Green's function estimates and the Poisson equation

Ovidiu Munteanu (Storrs, Connecticut)

Abstract: The Green's function of the Laplace operator has been 
widely studied in geometric analysis. Manifolds admitting a 
positive Green's function are called nonparabolic. By Li and Yau, 
sharp pointwise decay estimates are known for the Green's function 
on nonparabolic manifolds that have nonnegative Ricci curvature. 
The situation is more delicate when curvature is not nonnegative 
everywhere. While pointwise decay estimates are generally not 
possible in this case, we have obtained sharp integral estimates 
for the Green's function on manifolds admitting a Poincare 
inequality and an appropriate (negative) lower bound on Ricci 
curvature. This has applications to solving the Poisson equation, 
and to the study of the structure at infinity of such manifolds.