Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01c247dw26c
 Title: Min-max minimal hypersurfaces in higher dimensions Authors: Li, Yangyang Advisors: Marques, Fernando C. Contributors: Mathematics Department Keywords: generic regularitymin-max theoryminimal hypersurfacessingularity Subjects: Mathematics Issue Date: 2022 Publisher: Princeton, NJ : Princeton University Abstract: In the recent decade, the Almgren-Pitts min-max theory has advanced the existence theory of minimal hypersurfaces in a closed Riemannian manifold $(M^{n+1}, g)$. When $3 \leq n+1 \leq 7$, many properties of these minimal hypersurfaces, such as areas, Morse indices, multiplicities, and spatial distributions, have been well studied. However, in higher dimensions ($n+1 \geq 8$), min-max minimal hypersurfaces may contain singularities. This phenomenon invalidates many helpful techniques in the low dimensions to investigate these geometric objects. I will show how one can utilize various deformation arguments to overcome the obstacles and prove generic abundance, index estimates, and most of the geometric properties of min-max minimal hypersurfaces. In particular, in dimension eight, en route to obtaining generic results, in joint work with Zhihan Wang, we prove generic regularity of minimal hypersurfaces. URI: http://arks.princeton.edu/ark:/88435/dsp01c247dw26c 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