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arXiv · 1212.4041

Complete self-shrinkers confined into some regions of the space

Stefano Pigola·Michele Rimoldi
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Abstract

We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H. D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented.

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Stefano Pigola, Michele Rimoldi. 2012-12-17. Complete self-shrinkers confined into some regions of the space. https://doi.org/10.1007/s10455-013-9387-8

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