Calculus of Variations and Geometric Measure Theory

A. Magni

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

created by magni on 12 May 2014
modified on 05 Nov 2014


Accepted Paper

Inserted: 12 may 2014
Last Updated: 5 nov 2014

Journal: Geometric Flows
Year: 2014


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.