Asymptotically Extrinsic Tamed Submanifolds

We study, from the extrinsic point of view, the structure at infinity of open submanifolds, φ:Mm↪Mn(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :M^m \hookrightarrow \mathbb {M}^{n}(\kappa )$$\end{document} isometrically immersed in the real space forms of constant sectional curvature κ≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa \le 0$$\end{document}. We shall use the decay of the second fundamental form of the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in Greene et al. (Int Math Res Not 1994:364–377, 1994) and Petrunin and Tuschmann (Math Ann 321:775–788, 2001) and an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls.


Introduction
The geometry and the topology in the large of non-compact Riemannian manifolds is controlled by their curvature behavior at infinity, so that one can expect, if the manifold becomes nearly flat at infinity, i.e., out of the compact sets, that it shares some essential features with the Euclidean space R n . This fundamental idea, together with the analysis of asymptotically non-negative curved spaces, was developed in the seminal works [1,8,17,18].
In particular, it has been proved in [17], (resp. in [18]), that a complete non-compact Riemannian manifold M n with zero sectional curvature outside a compact set, (resp. with non-negative curvature outside a compact set), contains another compact K ⊆ M such that M \ K is a finite union of "conical ends," each of the form N × R + , being N a connected and compact (n − 1)-dimensional manifold.
From this point of view, it seems natural to think that it is possible to extract some similar description at infinity of a Riemannian manifold by replacing the flatness of the manifold outside a compact set by a weaker hypothesis. For instance, we can assume, as in [14], that the Riemannian manifold M has faster-than-quadratic-curvature-decay, namely, that there exists some > 0 and some c > 0 such that where K x is the supremum of sectional curvatures of the tangent 2-planes of T x M and ρ M(x) = dist M (x 0 , x) is the distance to a fixed base point x 0 ∈ M.
Then, it was proved in [14, Theorem 1] that if M is a complete, connected, and non-compact Riemannian manifold with faster-than-quadratic-curvature-decay, the manifold contains a connected open subset D ⊆ M with compact closure and smooth boundary such that the complement M \ D is a finite union of "conical ends" M i ≡ N i × R + described as before. Moreover, if the tangent bundle of each N i is non-trivial and its fundamental group is finite, the volume growth of the conical end M i has Euclidean order.
A slightly more general concept is the notion of asymptotically flatness. We say that a complete non-compact Riemannian manifold (M, g) is said to be asymptotically flat Assuming that the manifold M has cone structure at infinity, namely, that the pointed Gromov-Hausdorff limit of a decreasing-to-zero sequence of re-escaled metrics on (M, g) is a metric cone C with vertex o, and is asymptotically flat, A. Petrunin and W. Tuschmann proved in [26] an structural result in the line of [14,Theorem.1], namely, that there exists an open ball B R ( p) ⊂ M such that M \ B R ( p) is a disjoint union ∪ i N i of a finite number of ends, i.e., N i is a connected topological manifold with closed boundary ∂ N i which is homeomorphic to ∂ N i × [0, ∞). Moreover, if the end N i is simply connected, then N i is homeomorphic to S m−1 × [0, ∞).
Note that non-compact manifolds with faster-than-quadratic-curvature-decay or with non-negative curvature has cone structure at infinity, see [20] and [26].
We are going, in this paper, to study the structure at infinity of complete noncompact Riemannian manifold isometrically immersed ϕ : M m → M n (κ), in the real space forms of constant sectional curvature κ ≤ 0, from an extrinsic point of view. We shall use hence an extrinsic approach, preserving an extrinsic curvature decay condition satisfied by the so-called tamed immersions (see Definition 2.6), given in terms of two extrinsic invariants a(M) and b(M). These invariants describe the decay of the second fundamental form α of the submanifold M. We ignore in this extrinsic context the existence of the cone structure at infinity, to obtain a description at infinity of the submanifold in the line of the structural results in [14] and [26] and estimating from below the number of its ends in terms of the volume growth of an special class of extrinsic domains, the extrinsic balls We observe here that structural statement (1) in Theorems 1.1 and 1.2 comes directly from the following Theorem A, first stated in [4] for the case κ = 0, in [3] for the case κ < 0 and then in [12] it was given an extension of it to complete ambient manifolds with a pole and bounded radial curvatures. Theorem A constitutes an extrinsic version of the structural assertion in [14,Thm.1] and in [26,Thm. A] for the special class of submanifolds in M n (κ) called tamed submanifolds. Theorem A [3,4,12] Let ϕ : M m → M n (κ) be an isometric immersion of a complete Riemannian m-manifold M into an n-dimensional space form M n (κ) with constant sectional curvature κ ≤ 0. Let us suppose that Then: (1) ϕ is proper.
In the main theorem of [26], above mentioned, it was also proved that if M m , m ≥ 3, has cone structure at infinity, is asymptotically flat, and is simply connected with non-negative sectional curvature then M is isometric to R m . This gap result for manifolds with non-negative sectional curvatures, gives a partial answer (assuming the additional hypothesis that the manifold has cone structure at infinity) to the problem posed by M. Gromov in [2]: If M is simply connected of dimension n ≥ 3 and asymptotically flat with nonnegative curvature, show that M is isometric to R n .
Greene and Wu [18], addressed this question when the manifold M has a pole showing that in this case and when M has faster-than-quadratic-curvature-decay, the manifold is isometric to R n . From an extrinsic point of view, Kasue and Sugahara [21], established the following gap result: Assuming that the submanifold M is simply connected, we can state the following gap-type result that improves the decay conditions of Kasue-Sugahara's results in [21] and extends Greene-Wu's gap to submanifolds of Hyperbolic space. This theorem is proved as a corollary of the proofs of Theorems 1.1 and 1.2. Concerning the assertions (2) and (3) in Theorems 1.1 and 1.2, V. Gimeno and V. Palmer in [12], proved that there is a deep relation between the volume growth of the extrinsic spheres and the number of ends of extrinsic asymptotically flat submanifolds of rotationally symmetric spaces. In the particular setting of minimal immersions of the Euclidean space they showed that

If M has only one end then M is isometric to R m .
Theorem C shows a relation between the volume growth of the extrinsic balls and the number of ends, and moreover one deduce a gap-type theorem first stated by A. Kasue and K. Sugahara in [21].
However, this gap result does not hold for minimal submanifolds of the Hyperbolic space, as we can see in the following example, given in [22].
In the particular setting of minimal immersions of Hyperbolic space, using the lower bounds for the Cheeger constant and the fundamental tone for minimal submanifolds in H n (κ) given in [13], we can state an improved version of the theorems [12, Thm.

) M has finite topological type and the (finite) number of ends E(M) is bounded from below by
(3) The Cheeger constant satisfies

Outline of the Paper
The structure of the paper is as follows: In the preliminaries, Sect. 2, Subsect. 2.1, we recall the preliminary concepts and properties of extrinsic distance function. In Subsect. 2.2, it is presented and studied the notion of tamed submanifold and we finish the preliminaries establishing lower and upper bounds for the sectional curvatures of the boundary of an end in a tamed submanifold, in Subsect. 2.3. We shall prove Theorem 1.1 in Sect. 3, obtaining as a result of that proof Corollaries 3.1 and 3.2 which deal about several topological properties of the ends of the submanifold, such as vanishing first Betti number. We prove Theorem 1.2 in Sect. 4, obtaining Corollaries 4.1 and 4.2 in the same way as in Sect. 3. Finally, in Sect. 5 the gap-type result, Theorem 1.3, is proved.

Analysis of the Extrinsic Distance Function Defined on a Submanifold
We start presenting some standard definitions and results that we can find in previous works (see e.g., [12,25]). We assume throughout the paper that ϕ : (o, x), 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 | M or by r the composition r • ϕ : M → R + ∪ {0}. This composition is called the extrinsic distance function from o in M. The gradients of r in M n (κ) and of r | M in M are denoted by ∇ M n (κ) r and ∇ M r , respectively. Then we have the following basic relation, by virtue of the identification, given any point where B where α is the second fundamental form of the immersion.
On the other hand, the Hessian of the distance function r : M n (κ) \ {0} → R at a point p ∈ M n (κ) is given by where the function S κ is given by Let us recall that if ϕ : We have the following technical result in this context: (2.5) Here δ(t) is a decreasing function such that δ → 0 when t → ∞.

Tamed Submanifolds. Some Examples
The extrinsic decay conditions in the results stated above, can be described more carefully in the following way: However, the opposite implication it is not true in general, as we shall show below.
is finite when ρ M (x) goes to infinity, so α(x) goes very fast to zero when ρ M (x) goes to infinity and this implies, as Example 2. 7 We have seen that when we consider an isometric immersion ϕ : M → R n then a(M) = 0 implies that A(M) = 0. However, the opposite implication it is not true in general. If we consider the cylinder C = {(x, y, z) ∈ R 3 /x 2 + y 2 = 1} ⊆ R 3 isometrically immersed by the inclusion map, we know that its sectional curvature is K C p (σ ) = 0 for all points p ∈ C and all tangent planes σ ⊆ T p C. Hence, A(C) = 0. On the other hand, the norm of its second fundamental form α C = constant, so a(C) = ∞. Example 2. 9 We are going to present, following the construction given in [10], a rotation hypersurface M n of H n+1 (−1), n ≥ 2, with b(M) < ∞. For that, let us consider first the Hyperbolic space H n+1 (−1) as a hypersurface of the Lorentzian space L n+2 , with Lorentzian metric g −1 .
Let us choose P 2 a 2-dimensional plane in L n+1 , passing through the origin and such that the restriction g −1 | P 2 is Lorentzian. Let us denote as O(P 2 ) the set of all orthogonal transformations of L n+2 with positive determinant and such that P 2 is invariant by these transformations. Then, let us consider now a subspace P 3 ⊆ L n+2 such that P 2 ⊆ P 3 and P 3 ∩ H n+1 (−1) = ∅ and finally, let C be a regular curve in P 3 ∩ H n+1 (−1) that does not meet P 2 . With all this elements in hand, we define the rotation hypersurface M n ⊆ H n+1 (−1) as the orbit of C under the action of O(P 2 ).
Let us consider ϕ(t 1 , ..., t n−1 ) = (ϕ 1 (t 1 , ..., t n−1 ), ..., ϕ n (t 1 , ..., t n−1 )) an orthogonal parametrization of the unit sphere of {e 1 , ..., e n } . Then, is the parametrization of the rotation hypersurface M n generated by the curve C around P 2 . In Proposition 3.2 in [10], it is proved that the principal curvatures of M along the principal directions given by the coordinate curves t i , (i = 1, ..., n − 1) are and the principal curvature along the coordinate curve s is (2.10) When x 1 (s) = (a cosh(2s) − 1 2 ) 1 2 , with a ∈ R, a > 1 2 , it is straightforward to check that λ = −μ, so when the dimension of the submanifold M n is n = 2, then M is minimal, and, computing the norm of the second fundamental form of M 2 , denoted as α M 2 , we have, see [10], that

Sectional Curvature of the Extrinsic Spheres in Tamed Submanifolds
Let for all t > t 0 .
Proof Suppose that e i , e j are two orthonormal vectors of T p ∂ V (t) at a point p ∈ ∂ V (t). Then the sectional curvature K ∂ V (t) (π ) of the plane π expanded by e i , e j is, using Gauss formula in the situation ∂ V ⊆ M → M n (κ), see [12]: Since the immersion is tamed, we have, for t large enough Therefore the upper bounds on the statement of the proposition follows directly from the identity (2.18). In order to obtain the lower bounds, observe that from equality (2.18) (2.20) And the proposition follows.

Proof of Theorem 1.1
As we have observed in the Introduction, the assertion (1) in Theorem 1.1 follows from Theorem A. In order to prove the assertion (2), let us remind that by Theorem A, since ϕ : M → R n is a tamed immersion, there exists R 0 > 0 such that M has finitely many ends V k ∈ M \ D R 0 and we can work on each end separately. Let us denote Applying again Theorem A, we have that for any t > R 0 the extrinsic distance function has no critical points in so using basic Morse theory (see [27,Theorem 2.3] and [23] ), we know that for any t > R 0 . Hence, by statement (4) of Theorem A for any t ≥ R 0 Since a(M) < 1 2 , then a(M) < 1, so using Proposition 2.10, there exists t 0 > 0 such that the sectional curvatures of the tangent planes π to ∂ V k (t) (for all t > t 0 ) are bounded below and above by 1 On the other hand, as a(M) < 1 2 , let us consider now a quantity c ∈ (a(M), 1 2 ). From the definition of a(M) there exists t c such for all t > t c . Therefore, for any t > max{t c , Taking into account that t α ≤ c, the inequalities 3.4 yield for any t > max{t c , t 0 , R 0 }. Applying Lemma 2.4 for G(t) = c/t we have, for any t > max{t c , t 0 , R 0 }, that We are dealing with lower bounds in order to prove statement (2) of Theorem 1.1. Since δ(t) → 0, when t → ∞, and c < 1 2 , there exists t 1 > max{t c , t 0 , R 0 } such that for any t > t 1 .
Defining the function 0 : the lower bound for K ∂ V k (t) in inequality (3.7) can be therefore written as for any t > t 1 . Now, we apply Bishop's volume comparison theorem (see [6] or [7]), taking into account that the above inequality implies for any unit vector ξ ∈ T p ∂ V k (t), for any t > t 1 . Applying coarea formula (see for instance [27]) to the extrinsic annuls A t 1 ,t := D t \ D t 1 , and using inequality (3.6) vol Taking limits in inequalities (3.14) and (3.16) Letting t 1 → ∞ and taking into account 0 (3.18) Since the above inequalities are true for any c ∈ (a(M), 1 2 ) the desired inequalities of statement (2) of the Theorem 1.1 follow when c goes to a(M).
In order to prove statements (3) and (4) of the theorem, let us define, for t > 0 and for all c ∈ [a(M), 1 2 ) and K min (t) : We shall prove that for t large enough and when a(M) < 23− √ 337 32 1 2 < 1 2 , the sectional curvatures of the boundary of each end, ∂ V k (t) := ∂ D t ∩ V k satisfy the pinching: Then, we apply Synge's Theorem and either the Rauch-Berger Sphere Theorem or the Brendle-Schoen differentiable sphere theorem, if m ≥ 5, splitting the proof in two cases, according to the parity of the dimension of ∂ V k (t). First of all, we know that, in all cases, ∂ V k (t) is orientable, because there exists a everywhere non-vanishing smooth normal vector field ∇ M r |∇ M r | globally defined on ∂ V k (t). In assertion (3), we assume that the dimension m of the submanifold M is odd, so ∂ V k (t) is even dimensional. By Synge's Theorem (see [9, Corollary 3.10, Chap. 9]), ∂ V k (t) is simply connected. Taking into account the inequality (3.19) and the Rauch-Berger Sphere Theorem (see [9,Theorem 1.1,Chap. 13]), ∂ V k (t) is homeomorphic to S m−1 . If m − 1 ≥ 4, we apply Brendle-Schoen Differentiable Sphere Theorem (see [5]) and then ∂ V k (t) is diffeomorphic to S m−1 . Since V k diffeo.
In assertion (4), we assume that dimension m of the submanifold M is even, so is also orientable, we apply either the Rauch-Berger Sphere Theorem or Brendle-Schoen Differentiable Sphere Theorem, observing that in order to have m − 1 ≥ 4 and m even then m ≥ 6, to obtain the proof of assertion (4).
We are going now to prove that the sectional curvatures of ∂ V k (t) are pinched as in (3.19). First of all, observe that, given any c ∈ [a(M), 1 2 ), as a(M) ≤ c, there exist t c such t α ≤ c, (3.20) for all t > t c . Therefore, for any t > max{t c , t 0 , R 0 }, we can repeat the argument that leads to inequalities (3.7) to obtain, for all c ∈ [a(M), 1 2 ): Hence, by inequality (3.21) for all t > max{t c , t 0 , R 0 }. On the other hand, we have that for all c ∈ [a(M), 1 2 ), 1 2 ). On the other hand, thus, for any t > max{t , t c 0 , t 0 , R 0 }, we have Let us choose and, for this , for t large enough, we have, from (3.24), (3.25) and the sectional curvature pinching of ∂ V k (t) is proven. Moreover, if a(M) < 1 2 and m = 3 by using inequality (3.10) and the Gauss-Bonnet Theorem, since the surface ∂ V k (t) has positive curvature, the surface ∂ V k (t) is homeomorphic to a sphere, and we can state Actually, if a(M) < 1 2 inequality (3.10) implies for any dimension m > 2 that the sectional curvature of ∂ V k (t) is positive for a sufficiently large t. By the first Betti number Theorem (see [19]) we can state Remark g Note that b 1 (∂ V k ) = 0 implies that ∂ V k is not homeomorphic to the m − 1 torus T m−1 .

Proof of Theorem 1.2
This proof follows the lines of the proof in Sect. 3. As we have observed in Sect. 3, assertion (1) of the Theorem follows from Theorem A.
In order to prove assertion (2), we have, as before, a finite number of ends V k ∈ M \ D R 0 with boundaries ∂ V k (t) := ∂ D t ∩ V k and we work on each end separately. Taking into account that for any t > R 0 the extrinsic distance function has no critical points in we use Morse theory (see [27,Theorem 2.3] and [23]), to have that A k R 0 ,t is diffeomorphic to ∂ V k (R 0 ) × [R 0 , t]. In particular, ∂ V k (t) is diffeomorphic to ∂ V k (R 0 ) for any t > R 0 . Hence, by statement (4) of theorem A for any t ≥ R 0 The function u c (t) := δ(t)+c t−R 0 S κ (t) is decreasing for a sufficiently large t, because δ(t) is decreasing and γ (t) is also a decreasing function for a sufficiently large t. Moreover, γ (t) goes to 0 when t goes to ∞, so u c (t) goes to 0 when t goes to ∞ . Hence 1 − u 2 c (t) > 0 for all sufficiently large t and 2u c (t)c 1−u 2 c (t) is decreasing and goes to 0 when t goes to ∞. Therefore, using inequalities (4.6) and (4.7), for all t > max{t c , and hence, from (4.5) . (4.9) As c C κ (t) is decreasing and goes to zero too when t goes to ∞, we can conclude that the function is increasing and satisfies that lim t→∞ c (t) = 1, so there exists t 1 such that for all t > t 1 we have c (t) ≥ c (t 1 ) > 0. Therefore, there exists t 1 > max{t c , t 0 , R 0 } such that for all t > t 1 we have (4.10) By using the Bishop Volume Comparison Theorem, one concludes that for all t > t 1 vol(∂ V k (t)) ≤ 1 obtaining therefore the following lower estimate for the number of ends Similarly that in 3 by using again the coarea formula (4.13) And therefore, taking limits in inequalities (4.12) and (4.13) (4.14) Letting t 1 → ∞ in the above inequalities and taking into account that  In order to obtain the upper bound for the fundamental tone we only have to take into account that the geodesic ball B M t of radius t is a subset of the extrinsic ball D t of the same radius t because the extrinsic distance is always less or equal to the geodesic intrinsic distance in isometric immersions. Hence, by inequality (4. Finally by using [15, inequality (10.3)] the upper bound is proved. Moreover and in the same way than in Theorem 1.1, if b(M) < ∞ and m = 3 by using inequality (4.10) and the Gauss-Bonnet Theorem, since the surface ∂ V k (t) has positive curvature for a sufficiently large t, the surface ∂ V k (t) is homeomorphic to an sphere, and we can state Actually, if b(M) < ∞ then inequality (4.10) implies for any dimension m > 2 that the sectional curvature of ∂ V k (t) is positive for a sufficiently large t. As in Corollary 3.2, by the first Betti number Theorem (see [19]) we can state Remark h Note that b 1 (∂ V k ) = 0 implies that ∂ V k is not homeomorphic to the m − 1 torus T m−1 .

Proof of Theorem 1.3
The submanifold M m is simply connected and it has sectional curvatures bounded from above by K M ≤ k ≤ 0. Then, all the points p ∈ M are poles of M and the number of ends of M is E(M) = 1. We apply Bishop-Günther's Theorem (see for instance [7]), so we have, for any geodesic ball B M R ( p) of radius R on M, for any R ∈ R + and that completes the proof of the corollary, because then, for all the points p ∈ M, any geodesic R-ball B M R ( p) is isometric B R n R . When k < 0, we argue exactly in the same way, but using now inequality (4.13).