A New Geometric Flow on 3-Manifolds: the $K$-Flow
Abstract
We define a new geometric flow, which we shall call the -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 -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 -flow using a DeTurck-type argument.
Keywords
Cite
@article{arxiv.2308.01845,
title = {A New Geometric Flow on 3-Manifolds: the $K$-Flow},
author = {Kezban Tasseten and Bayram Tekin},
journal= {arXiv preprint arXiv:2308.01845},
year = {2023}
}
Comments
42 pages, 7 figures, a reference along with its explanation regarding the short-time existence and the uniqueness of the flow is added, v3 matches the published version