The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

Abstract

We state and prove a Chern-Osserman-type Inequality in terms of the volume growth for minimal surfaces S which have finite total extrinsic curvature and are properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0 and such that they are not too curved (on average) with respect to the Hyperbolic space with constant sectional curvature given by the upper bound b. We have also proven the same Chern-Osserman-type Inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0.2000 Mathematics Subject Classification. Primary 53A15, 53C20; Secondary 53C42.is the geodesic t-ball centered at the pole o in the ambient space N , and B b,2 t denotes the geodesic t-ball in H 2 (b). Remark 1.2. The main theorem in [7] is a corollary of Theorem 1.1. In fact, note that condition (1.4) is superfluous when the ambient manifold is H n (b). On the other hand, when the ambient manifold is H n (b), then condition (1.3) implies that A S (q) goes

On the Fundamental Tone of Minimal Submanifolds with Controlled Extrinsic Curvature

“…In particular (in view of [9]), for every 2-dimensional Cartan-Hadamard manifold M which is asymptotically locally κ-hyperbolic of order 2 (see [18], [9]) and with sectional curvatures bounded from above by K M ≤ κ < 0, the fundamental tone satisfies…”

“…Remark b. In view of the intrinsic version of the results of [9] for complete, simply connected, 2-dimensional manifold M 2 with Gaussian curvature K M bounded from above by K M ≤ κ < 0, the condition (1.8) can be achieved provided the following…”

Mean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold

“…As a corollary of Theorem 3.2, we have the following result, which is a generalization of the main theorem in [7], when we consider connected and minimal surfaces in a Cartan-Hadamard manifold, (see also [9]): Corollary 3.4. Let S 2 be a complete, connected and minimal surface properly immersed in a Cartan-Hadamard manifold N , with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0.…”

“…, H S Proposition 4.2. (See[7] and[9]) Let S 2 be a complete, non-compact, and properly immersed surface in a Cartan-Hadamard manifold N n . Let us consider {D t } t>0 an exhaustion of S by extrinsic balls.…”

Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to ODEs

“…We refer to subsequent work of Jorge and Meeks [14], and Kobuku et al [15] (among others). We also refer to the discussion in Chen and Cheng [4] or Seo [21] where the ambient space is H n , to Esteve and Palmer [10] where the ambient manifold is a Cartan-Hadamard manifold, and to Ma [18] and Ma, Wang, and Wang [19] where the ambient space is Lorentzian.…”