Skip navigation
Please use this identifier to cite or link to this item:
Title: Some results in the variational theory of the area and related functionals
Authors: Dey, Akashdeep
Advisors: Marques, Fernando C.
Contributors: Mathematics Department
Keywords: Allen-Cahn equation
CMC hypersurface
Min-max method
Minimal hypersurface
Subjects: Mathematics
Issue Date: 2022
Publisher: Princeton, NJ : Princeton University
Abstract: In 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.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Mathematics

Files in This Item:
File Description SizeFormat 
Dey_princeton_0181D_14119.pdf987.88 kBAdobe PDFView/Download

Items in Dataspace are protected by copyright, with all rights reserved, unless otherwise indicated.