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The geometric triharmonic heat flow of immersed surfaces near spheres

Journal Article


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Abstract


  • We consider closed immersed surfaces in R^3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L^2. We further use an {\epsilon}-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-umbilic embedded stationary surface. This allows us to conclude that any solution with initial L^2-norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to the cube root of 3V_0/4{\pi}, where V_0 denotes the signed enclosed volume of the initial data.

Publication Date


  • 2017

Citation


  • McCoy, J., Parkins, S. & Wheeler, G. (2017). The geometric triharmonic heat flow of immersed surfaces near spheres. Nonlinear Analysis, 161 44-86.

Scopus Eid


  • 2-s2.0-85020904472

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1370&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/369

Number Of Pages


  • 42

Start Page


  • 44

End Page


  • 86

Volume


  • 161

Place Of Publication


  • United Kingdom

Abstract


  • We consider closed immersed surfaces in R^3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L^2. We further use an {\epsilon}-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-umbilic embedded stationary surface. This allows us to conclude that any solution with initial L^2-norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to the cube root of 3V_0/4{\pi}, where V_0 denotes the signed enclosed volume of the initial data.

Publication Date


  • 2017

Citation


  • McCoy, J., Parkins, S. & Wheeler, G. (2017). The geometric triharmonic heat flow of immersed surfaces near spheres. Nonlinear Analysis, 161 44-86.

Scopus Eid


  • 2-s2.0-85020904472

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1370&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/369

Number Of Pages


  • 42

Start Page


  • 44

End Page


  • 86

Volume


  • 161

Place Of Publication


  • United Kingdom