Mean curvature, volume and properness of isometric immersions

  title={Mean curvature, volume and properness of isometric immersions},
  author={Vicent Gimeno and Vicente Palmer},
  journal={arXiv: Differential Geometry},
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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