Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp017d278x189
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dc.contributor.authorDey, Akashdeep
dc.contributor.otherMathematics Department
dc.date.accessioned2022-06-16T20:34:23Z-
dc.date.available2022-06-16T20:34:23Z-
dc.date.created2022-01-01
dc.date.issued2022
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/dsp017d278x189-
dc.description.abstractIn this thesis, we prove some results in the variational theory of the minimal hypersurfaces, constant mean curvature (CMC) hypersurfaces and the Allen-Cahn equation. In Chapter 1, we summarize the main results. In Chapter 2, we show that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold tends to infinity as $c$ tends to $0^+$. In Chapter 3, we show that the space of closed singular minimal hypersurfaces (in a closed Riemannian manifold), whose areas are uniformly bounded from above and the $p$-th Jacobi eigenvalues are uniformly bounded from below, is sequentially compact. In Chapter 4, we prove a sub-additive inequality for the volume spectrum of a closed Riemannian manifold. In Chapter 5, we prove two results related to the question to what extent the Almgren-Pitts min-max theory and the Allen-Cahn min-max theory agree. In Chapter 6, we prove the existence of finite energy min-max solutions to the Allen-Cahn equation on a complete Riemannian manifold of finite volume.
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherPrinceton, NJ : Princeton University
dc.relation.isformatofThe Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: <a href=http://catalog.princeton.edu>catalog.princeton.edu</a>
dc.subjectAllen-Cahn equation
dc.subjectCMC hypersurface
dc.subjectMin-max method
dc.subjectMinimal hypersurface
dc.subject.classificationMathematics
dc.titleSome results in the variational theory of the area and related functionals