Formal asymptotics for blowup in the Willmore flow - Eurandom
Formal asymptotics for blowup in the Willmore flow - Eurandom Formal asymptotics for blowup in the Willmore flow - Eurandom
Curvatures Let κ1 and κ2 be the maximal and minimal curvatures of a surface at a particular point. The mean and Gaussian curvatures at that point are given by H = 1 2 (κ1 + κ2), K = κ1κ2. Willmore flow : ¯φt = −∆H − 2H(H 2 − K).
The sphere Consider a sphere of radius R. If we choose the normal such that it points outwards, then everywhere. Hence, H = − 1 R on the whole sphere. We see immediately that κ1 = κ2 = − 1 R , 1 and K = , R2 ¯φt = −∆H − 2H(H 2 − K) = 0.
- Page 1 and 2: Formal asymptotics for blowup in th
- Page 3 and 4: Introduction Consider an elastic su
- Page 5 and 6: The Willmore flow I Since the integ
- Page 7: The Willmore flow II The Willmore f
- Page 11 and 12: The sphere and more ◮ The Willmor
- Page 13 and 14: Problem Can the Willmore flow creat
- Page 15 and 16: Numerical computations Numerical co
- Page 17 and 18: Matched asymptotics On every region
- Page 19 and 20: Results ◮ Matching gives (T − t
Curvatures<br />
Let κ1 and κ2 be <strong>the</strong> maximal and m<strong>in</strong>imal curvatures of a surface<br />
at a particular po<strong>in</strong>t. The mean and Gaussian curvatures at that<br />
po<strong>in</strong>t are given by<br />
H = 1<br />
2 (κ1 + κ2),<br />
K = κ1κ2.<br />
<strong>Willmore</strong> <strong>flow</strong> : ¯φt = −∆H − 2H(H 2 − K).
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