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

Barbosa, Ezequiel; Sharp, Ben; Wei, Yong

doi:10.1090/proc/13628

Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index
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by Ezequiel Barbosa, Ben Sharp and Yong Wei PDF

Proc. Amer. Math. Soc. 145 (2017), 4945-4961 Request permission

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-Émery 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.

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