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arXiv · 1503.01945

Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index

Ezequiel Barbosa·Ben Sharp·Yong Wei
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Abstract

Let $(M^{n+1},g,e^{-f}d\mu)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-\'{E}mery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in $\mathbb{R}^{n+1}$ with $2\leq n\leq 6$. We also prove some estimates on the $f$-index of $f$-minimal hypersurfaces, and give a conformal structure of $f$-minimal surface with finite $f$-index in three-dimensional smooth metric measure space.

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Ezequiel Barbosa, Ben Sharp, Yong Wei. 2015-03-06. Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index. https://arxiv.org/abs/1503.01945

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