자료유형  

 학위논문

Control Number  

0016932463

International Standard Book Number  

9798379717872

Dewey Decimal Classification Number  

510

Main Entry-Personal Name  

Wang, Zhihan.

Publication, Distribution, etc. (Imprint  

[S.l.] : Princeton University., 2023

Publication, Distribution, etc. (Imprint  

Ann Arbor : ProQuest Dissertations & Theses, 2023

Physical Description  

1 online resource(334 p.)

General Note  

Source: Dissertations Abstracts International, Volume: 84-12, Section: B.

General Note  

Advisor: Marques, Fernando Coda.

Dissertation Note  

Thesis (Ph.D.)--Princeton University, 2023.

Restrictions on Access Note  

This item must not be sold to any third party vendors.

Summary, Etc.  

요약Regularity theory of minimal hypersurfaces has long been studied since last century. A satisfactory regularity result for area-minimizing and stable minimal hypersurfaces in a manifold of dimension ≤ 7 has been established over 40 years ago. In higher dimensional spaces, area-minimizing or stable minimal hypersurfaces are known to potentially have singular sets, and the wild behavior of these singularities obstructs us from further exploration of global geometric behaviors of minimal hypersurfaces as well as applications in other fields. In this thesis, we will discuss how singular sets affect the global behavior of minimal hypersurfaces.First, we generalize a Theorem of Hardt-Simon [HS85] to approximate any minimizing (resp. viscosity mean convex) hypercone by complete smooth minimizing (resp. strictly mean convex) hypersurfaces. This suggests that singularity models for minimizing hypersurfaces may all be smoothable.Second, we develop the spectral analysis of the Jacobi operator on a closed locally stable minimal hypersurface with strongly isolated singularities. This in particular applies to all min-max minimal hypersurfaces in an eight manifold. We include primary application of this linear theory to obtain a local exitence and a local minimizing property of minimal hypersurface under reasonable spectral assumptions.Finally, in an ambient manifold of dimension 8, by further exploiting the linear analysis of the Jacobi operators, joint with Yangyang Li, we prove that under generic metric, every minimal hypersurface constructed by min-max approach are entirely smooth.

Subject Added Entry-Topical Term  

Mathematics.

Subject Added Entry-Topical Term  

Theoretical mathematics.

Index Term-Uncontrolled  

Generic regularity

Index Term-Uncontrolled  

Geometric variational problem

Index Term-Uncontrolled  

Minimal hypersurfaces

Index Term-Uncontrolled  

Singularity

Index Term-Uncontrolled  

Tangent cone

Added Entry-Corporate Name  

Princeton University Mathematics

Host Item Entry  

Dissertations Abstracts International. 84-12B.

Host Item Entry  

Dissertation Abstract International

Electronic Location and Access  

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Control Number  

joongbu:642773