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Finite time singularities for the locally constrained Willmore flow of surfaces

Journal Article


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Abstract


  • In this paper we study the steepest descent L2-gradient flow of the functional W, which is the the sum of the Willmore energy, weighted surface area, and 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.

Publication Date


  • 2016

Citation


  • 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.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=7166&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/6136

Number Of Pages


  • 43

Start Page


  • 843

End Page


  • 886

Volume


  • 24

Issue


  • 4

Place Of Publication


  • United Kingdom

Abstract


  • In this paper we study the steepest descent L2-gradient flow of the functional W, which is the the sum of the Willmore energy, weighted surface area, and 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.

Publication Date


  • 2016

Citation


  • 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.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=7166&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/6136

Number Of Pages


  • 43

Start Page


  • 843

End Page


  • 886

Volume


  • 24

Issue


  • 4

Place Of Publication


  • United Kingdom