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 $\operatorname {Ric}_{f}\!\ge -(n-1)$ in a geodesic ball $B_{o}(R)$ with radius $R$ and center $o\in M^n$. When $\operatorname {Ric}_{f}\ge 0$ 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 $\operatorname {Ric}_{f}\ge \frac 12$ and $f(x)\le \frac 14d^2(x,B_{o}(R))+c$ for some constant $c>0$ outside $B_{o}(R)$, we can also get the same gap conclusion.