Mean curvature, volume and properness of isometric immersions

@article{Gimeno2015MeanCV,
  title={Mean curvature, volume and properness of isometric immersions},
  author={Vicent Gimeno and Vicente Palmer},
  journal={arXiv: Differential Geometry},
  year={2015}
}
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its… Expand
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References

SHOWING 1-10 OF 40 REFERENCES
AN ESTIMATE FOR THE CURVATURE OF BOUNDED SUBMANIFOLDS
1. Statement of the result. All manifolds considered in this paper shall be connected, of class C"' (smooth), and dimension at least 2. Im- mersions will also be smooth, and of codimension at leastExpand
Compactification of minimal submanifolds of hyperbolic space
In this paper we study the geometry of complete minimal submanifolds of hyperbolic space HI. Specifically, we are interested in m-dimensional submanifolds whose second fundamental form A satisfies fMExpand
Total curvatures and minimal areas of complete surfaces
Minimal areas for certain classes of finitely connected complete open surfaces are obtained by using a Bonnesen-style isoperimetric inequality for large balls on the surfaces. In particular, theExpand
The minimal lamination closure theorem
We prove that the closure of a complete embedded minimal surface M in a Riemannian three-manifold N has the structure of a minimal lamination, when M has positive injectivity radius. When N is R, weExpand
Finiteness of index and total scalar curvature for minimal hypersurfaces
Let Mn, n > 3, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for n = 3, 4, 5, or 6, the index of Mn is finite if and only if the total scalar curvature ofExpand
CURVATURE AND FUNCTION THEORY ON RIEMANNIAN MANIFOLDS
Function theory on Euclidean domains in relation to potential theory, partial differential equations, probability, and harmonic analysis has been the target of investigation for decades. There is aExpand
On the volume growth and the topology of complete minimal submanifolds of a Euclidean space
Let M be a n-dimensional complete properly immersed minimal submanifold of a Euclidean space. We show that the number of the ends of M is bounded above by k = sup volume(M ∩B(t)) ωntn , where B(t) isExpand
Complete submanifolds with bounded mean curvature in a Hadamard manifold
Abstract Let Q be an m -dimensional Hadamard manifold and let M be an n -dimensional complete non-compact submanifold in Q . Let κ be a non-positive constant and suppose that the ( n − 1 ) -th RicciExpand
Complete submanifolds of $R^n$ with finite topology
We show that a complete m-dimensional immersed submanifold M of R with a(M) < 1 is properly immersed and have finite topology, where a(M) ∈ [0,∞] is an scaling invariant number that gives the rateExpand
Total curvature and L2 harmonic 1-forms on complete submanifolds in space forms
Let Mn be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., $${\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}$$, in an (n + p)-dimensional simply connected spaceExpand
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