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-Emery Ricci tensor $mathrm{Ric}_fge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $oin M^n$. When $mathrm{Ric}_fge 0$ and $f$ has some degeneration (including sublinear growth) outside $B_o(R)$, we show that there exists an $epsilon=epsilon(n,sup_{B_o(1)}|f|)$ such that such a manifold has at most two ends if $Rleepsilon$. When $mathrm{Ric}_fgefrac 12$ and $f(x)lefrac 14d^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion.