Complete self-shrinkers confined into some regions of the space

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.

21 pages

References in corpus (1)

• On a classification theorem for self-shrinkers

Cited by in corpus (5)

• Stability properties and topology at infinity of f-minimal hypersurfaces
• Halfspace type Theorems for Self-Shrinkers
• Bi-Halfspace and Convex Hull Theorems for Translating Solitons
• Rigidity and gap results for low index properly immersed self--shrinkers in ${\mathbb{R}}^{m+1}$
• A strong Frankel Theorem for shrinkers