A New Geometric Flow On Three Manifolds

Date

2023-8-23

Author

Taşseten Ata, Kezban

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In this thesis, we define a new geometric flow, which we shall call the K-flow, on 3-dimensional Riemannian manifolds; and study the behavior of Thurston’s model geometries under this flow both analytically and numerically. As an example, we show that an initially arbitrarily deformed homogeneous 3-sphere flows into a round 3-sphere and shrinks to a point in the unnormalized flow; or stays as a round 3-sphere in the volume normalized flow. The K-flow equation arises as the gradient flow of a specific purely quadratic action functional that has appeared as the quadratic part of New Massive Gravity in physics; and a decade earlier in the mathematics literature, as a new variational characterization of three-dimensional space forms. We show the short-time existence of the K-flow using a DeTurck-type argument.

Subject Keywords

gradient flow, geometric evolution, geometry and topology of 3-manifolds

URI

https://hdl.handle.net/11511/105494

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Graduate School of Natural and Applied Sciences, Thesis