A Convergence Result for the Gradient Flow of $\int |A|^2$ in Riemannian Manifolds

TL;DR

This paper investigates the gradient flow of the $L^2$-norm of the second fundamental form for surfaces in Riemannian manifolds, establishing lifespan estimates, blowup limits, and conditions for long-term convergence to critical immersions.

Contribution

It provides new lifespan estimates and convergence results for the gradient flow of the second fundamental form in Riemannian manifolds, extending analogous results from Willmore flow.

Findings

Lifespan estimates based on initial $L^2$-concentration.

Existence of blowup limits for the flow.

Long-time existence and subconvergence under specific conditions.

Abstract

We study the gradient flow of the $L^2-$norm of the second fundamental form of smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained for the Willmore flow in Riemannian manifolds, we prove lifespan estimates in terms of the $L^2-$concentration of the second fundamental form of the initial data and we show existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and subconvergence to a critical immersion.