Dissertation
Fully nonlinear equations in conformal geometry
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2025
DOI: 10.25820/etd.007867
Abstract
This thesis consists of two parts on the study of some fully nonlinear equations in conformal geometry.
In the first part, we study Liouville's equation and define a new conformal invariant on complete non-compact hyperbolic surfaces that can be conformally compactified to bounded domains in C. We study and compute this invariant up to doubly connected surfaces. Our results give a new geometric criterion for choosing canonical representations of bounded domains in C.
In the second part, we study the regularity of Lipschitz viscosity solutions to the sigma_k Yamabe problem in the negative cone case on n-dimensional manifolds with n>=3. If either k=n or the manifold is conformally flat and k>n/2, we prove that all Lipschitz viscosity solutions are smooth away from a closed set of measure zero. For the general k>n/2 case, under certain assumptions, we prove the existence of a Lipschitz viscosity solution that is smooth away from a closed set of measure zero.
Details
- Title: Subtitle
- Fully nonlinear equations in conformal geometry
- Creators
- Jinyang Wu
- Contributors
- Hao Fang (Advisor)Lihe Wang (Committee Member)Mohammad Farajzadeh Tehrani (Committee Member)Tong Li (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2025
- DOI
- 10.25820/etd.007867
- Publisher
- University of Iowa
- Number of pages
- vii, 70 pages
- Copyright
- Copyright 2025 Jinyang Wu
- Language
- English
- Date submitted
- 04/15/2025
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 61-70).
- Public Abstract (ETD)
- Conformal geometry studies geometric properties invariant under angle- preserving transformations. In many cases, problems in conformal geometry can be reduced to solving a fully nonlinear geometric PDE. This thesis con- sists of two parts, focusing on the study of some fully nonlinear equations in conformal geometry. In the rst part, we study Liouville's equation, which is a partial dierential equation in dimension two. We explore the geometric meaning of a quantity dened as the integral of the rst global term in the asymptotic expansion of the solution to Liouville's equation near the boundary of the domain, and dene a new conformal invariant on complete non-compact hyperbolic surfaces that can be conformally compactied to bounded domains in C. We study and compute this invariant up to doubly connected surfaces. In the second part, we study the regularity of Lipschitz viscosity solu- tions to the σk Yamabe problem in the negative cone case on n-dimensional manifolds with n ≥ 3. A viscosity solution is a type of weak solution. It is important to know where a viscosity solution is regular, in other words, physically/geometrically meaningful. Our results show that for all known Lip- schitz viscosity solutions when k > n/2, the solution is smooth, hence regular, outside a closed measure zero set.
- Academic Unit
- Mathematics
- Record Identifier
- 9984830729302771
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