Complete Non-compact Riemannian Manifold On The Maximum Principle

In this paper we discuss the maximum principles on noncompact manifolds analogues to compact case.It is important to control the growth of the curvature as the distance tends to infinity.We extend the maximum principle to the quasi-linear versions of the heat equation.Also we discuss the tensor version of the maximum principles and Weinberger-Hamilton's maximum principles.Since the curvatures evolve as heat-type parabolic equations under Ricci flow,we discuss some applications of the maximum principles in Ricci flow at last.