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 d v stretchy="false">) (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor alttext="upper R i c Subscript greater-than-or-equal-to minus left-parenthesis 1 Ric ≥ = movablelimits="true" form="prefix">sup stretchy="false">| encoding="application/x-tex">\epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such at most if less-than-or-equal-to epsilon"> ≤<!-- ≤ encoding="application/x-tex">R\le \epsilon one half"> 2 \frac 12 alttext="f x fourth squared plus c"> x 4 + c encoding="application/x-tex">f(x)\le 14d^2(x,B_{o}(R))+c constant alttext="c greater-than &gt; encoding="application/x-tex">c&gt;0 can also get same conclusion.