Gap theorems for ends of smooth metric measure spaces

Abstract

In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n,g,e^{-f}dv)$ with the Bakry-Émery Ricci tensor ${\mathrm{Ric}}_f\ge -(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When ${\mathrm{Ric}}_f\ge 0$ and $f$ has some degeneration outside $B_o(R)$, we show that there exists an $\epsilon =\epsilon (n,{\sup }_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\le \epsilon$. When ${\mathrm{Ric}}_f\ge \frac{1}{2}$ and $f(x)\le \frac{1}{4}{d}^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion.

Proc AMS, final version

References in corpus (1)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds