On the variational methods for minimal submanifolds
Abstract/Contents
- Abstract
- In this thesis, we study two problems concerning variational methods for minimal submanifolds. Specifically we consider min-max theory for minimal surfaces. We construct the "saddle point" type critical points for the area functional, and study the geometrical properties of these. In the first part, we consider the mapping problem for the min-max theory. In particular, we prove an existence theorem for min-max minimal surfaces of arbitrary genus g ≥ 2 by variational methods. We show that the min-max critical value for the area functional can be achieved by the bubbling limit of branched minimal surfaces with nodes of genus g together with possibly finitely many branched minimal spheres. We also prove a strong convergence theorem similar to the classical mountain pass lemma. In the second part, we consider the geometric measure theory approach to the min-max theory. We study the shape of the min-max minimal hypersurface constructed by Almgren-Pitts corresponding to the fundamental class of a Riemannian manifold (M, g) of dimension n + 1 with positive Ricci curvature and 2 ≤ n ≤ 6. We characterize the Morse index, area and multiplicity of this min-max hypersurface. Precisely, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2013 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Zhou, Xin |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Schoen, Richard (Richard M.) |
| Thesis advisor | Schoen, Richard (Richard M.) |
| Thesis advisor | Brendle, Simon, 1981- |
| Thesis advisor | White, Brian, 1957- |
| Advisor | Brendle, Simon, 1981- |
| Advisor | White, Brian, 1957- |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Xin Zhou. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2013. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2013 by Xin Zhou
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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