 
Summary: A Neumann Type Maximum Principle for the
Laplace Operator on Compact Riemannian
Manifolds
Guofang Wei and Rugang Ye
Department of Mathematics
University of California, Santa Barbara
Abstract
In this paper we present a proof of a Neumann type maximum principle
for the Laplace operator on compact Riemannian manifolds. A key p oint is
the simple geometric nature of the constant in the a priori estimate of this
maximum principle. In particular, this maximum principle can be applied to
manifolds with Ricci curvature bounded from below and diameter bounded
from above to yield a maximum estimate without dependence on a positive
lower bound for the volume.
1 Introduction
The main purpose of this paper is to present a proof of a Neumann type maximum
principle for the Laplace operator on a closed Riemannian manifold. As a key fea
ture of this maximum principle, the constant in the maximum estimate depends on
the Riemannian manifold only in terms of the dimension and the volumenormalized
Neumann isoperimetric constant. This allows us to apply it to manifolds with Ricci
