On Deciding Whether a Submanifold Is Parabolic or Hyperbolic Using Its Mean Curvature

2014

Ark. Mat.

Self Cite

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

“…As a consequence of this result, we have the following Laplacian inequalities (see [20], [26], or [13] for detailed developments):…”

Section: Hessian Analysis Gauss-bonnet Theorem and Estimates For Thmentioning

confidence: 76%

Section: Hessian Analysis Gauss-bonnet Theorem and Estimates For Thmentioning

confidence: 97%

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

Esteve

2014

Ark. Mat.

Self Cite

“…The 2nd order analysis of the restricted distance function r |S is governed by the Hessian comparison Theorem A in [10]. A corollary of this result is the following proposition (see [14] or [25] for further details):…”

Section: Hessian Analysis Gauss-bonnet Theorem and Estimates For Thementioning

confidence: 97%

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

Esteve

Communications in Analysis and Geometry

2014

Self Cite

“…In this situation, the Hessian in P depends on the Hessian in M and the second fundamental form of P . As in the Riemannian setting, see for instance [39], we may assume suitable bounds on weighted or unweighted sectional curvatures in (M, g, e h ) to control the weighted Laplacian ∆ h P of radial functions in P by means of our previous analysis on Hess r, and from radial bounds on the term ∇h + H h P , ∇r , which involves the radial derivatives of h and the weighted mean curvature vector H h P defined in Section 5.1. In the particular case of weighted rotationally symmetric manifolds with a pole, the sectional curvature is a radial function from the pole, and the Laplacian ∆ h P can be explicitly computed as we did in [19].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Hurtado

Journal of Mathematical Analysis and Applications

Rosales

2020

Let (M, g) be a complete non-compact Riemannian manifold together with a function e h , which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of M to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in M centered at a point o ∈ M . As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from o. The technique extends to non-compact submanifolds properly immersed in M under certain control on their weighted mean curvature.