Brian White - Mean Curvature Flow (math 258) Lecture Notes Notes by Otis Chodosh
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چکیده
1. Overview 2 1.1. Curve shortening flow 2 1.2. Flow of hypersurfaces 5 1.3. Mean convex surfaces 6 2. The maximum principle 7 3. Unparameterized mean curvature flow 8 3.1. Graphs 8 4. Short-time existence and smoothing 9 5. Long term behavior of mean curvature flow 9 6. Renormalized mean curvature flow 11 7. The level set approach to weak limits 13 8. Weak compactness of submanifolds 16 8.1. Examples of varifold convergence 17 8.2. The pushforward of a varifold 18 8.3. Integral varifolds 18 8.4. First variation of a varifold 18 9. Brakke flow 21 9.1. A compactness theorem for integral Brakke flows 23 9.2. Self shrinkers 24 9.3. Existence by elliptic regularization 25 9.4. Why doesn’t the flow disappear immediately? 28 9.5. Ilmanen’s enhanced flow 28 10. Monotonicity and entropy 29 10.1. Monotonicity for mean curvature flow 30 10.2. Gaussian density 30 11. A version of Brakke’s regularity theorem 32 12. Stratification 34 12.1. Ruling out the worst singularities 37 13. Easy parity theorem 37 14. The maximum principle 38 15. Mean convex flows 40 15.1. Curvature estimates along the flow 40 15.2. One-sided minimization 40 15.3. Blow-up limits of mean convex flows 41 References 43
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تاریخ انتشار 2015