On a free boundary problem for embedded minimal surfaces and instability theorems for manifolds with positive isotropic curvature

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In this thesis, we describe a min-max construction of embedded minimal surfaces satisfying the free boundary condition in any compact 3-manifolds with boundary. We also prove the instability of minimal surfaces of certain conformal type in 4- manifolds with positive isotropic curvature. Given a compact 3-manifold M with boundary [d̳]M, consider the problem of find- ing an embedded minimal surface [Sigma] which meets [d̳]M orthogonally along [d̳][Sigma]. These surfaces are critical points to the area functional with respect to variations preserving [delta]M. We will use a min-max construction to construct such a free boundary solution and prove the regularity of such solution up to the free boundary. An interesting point is that no convexity assumption on [d̳]M is required. We also discuss some geometric properties, genus bounds for example, for these free boundary solutions. Just as positive sectional curvature tends to make geodesics unstable, positive isotropic curvature tends to make minimal surfaces unstable. In the second part of this thesis, we prove a similar instability result in dimension 4. Given a compact 4- manifold M with positive isotropic curvature, we show that any complete immersed minimal surface [Sigma] in M which is uniformly conformally equivalent to the complex plane is unstable. The same conclusion holds in higher dimensions as well if we assume that the manifold has uniformly positive complex sectional curvature. The proof uses the H ̈ormander's weighted L^2 method and the stability inequality to derive a contradiction.


Type of resource text
Form electronic; electronic resource; remote
Extent 1 online resource.
Publication date 2011
Issuance monographic
Language English


Associated with Li, Man Chun
Associated with Stanford University, Department of Mathematics
Primary advisor Schoen, Richard (Richard M.)
Thesis advisor Schoen, Richard (Richard M.)
Thesis advisor Simon, L. (Leon), 1945-
Thesis advisor White, Brian, 1957-
Advisor Simon, L. (Leon), 1945-
Advisor White, Brian, 1957-


Genre Theses

Bibliographic information

Statement of responsibility Man Chun Li.
Note Submitted to the Department of Mathematics.
Thesis Ph.D. Stanford University 2011
Location electronic resource

Access conditions

© 2011 by Man Chun Li
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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