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arXiv · 2308.01845

A New Geometric Flow on 3-Manifolds: the $K$-Flow

Kezban Tasseten·Bayram Tekin
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Abstract

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.

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BibTeXRIS

Kezban Tasseten, Bayram Tekin. 2023-08-03. A New Geometric Flow on 3-Manifolds: the $K$-Flow. https://doi.org/10.1007/jhep10(2023)114

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