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On the rate of convergence of the rescaled mean curvature flow and other geometric problems
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Martin-Hagemeyer, Rory. On the rate of convergence of the rescaled mean curvature flow and other geometric problems. Retrieved from https://doi.org/doi:10.7282/t3-2d4s-z438

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TitleOn the rate of convergence of the rescaled mean curvature flow and other geometric problems
NameMartin-Hagemeyer, Rory (author); Sesum, Natasa (chair); Woodward, Christopher (member); Li, Yanyan (member); Knopf, Dan (member); Rutgers University; School of Graduate Studies
Date Created2022
Other Date2022-10 (degree)
SubjectMathematics, Geometric analysis, Geometric flows, Mean curvature flow
Extent58 pages
DescriptionIn this thesis, we present two results from fields situated at two different extremities of the broader subject of (semi)-Riemannian geometry: mean curvature flow, and the investigation of algebraic properties of curvature tensors and their derivatives. First, in the field of mean curvature flow, I show that the rate of convergence of any solution to rescaled mean curvature flow which is not a shrinker moving by diffeomorphisms is at most some fixed exponential rate. We prove this by defining a new flow, the normal rescaled mean curvature flow, which is equivalent up to tangential diffeomorphisms to the rescaled mean curvature flow. In this new flow, evolution equations are very easy to compute as it has no tangential component. We calculate the evolution equations of key geometric quantities in this new flow and use the calculations to define a suitable energy functional and compute its evolution equations. With these evolution equations, we define a Dirichlet Einstein quotient and prove its boundedness for all time. Next, we show that the standard representation of GL(n) on the space of algebraic covariant derivative curvature tensors is faithful. This theorem was part of a 2018 paper joint with Corey Dunn and Darrien Elderfield in which it was also shown that the standard representation of $GL(n)$ on the space of algebraic curvature tensors is almost faithful, with Kernel ±I. Both results are proven via the Jordan normal form, we shall give a full proof of the former result.
NotePh.D.
NoteIncludes bibliographical references
Genretheses
Persistent URLhttps://doi.org/doi:10.7282/t3-2d4s-z438
LanguageEnglish
CollectionSchool of Graduate Studies Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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