Almost-Rigidity and the Geometric Analysis of Lower Curvature Bounds Hunter Stufflebeam

Format:

Book

Thesis/Dissertation

Author/Creator:

Stufflebeam, Hunter, author.

Contributor:

University of Pennsylvania. Mathematics., degree granting institution.

Language:

English

Subjects (All):

Mathematics.

Mathematics education.

0405.

0280.

Local Subjects:

Mathematics.

Mathematics education.

0405.

0280.

Physical Description:

1 electronic resource (215 pages)

Contained In:

Dissertations Abstracts International 86-12A

Place of Publication:

Ann Arbor : ProQuest Dissertations and Theses, 2025

Language Note:

English

Summary:

A key paradigm in geometry holds that control on the curvature of a space yields control on other geometric and topological invariants, and engenders new relationships between them. In this dissertation, we prove new results to this end.Our main invariant of focus is the min-max width originally studied by Birkhoff (1917); Almgren (1962); Smith (1983), et cetera Over the last century, a long roster of researchers has studied this invariant in its many forms, and how it interacts with curvature bounds. In a beautiful paper, Marques and Neves (2012) investigated how the min-max width of a Riemannian 3-sphere, realized as the area of an unstable minimal 2-sphere inside of it, is controlled in terms of a scalar curvature lower bound. They show that this width has a sharp upper bound which, when achieved, implies that the entire 3-sphere must be isometric to the standard round 3-sphere.Our main results here seek to stabilize this theorem in various ways. Indeed, we ask what can be said of the 3-sphere when its width is almost-maximal, providing an effective generalization of Marques-Neves' rigidity theorem. In Stufflebeam and Sweeney Jr. (2024), reported in Chapter 3.3, we thereby address a conjecture of Marques-Neves (compare Sormani (2017a)). In Chapter 3.4 we outline a framework for fully resolving another form of the conjecture. In Maximo and Stufflebeam (2025), reported in Chapter 3.2, the author and Davi Maximo fully resolve the conjecture in 2-dimensions, thereby stabilizing a celebrated result of Toponogov. In the final Chapter 4 reporting our work in Stufflebeam (2023), we investigate a related problem informed by mathematical general relativity; we prove a stable version of the famous Min-Oo Conjecture in the only case where it is generally true-in dimension 2

Notes:

Source: Dissertations Abstracts International, Volume: 86-12, Section: A.

Advisors: Maximo, Davi Committee members: Ziller, Wolfgang; Hynd, Ryan

Ph.D. University of Pennsylvania 2025

Local Notes:

School code: 0175

ISBN:

9798280759992

Access Restriction:

Restricted for use by site license