Hessian estimate for heat equation
The author derives an interior Hessian estimate for the scalar heat equation on a complete Riemannian manifold \((M^n,g)\) which is independent of the covariant derivatives of the curvature. More precisely, assume that the curvature operator is bounded, \(|\mathrm{Rm}|\leq K\) on the geodesic ball \(B_p(\rho)\) centered at \(p\) and with radius \(\rho,\) and consider a function \(u\) on \(B_p(\rho)\times[0,2T]\) that solves the heat equation \(\Delta u=u_t\) on \((M^n,g).\) Then the Hessian of \(u\) satisfies the bound \[ \sup_{B_p(\rho/2),T\leq t\leq 2T}|\nabla \mathrm{d}u|(x,t)\leq C(K,T,\rho) \sup_{B_p(\rho),0\leq t\leq 2T}|u|(x,t). \] The proof relies on the fact that the differential \(\mathrm{d}u\) is a solution of the heat equation for the Hodge Laplacian on one-forms, and on suitable combination of the Bochner-Weitzenbock formula with maximum principle arguments.
scientific article; zbMATH DE number 6859108