GRADIENT ESTIMATES OF A NONLINEAR LICHNEROWICZ EQUATION UNDER THE FINSLER-GEOMETRIC FLOW

Let (M-n, F(t), m), t E [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow eth g(x,t)/ eth t = 2h(x, t), where g(t) is the symmet-ric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we study gradient estimates for positive solutions of a nonlinear Lichnerowicz equation eth (t)u(x, t) = ?(m)u(x, t) + c(u(x, t))(p), (x, t) is an element of M x [0, T],under Finsler-geometric flow eth g(x,t)/ eth t = 2h(x, t), where c, p are two constants and p > 0. We obtain a global estimate and a Harnack estimate for positive solutions. Our results are also natural extension of similar results on Riemannian-geometric flow.