Volume growth, eigenvalue and compactness for self-shrinkers

Abstract

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\ir{3}$ obtained by Colding-Minicozzi under weaker conditions.

17 pages

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• Stability and compactness for complete $f$-minimal surfaces
• Eigenvalues of the drifted Laplacian on complete metric measure spaces
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• The rigidity theorems for Lagrangian self shrinkers
• The Rigidity Theorems of Self Shrinkers via Gauss maps
• Estimates for eigenvalues of $\mathfrak{L}$ operator on Self-Shrinkers
• Complete self-shrinkers of the mean curvature flow
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• Hamiltonian F-stability of complete Lagrangian self-shrinkers
• A gap theorem of self-shrinkers
• Some results on space-like self-shrinkers
• A Bernstein type theorem for graphic self-shrinkers with flat normal bundle