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

TL;DR

This paper introduces the $K$-flow, a new geometric flow on 3-manifolds, analyzing its behavior on Thurston geometries and demonstrating convergence to round spheres, with connections to physics and variational principles.

Contribution

It defines the $K$-flow, proves short-time existence, and explores its effects on Thurston geometries, linking geometric analysis with physics-inspired functionals.

Findings

Homogeneous 3-sphere flows into a round sphere and shrinks to a point.

Round 3-sphere remains unchanged under volume-normalized flow.

The $K$-flow is the gradient flow of a quadratic action functional related to physics.

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.