Dissertation
Analysis on geometric singularities
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2022
DOI: 10.25820/etd.006513
Abstract
This thesis consists of 4 parts on the analysis of singularities that arise from conformal geometry, geometric flows, and geometric inequalities.
In the first part, we study the constant Q-curvature metrics on conic 4 manifolds. We introduce 3 different types of configurations of conic singularities: stable, semi-stable, and unstable. In the stable case, we prove the existence of a constant Q-curvature metric in the conformal class. For conic 4-sphere with at most 2 singularities, we classify the solutions. They are rotational symmetric metrics on conic 4-sphere with identical cone type at each singularity and they correspond to the semi-stable configuration. Along the study, we derive an asymptotic expansion of the constant Q-curvature metric near conic singularities which is of independent interest.
In the second part, we study the curve-shortening-flow (CSF) on Riemann surfaces with possible ambient conic singularities. We generalize a comparison function due to Huisken which was originally defined for CSF on plane to surfaces. We obtain a new proof of the classic Gage-Hamilton-Grayson theorem on surfaces by studying the first time singularities of the flow. If the ambient Riemann surface admits conic singularities, we prove that under the flow, embedded simple closed curves cannot touch conic singularities with cone angles less than or equal to π.
In the third part, we generalize the Andrews’ inequality to manifolds with conic singularities and with boundary. We establish the rigidity result when the equality holds.
In the last part of the thesis, we prove a Liouville type theorem for an one parameter family of 2-Hessian type fully non-linear elliptic equations on R^4 . When the parameter ρ runs from 1 to 0, the equations interpolate between the σ2 -Yamabe equation and the 2-Hessian equation. Our theorem holds for ρ > 0. Our approach contains 2 important ingredients: one is a monotonic quasi-local mass type quantity defined on the level sets of the solution; another is a series of estimates which controls the asymptotic shape of level sets of u at infinity.
Details
- Title: Subtitle
- Analysis on geometric singularities
- Creators
- Biao Ma
- Contributors
- Hao Fang (Advisor)Celal Oguz Durumeric (Committee Member)Mohammad Tehrani (Committee Member)Lihe Wang (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2022
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.006513
- Number of pages
- x, 190 pages
- Copyright
- Copyright 2022 Biao Ma
- Language
- English
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 170-190).
- Public Abstract (ETD)
This thesis consists of 4 parts on the analysis of singularities. We mainly consider the cone-like singularities and their effects on the local and global geometry.
In the first part, we study the constant Q-curvature metrics on 4-manifolds with cone-like singularities. We define 3 types of configurations of singularities: stable, semi-stable, and unstable. In the stable case, we prove the existence of a constant Q-curvature metric. For 4-sphere with at most 2 singularities, we classify the solutions: they are semi-stable and look like an American football.
In the second part, we study the curve-shortening-flow on surfaces with possible ambient singularities. We generalize a comparison function due to Huisken which was originally defined on plane. If the ambient surface admits cone-like singularities, we prove that under the flow, simple curves cannot touch conic singularities with small angles (≤ π).
In the third part, we generalize the Andrews’ inequality to manifolds with cone-like singularities and with boundary. We establish the rigidity result when the equality holds.
In the last part of the thesis, we prove that an one parameter family of fully non-linear elliptic equations on R4 only admits radially symmetric solutions.
- Academic Unit
- Mathematics
- Record Identifier
- 9984271155402771
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