48 research outputs found
Parabolicity criteria and characterization results for submanifoldsof bounded mean curvature in model manifolds with weights
Let P be a submanifold properly immersed in a rotationally symmetric manifold
having a pole and endowed with a weight e
h. The aim of this paper is twofold. First, by assuming certain control on the h-mean curvature of P, we establish comparisons for the h-capacity of
extrinsic balls in P, from which we deduce criteria ensuring the h-parabolicity or h-hyperbolicity
of P. Second, we employ functions with geometric meaning to describe submanifolds of bounded
h-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results
apply for some relevant h-minimal submanifolds appearing in the singularity theory of the mean
curvature flow
Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces
We study the topology of (properly) immersed complete minimal surfaces P 2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in â n and in â n (b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature
A note on the p-parabolicity of submanifolds
We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for pââ„â2
Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below
We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above
Volume growth, number of ends and the topology of complete submanifolds
eprint de ArXIV. Pendent de publicar a Journal of Geometric Analysis, 2012Given a complete isometric immersion in an ambient Riemannian manifold with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space , we determine a set of conditions on the extrinsic curvatures of that guarantees that the immersion is proper and that has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. 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 Anderson's paper "The compactification of a minimal submanifold by the Gauss Map", (Preprint IEHS, 1984), for complete and minimal submanifolds in \erre^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 due to A. Kasue and K. Sugahara "Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space forms", (Osaka J. Math. 24,1987)
Mean curvature and compactification of surfaces in a negatively curved CartanâHadamard manifold
We state and prove a ChernâOsserman-type inequality in terms
of the volume growth for complete surfaces with controlled mean
curvature properly immersed in a CartanâHadamard manifold N
with sectional curvatures bounded from above by a negative quantity
KN †b < 0
Estimates of the first Dirichlet eigenvalue from exit time moment spectra
We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and CheungâLeung concerning the fundamental tones of CartanâHadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.Supported by the Spanish Mineco-FEDER grant MTM2010-21206-C02-01 and Junta de Andalucia grants
FQM-325 and P09-FQM-5088. Supported by the Spanish Mineco-FEDER grant MTM2010-21206-C02-02 and by the Pla de Promocio de la InvestigaciĂł de la Universitat Jaume
Comparison results for capacity
postprint de l'autor en arXiv: http://arxiv.org/abs/1012.0487We obtain in this paper bounds for the capacity of a compact set . If is contained in an -dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of are larger than or equal to , then . When is contained in an -dimensional manifold with non-negative Ricci curvature, has smooth boundary, and the mean curvature of is smaller than or equal to , we prove the inequality . In both cases we are able to characterize the equality case. Finally, if is a convex set in Euclidean space which admits a supporting sphere of radius at any boundary point, then we prove and that equality holds for the round sphere of radius
Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds
Let (M, g)be a complete non-compact Riemannian manifold together with a function eh, 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 Mto deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in Mcentered at a point o âM. 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 Munder certain control on their weighted mean curvature
Comparison theory of Lorentzian distance with applications to spacelike hypersurfaces
In this paper we summarize some comparison results for the Lorentzian distance function
in spacetimes, with applications to the study of the geometric analysis of the Lorentzian distance
on spacelike hypersurfaces. In particular, we will consider spacelike hypersufaces whose image
under the immersion is bounded in the ambient spacetime and derive sharp estimates for the mean
curvature of such hypersurfaces under appropriate hypotheses on the curvature of the ambient
spacetime. The results in this paper are part of our recent work [1], where complete details and
further related results may be found