Geometric evolution equations and p-harmonic theory with applications in differential geometry.
| dc.contributor.advisor | Wei, Shihshu, | en_US |
| dc.contributor.author | Li, Jun-fang. | en_US |
| dc.date.accessioned | 2013-08-16T12:20:17Z | |
| dc.date.available | 2013-08-16T12:20:17Z | |
| dc.date.issued | 2006 | en_US |
| dc.description.abstract | 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. | en_US |
| dc.format.extent | ix, 77 leaves ; | en_US |
| dc.identifier.uri | http://hdl.handle.net/11244/1053 | |
| dc.note | Adviser: Shihshu Wei. | en_US |
| dc.note | Source: Dissertation Abstracts International, Volume: 67-05, Section: B, page: 2591. | en_US |
| dc.subject | Differential equations, Partial. | en_US |
| dc.subject | Riemannian manifolds. | en_US |
| dc.subject | Ricci flow. | en_US |
| dc.subject | Mathematics. | en_US |
| dc.thesis.degree | Ph.D. | en_US |
| dc.thesis.degreeDiscipline | Department of Mathematics | en_US |
| dc.title | Geometric evolution equations and p-harmonic theory with applications in differential geometry. | en_US |
| dc.type | Thesis | en_US |
| ou.group | College of Arts and Sciences::Department of Mathematics | |
| ou.identifier | (UMI)AAI3218125 | en_US |
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