Refined gradient bounds, Poisson equations and some applications to open Kähler manifolds

Abstract

Under some natural curvature assumptions on noncompact manifolds, we prove that the Poisson and Poincare-Lelong equation Au = / and yf^lddu = p can be solved when / and p are in the long range, i.e. when they decay at a slower rate than l/d(x) near infinity.This extends, to the long range case, earlier results in [MSY] and [NST] which treated the case when / decays faster than l/d(x).The improvement is based on a refined gradient estimate for harmonic and caloric functions.Some applications to the problems of curvature characterization of Stein manifolds are given.

Locations

• Asian Journal of Mathematics - View - PDF