Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01c247dw26c
DC FieldValueLanguage
dc.contributor.authorLi, Yangyang
dc.contributor.otherMathematics Department
dc.date.accessioned2022-06-16T20:34:49Z-
dc.date.available2022-06-16T20:34:49Z-
dc.date.created2022-01-01
dc.date.issued2022
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/dsp01c247dw26c-
dc.description.abstractIn 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.
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.subjectgeneric regularity
dc.subjectmin-max theory
dc.subjectminimal hypersurfaces
dc.subjectsingularity
dc.subject.classificationMathematics
dc.titleMin-max minimal hypersurfaces in higher dimensions