Green functions on stationary varifolds

Abstract

We establish a local Harnack inequality in a neighborhood of an indecomposable singular point of a stationary integral varifold. Extending the method of Gr\"uter and Widman \cite{gruter1982green}, we construct the Green function on a stationary integral varifold with Euclidean volume growth, allowing the pole to be any point in the support. Using the local Harnack inequality, we show that if a sequence of stationary varifolds converges with multiplicity one, then the corresponding Green functions converge as well. As applications, we determine the asymptotic behavior of the Green function both near an indecomposable pole and at infinity. We further establish global lower and upper bounds for the Green function and present an application of these estimates. Finally, we analyze the behavior of the Green function when the varifold is decomposable at a point, including examples illustrating the possible phenomena.