall see in the corollaries, when the submanifold is a minimal immersion in the Euclidean space or when we are dealing with minimal surfaces in the Euclidean or the Hyperbolic space, this crucial fact relates Cheeger's constant I ∞ (P ) with the total extrinsic curvature of the submanifold P B P m dσ, in the sense that the finiteness of this total extrinsic curvature implies the upper bounds for Cheeger's constant, using the results in [1], [6] and [8].
These lower and upper bounds of I ∞ (P ) given in Theorems 3.2 and 3.3 come from comparisons for the Laplacian of the extrinsic distance defined on the submanifold, and the techniques used to obtain these comparisons are based on the Hessian analysis of this restricted distance function. When the extrinsic curvature of the submanifold is bounded (from above or from below), this analysis focuses on the relation, given in [10], between the Hessian of this function and these (extrinsic) curvature bounds, thus providing comparison results for the Hessian and the Laplacian of the distance function in the submanifold.
The model used in these comparisons is constructed from the corresponding values for these operators computed for the intrinsic distance of a rotationally symmetric space whose sectional curvatures bound the corresponding curvatures of the ambient manifold.
We shall see that the Cheeger constant I ∞ (P ) is bounded by the limit of some isoperimetric quotient determined by the geodesic r-balls in these model spaces, which involves the mean curvature of the submanifold.
1.1. Outline of the paper. In section §.2 we present the basic definitions and facts concerning the extrinsic distance restricted to a submanifold, and about the rotationally symmetric spaces used as a model for comparison. We also present the basic results regarding the Hessian comparison theory of restricted distance function that will be used. This section finishes with the description of the isoperimetric context where the results hold. Section §.3 is devoted to the statement and proof of the two main Theorems 3.2 and 3.3 and three corollaries are stated and proven in the final section §.4.

PRELIMINAIRES
2.1. The extrinsic distance. We assume throughout the paper that P m is a complete, non-compact, properly immersed, m-dimensional submanifold in a complete Riemannian manifold N n which possesses at least one pole o ∈ N . Recall that a pole is a point o such that the exponential map , and this distance is realized by the length of a unique geodesic from o to x, which is the radial geodesic from o. We also denote by r the restriction r| P : P → R + ∪ {0}. This restriction is called the extrinsic distance function from o in P m . The gradients of r in N and P are denoted by ∇ N r and ∇ P r, respectively. Let us remark that ∇ P r(x) is just the tangential component in P of ∇ N r(x), for all x ∈ S. Then we have the following basic relation: Definition 2.1. Given a connected and complete submanifold P m properly immersed in a manifold N n with a pole o ∈ N , we denote the extrinsic metric balls of radius t > 0 and center o ∈ N by D t (o). They are defined as the intersection denotes the open geodesic ball of radius t centered at the pole o in N n . Remark a. The extrinsic domains D t (o) are precompact sets (because in the definition above it was assumed that the submanifold P is properly immersed), with smooth boundary ∂D t (o). The assumption on the smoothness of ∂D t (o) makes no restriction. Indeed, the distance function r is smooth in N \ {o} since N is assumed to possess a pole o ∈ N . Hence the restriction r| P is smooth in P and consequently the radii t that produce smooth boundaries ∂D t (o) are dense in R by Sard's theorem and the Regular Level Set Theorem.
We now present the curvature restrictions which constitute the geometric framework of our study.

Definition 2.2. Let o be a point in a Riemannian manifold N and let
In order to control the mean curvatures H P (x) of P m at distance r from o in N n we introduce the following definition: The o-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:

Model Spaces.
The model spaces M m w are rotationally symmetric spaces which serve as comparison controllers for the radial sectional curvatures of the ambient space N n . Definition 2.4 (see [11], [10]). A w−model M m w is a smooth warped product with base Note that for b > 0 the function w b (r) admits a smooth extension to r = π/ √ b. For b ≤ 0 any center point is a pole.
(see [10], [11] [18]). Moreover, the mean curvature of the distance sphere of radius r from the center point is [19]): In particular, in [16] we introduced, for any given warping function w(r) , the isoperimetric quotient function q w (r) for the corresponding w−model space M m w as follows: where B w r and S w r denotes the metric r−ball and the metric r−sphere in M m w respectively.

Hessian comparison analysis of the extrinsic distance.
This subsection offers a corollary of the Hessian comparison Theorem A in [10], which concerns the bounds for the Laplacian of a radial function defined on the submanifold (see [13] and [20] for detailed computations, see also [14]).
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 .

The Isoperimetric Comparison space.
We are going to define a new kind of model of the quotient determined by its warping function (this quotient is given in terms of the mean curvature of the geodesic spheres in M m W and the bounds on the mean curvature of the submanifold P ) will serve as estimate for the isoperimetric constant I ∞ (P ).
and the following boundary condition: By using equation (2.8), it is straightforward to see that W (r) = 0 only at r = 0, so M m W has a well-defined pole o W at r = 0. Moreover, W (r) > 0 for all r > 0.
Note that when h(r) = 0 for all r, then W (r) = w(r) for all r, so M m W becomes a model space with warping function w, M m w .
Definition 2.7. The model space M m W is w−balanced from above (with respect to the intermediary model space M m w ) iff the following holds for all r ∈ [ 0, R ]: so M m w is w-balanced from below. However, it is easy to see that η w (r) = 2re r 2 +1 e r 2 +r−1 is an increasing function from a given value r 0 > 0 and, hence, does not satisfy second inequality in (2.9) and is therefore not w-balanced from above.
, be the Euclidean or Hyperbolic space, with warping function w b (r). Let us consider h(r) = 0 ∀r. In this context, these spaces are isoperimetric spaces with themselves as intermediary spaces, and satisfy both balance conditions given in definitions 2.7 and 2.8 (see [16]).

Comparison Constellations.
We now present the precise settings where our main results take place, and introduce the notion of comparison constellations. It is straigthforward to see that, under these restrictions, W = w b and, hence, is w b -balanced from above and from below.
Let us consider again a Cartan-Hadamard manifold N , with sectional curvatures bounded above by a ≤ 0. Let P m be a properly immersed submanifold in N such that where, by fixing is w a -balanced from above and from below.

MAIN RESULTS
Before stating our main theorems, we find the upper bounds for the isoperimetric quotient defined as the volume of the extrinsic sphere divided by the volume of the extrinsic ball, in the setting given by the comparison constellations. Theorem 3.1. (see [13], [19], [15]) Consider an isoperimetric comparison constellation {N n , P m , M m W }. Assume that the isoperimetric comparison space M m W is w-balanced from below. Then .

Furthermore, the function f (t) = Vol(Dt)
Vol(B W t ) is monotone non-decreasing in t. Moreover, if equality holds in (3.1) for some fixed radius t 0 > 0, then D t0 is a cone in the ambient space N n .
The following is the upper bound for the Cheeger constant of a submanifold P : .
In particular, let P m be a complete and minimal submanifold properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above as K N ≤ b ≤ 0, and suppose that Sup t>0 ( Vol(Dt) Proof. Let us define By the co-area formula and applying Theorem 3.1 it is easy to see that F (t) is a nonnegative function. Moreover, Vol(Dt) Vol(B W t ) is non-decreasing (see [15]). Integrating between t 0 > 0 and t > t 0 : But on the other hand, from hypothesis (2) and the fact that Vol(Dt) and hence there is a monotone increasing sequence {t i } ∞ i=0 tending to infinity, such that: Let us consider now the exhaustion {D ti } ∞ i=1 of P by these extrinsic balls. By using equation (1.1), we have that, On the other hand, since lim i→∞ F (t i ) = 0, then 3) follows inmediately taking into account that, as was shown in the examples above, when P is minimal in a Cartan-Hadamard manifold, then considering h(r) = 0 ∀r and considering w(r) = w b (r), we have that {N n , P m , K m (b)} is a comparison constellation, with K m (b) w b -balanced from below.
As by hypothesis, Sup t>0 ( Vol(Dt) Vol(B b,m t ) ) < ∞ and we have that, we now apply inequality (3.2). Now, we have the following result, which is a direct extension to Yau's classical result (see [21]) on minimal submanifolds, using the same techniques as in [4]: In particular, let P m be a complete and minimal submanifold properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above as Then Proof. From equation (2.7) in definition 2.6 of the isoperimetric comparison space, we have: On the other hand, from Theorem 2.5: (3.13) Then, applying (3.12) (3.14) Now, if we consider a domain Ω ⊆ P , which is precompact and with smooth closure, we have, given its outward unitary normal vector field, ν: ν, ∇ P r ≤ 1 hence by applying divergence Theorem, and taking into account that W ′ (r) W (r) is non-increasing for any domain Ω, we have the result. Inequality (3.11) follows inmediately taking into account that, as in the proof of Theorem 3.2 and in the examples above, when P is minimal in a Cartan-Hadamard manifold, then we have that {N n , P m , K m (b)} is a comparison constellation (h(r) = 0 ∀r and w(r) = w b (r)), with the isoperimetric comparison space used as a model M m  where χ(P ) is the Euler characteristic of P and Φ is the Gauss-Bonnet-Chern form on P , and S b,m−1 t denotes the geodesic t-sphere in K m (b).

Remark d.
Note that, on applying inequality (3.1) in Theorem 3.1 to the submanifold P the theorem above, we conclude that, under the assumptions of Theorem 4.1, we have the following bound for the volume growth On the other hand, we have that Chern-Osserman Inequality is satisfied by complete and minimal surfaces in a simply connected real space form with constant sectional curvature b ≤ 0, K n (b). Namely Theorem 4.2. (see [1], [6] and [8]. For an alternative proof, see [9]). Let P 2 be an complete minimal surface immersed in a simply connected real space form with constant sectional curvature b ≤ 0, K n (b). Let us suppose that P B P 2 dσ < ∞. Then (1) P has finite topological type.
where χ(P ) is the Euler characteristic of P .

The Corollaries.
In this subsection, we are going to state and prove the following results, which are direct consequences of the main theorems in Section §.3 and Theorems 4.1 and 4.2 in Subsection §4.1.
The first Corollary 4.3 is a direct application of Theorems 3.2 and 3.3.  Proof. This is a direct consequence of inequalities (3.3) and (3.11) in Theorem 3.  Proof. In this case, taking h(r) = 0 ∀r and w 0 (r) = r, we have that {R n , P m , R m } is a comparison constellation bounded from above, with R m w 0 -balanced from below. Hence, we apply Theorem 3.1 to obtain Hence, applying Theorem 3.2, I ∞ (P ) ≤ 0, so I ∞ (P ) = 0.
Corollary 4.4 can be extended to complete and minimal surfaces (properly) immersed in the Hyperbolic space, with finite total extrinsic curvature: Corollary 4.5. Let P 2 be a complete and minimal surface immersed in K n (b) with finite total extrinsic curvature P B P 2 dσ < ∞. Then Proof. As the total extrinsic curvature of P is finite, by applying Theorem 4.2 we have: Vol(B b,2 t ) ) < ∞ Then, apply Corollary 4.3 with m = 2.