Volume Growth, Number of Ends, and the Topology of a Complete Submanifold

Abstract

Abstract. Given a complete isometric immersion ϕ : P m −→ N n in an ambient Riemannian manifold N n with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space M n w , we determine a set of conditions on the extrinsic curvatures of P that guarantees that the immersion is proper and that P has finite topology in the line of the results in [24] and [25]. When the ambient manifold is a radially symmetric space, it is shown an inequality between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends which generalizes the classical inequality stated in [1] for complete and minimal submanifolds in R n . We obtain as a corollary the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in the Hyperbolic space together with Bernstein type results for such submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work [12].

Asymptotically Extrinsic Tamed Submanifolds

“…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] 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 nonnegative sectional curvature then M is isometric to R m .…”

“…Then the sectional curvature K ∂V (t) (π) of the plane π expanded by e i , e j is, using Gauss formula, see [12]:…”

“…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 Theorem C (See [12]).…”

“…In the particular setting of minimal immersions of the Euclidean space they showed that Theorem C (See [12]). Let ϕ : M m ֒→ R n be an isometric and minimal immersion of a complete Riemannian m-manifold M into the n-dimensional Euclidean space R n .…”

“…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 :…”