Viscosity solutions to second order parabolic PDEs on Riemannian manifolds

Abstract

In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold $M$ has nonnegative sectional curvature, we get the finest results. If one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, the assumptions on curvature can be thrown away.