The gradient flow of the $L^2$ curvature energy on surfaces

TL;DR

This paper studies the evolution of surface geometries under the gradient flow of the squared curvature energy, proving long-term existence and convergence to constant curvature metrics under certain energy conditions.

Contribution

It establishes long-time existence and exponential convergence of the flow on surfaces, extending understanding of curvature-driven geometric evolution.

Findings

Flow exists for all time from arbitrary initial data.

Flow converges exponentially to a constant scalar curvature metric.

Convergence depends on initial energy being below a specific threshold.

Abstract

We investigate the gradient flow of the $L^2$ norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature metric when the initial energy is below a constant determined by the Euler characteristic of the underlying surface.