104 research outputs found
On the fundamental tone of minimal submanifolds with controlled extrinsic curvature
The aim of this paper is to obtain the fundamental tone for minimal
submanifolds of the Euclidean or hyperbolic space under certain restrictions on
the extrinsic curvature. We show some sufficient conditions on the norm of the
second fundamental form that allow us to obtain the same upper and lower bound
for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient
manifold. As an intrinsic result, we obtain a sufficient condition on the
volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the
fundamental tone given by McKean.Comment: 10 pages, minor corrections, accepted to Potential Analysi
Conformal type of ends of revolution in space forms of constant sectional curvature
In this paper we consider the conformal type (parabolicity or
non-parabolicity) of complete ends of revolution immersed in simply connected
space forms of constant sectional curvature. We show that any complete end of
revolution in the -dimensional Euclidean space or in the -dimensional
sphere is parabolic. In the case of ends of revolution in the hyperbolic
-dimensional space, we find sufficient conditions to attain parabolicity for
complete ends of revolution using their relative position to the complete flat
surfaces of revolution.Comment: 22 pages, 5 figure
Ends, fundamental tones, and capacities of minimal submanifolds via extrinsic comparison theory
We study the volume of extrinsic balls and the capacity of extrinsic annuli
in minimal submanifolds which are properly immersed with controlled radial
sectional curvatures into an ambient manifold with a pole. The key results are
concerned with the comparison of those volumes and capacities with the
corresponding entities in a rotationally symmetric model manifold. Using the
asymptotic behavior of the volumes and capacities we then obtain upper bounds
for the number of ends as well as estimates for the fundamental tone of the
submanifolds in question.Comment: 21 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1104.5417 by other author
On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form
In this paper we show that a complete and non-compact surface immersed in the
Euclidean space with quadratic extrinsic area growth has finite total curvature
provided the surface has tamed second fundamental form and admits total
curvature. In such a case we obtain as well a generalized Chern-Osserman
inequality. In the particular case of a surface of nonnegative curvature, we
prove that the surface is diffeomorphic to the Euclidean plane if the surface
has tamed second fundamental form, and that the surface is isometric to the
Euclidean plane if the surface has strongly tamed second fundamental form. In
the last part of the paper we characterize the fundamental tone of any
submanifold of tamed second fundamental form immersed in an ambient space with
a pole and quadratic decay of the radial sectional curvatures.Comment: 19 pages. Title changed and several improvement of the main theorems
are done. arXiv admin note: text overlap with arXiv:0805.0323 by other
author
Lower bounds for the volume with upper bounds for the Ricci Curvature in dimension three
In this note, we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a 3‐dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature
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