A note on singularities in finite time for the L2 gradient flow of the Helfrich functional View Full Text


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Article Info

DATE

2019-02-01

AUTHORS

Simon Blatt

ABSTRACT

This work investigates the formation of singularities under the steepest descent L2-gradient flow of Wλ1,λ2 with zero spontaneous curvature, i.e., the sum of the Willmore energy, λ1 times the area, and λ2 times the signed volume of an immersed closed surface without boundary in R3. We show that in the case that λ1>1 and λ2=0, any immersion develops singularities in finite time under this flow. If λ1>0 and λ2>0, embedded closed surfaces with energy less than 8π+min16πλ13/3λ22,8πand positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than 8π, the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that λ2 is negative. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For λ1>0 and λ2≥0, they showed that embedded closed spheres with positive volume and energy close to 4π, i.e., close to the Willmore energy of a round sphere, converge to round points in finite time. More... »

PAGES

1-15

Journal

TITLE

Journal of Evolution Equations

ISSUE

N/A

VOLUME

N/A

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00028-019-00483-y

DOI

http://dx.doi.org/10.1007/s00028-019-00483-y

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1111841030


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