Gradient estimates of a nonlinear Lichnerowicz equation under the Finsler-geometric flow

Abstract

Let (Mn, F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow ∂g(x,t)∂t = 2h(x, t), where g(t) is the symmetric 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 ∂tu(x, t) = ∆mu(x, t) + c(u(x, t))p, (x, t) ∈ M × [0, T], under Finsler-geometric flow ∂g(x,t) ∂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.