Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp017d278x189
 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 equationCMC hypersurfaceMin-max methodMinimal 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. URI: http://arks.princeton.edu/ark:/88435/dsp017d278x189 Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: catalog.princeton.edu Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Mathematics