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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Acknowledgments
We would like to thank J.D. Streets for pointing out his work and solutions on the existence-uniqueness issues, and Sahin Ulas Koprucu for his help with the Matlab code.
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Tasseten, K., Tekin, B. A new geometric flow on 3-manifolds: the K-flow. J. High Energ. Phys. 2023, 114 (2023). https://doi.org/10.1007/JHEP10(2023)114
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DOI: https://doi.org/10.1007/JHEP10(2023)114