An alternative proof of lower bounds for the first eigenvalue on   manifolds
  Recently, Andrews and Clutterbuck [AC13] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [AC13]. More precisely, following Ni's method ([Ni13, Section 6]) we give an elliptic proof of this theorem. 
Yuntao Zhang and Kui Wang
math.DG math.AP