Finite time singularities for the locally constrained Willmore flow of surfaces

Authors

James A. McCoy, University of WollongongFollow
Glen Wheeler, Otto-Von-Guericke-UniversitatFollow

RIS ID

46130

Publication Details

McCoy, J. A. & Wheeler, G. E. (2016). Finite time singularities for the locally constrained Willmore flow of surfaces. Communications in Analysis and Geometry, 24 (4), 843-886.

Abstract

In this paper we study the steepest descent L²-gradient flow of the functional W_λ1,λ2, which is the the sum of the Willmore energy, λ1-weighted surface area, and λ2-weighted enclosed volume, for surfaces immersed in R³. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in L² we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time.

Grant Number

ARC/DP120100097

Grant Number

ARC/DP0556211