| In this dissertation, we consider parabolic (e.g. Ricci flow) and elliptic (e.g. p-harmonic equations) partial differential equations on Riemannian manifolds and use them to study geometric and topological problems. More specifically, to classify a special class of Ricci flow equations, we constructed a family of new entropy functionals in the sense of Perelman. We study the monotonicity of these functionals and use this property to prove that a compact steady gradient Ricci breather is necessarily Ricci-flat. We introduce a new approach to prove the monotonicity formula of Perelman's W -entropy functional and we construct similar entropy functionals on expanders from this new viewpoint. We prove that a large family of complete non-compact Riemannian manifolds cannot be stably minimally immersed into Euclidean space as a hypersurface which serves as a non-existence theorem considering the Generalized Bernstein Conjecture. We give another yet simpler proof for a theorem of do Carmo and Peng, concerning stable minimal hypersurfaces in Euclidean space with certain integral curvature condition. In the study of p-harmonic geometry, we develop a classification theory of Riemannian manifolds by using p-superharmonic functions in the weak sense. We gave sharp estimates as sufficient conditions for a p-parabolic manifold. By developing a Generalized Uniformization Theorem, a Generalized Bochner's Method, and an iterative method, we approach various geometric and variational problems in complete noncompact manifolds of general dimensions. |
