ELLIPTIC-TYPE GRADIENT ESTIMATES UNDER INTEGRAL RICCI CURVATURE BOUNDS

Let (M-n, g) be an n-dimensional complete Riemannian manifold. We prove that for any p > n/2, when k(p, 1) is small enough, two certain elliptic-type gradient estimates hold for any positive solutions of the equation u(t) =Delta u + au log u on geodesic balls B(O, r) in M-n with 0 < r <= 1. Here the assumption that k(p, 1) is small allows the situation where the manifold is collapsing. As applying, two forms of elliptic type gradient estimates to the heat equation on M-n under integral curvature conditions are obtained. Besides, each of the two forms of gradient estimate has its own advantages and differences (see Remark 4).

Automated summary

This paper proves the existence of specific elliptic-type gradient estimates that can be applied to equations on complete Riemannian manifolds. These gradient estimates are crucial for understanding the geometric properties of the manifold.