Gap theorems for ends of smooth metric measure spaces

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 $ε=ε(n,\sup_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\leε$. When $\mathrm{Ric}_f\ge\frac 12$ 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.

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