Evolving convex curves

Calculus of Variations and Partial Differential Equations

Volumn 7, Issue 4, 1998, Pages 315-371

  Andrews, Ben ^a,b  

^a STANFORD UNIVERSITY   (United States)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000342399     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s005260050111     Document Type: Article

References (36)

1. [A1] Andrews, B: Evolving convex hypersurfaces. PhD thesis, Australian National University, 1993.
2. [A2] Andrews, B: Harnack inequalities for evolving hypersurfaces. Math. Z. 217 (1994), 179-197.
3. [A3] Andrews, B: Entropy estimates for evolving hypersurfaces. Comm. Analysis and Geometry 2 (1994), 53-64.
4. [A4] Andrews, B: Contraction of convex hypersurfaces by their affine normal. J. Differential Geometry 43 (1996), 207-230.
5. [A5] Andrews, B: Instability of limiting shapes for curves evolving by curvature. Preprint.
6. [A6] Andrews, B: Limiting shapes of smooth and crystalline curvature flows. Preprint.
7. [A7] Andrews, B: Classification of limiting shapes for isotropic curve flows. Preprint.
8. [A8] Andrews, B: Evolution of hypersurfaces by Gauss curvature. Preprint.
9. [AG] Angenent, S., Gurtin, M.: Multiphase thermodynamics with interfacial structure 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989), 323-391.
10. [An] Angenent, S.: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 380 (1988), 79-96.
11. [Ch1] Chow, B.: Harnack's Inequality and Entropy for the Gaussian Curvature flow. Comm. Pure Appl. Math. 44 (1991), 469-483.
12. [Ch2] Chow, B.: Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations. Preprint, 19 pages.
13. [CG] Chow, B., Gulliver, R.: Aleksandrov reflection and nonlinear evolution equations I: The n-sphere and the n-ball. Calc. Var. Partial Differential Equations 4 (1996), 249-264.
14. [CHT] Taylor, J. E., Cahn, J. W., Handwerker, C. A.: Geometric models of crystal growth. Acta Metall. mater. 40 (1992), 1443-1474.
15. [CT] Chow, B., Tsai, D. H.: Geometric expansion of convex plane curves. J. Differential Geometry 44 (1996), 312-330.
16. [B] Blaschke, W.: Über Affine Geometrie I: Isoperimetrische Eigenschaften von Ellipse und Ellipsoid. Ber. Verh. Sächs. Akad. Leipzig, Math.-Phys. Kl. 68 (1916), 217-39.
17. [Fi] Firey, W. J.: Shapes of worn stones. Mathematika 21 (1974), 1-11.
18. [Ga1] Gage, M. E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50 (1983), 1225-1229.
19. [Ga2] Gage, M. E.: Curve shortening makes convex curves circular. Invent. Math. 76 (1984), 357-364.
20. [Ga3] Gage, M. E.: Positive centers and the Bonnesen inequality. Proc. Amer. Math. Soc. 110 (1990), 1041-1048.
21. [Ga4] Gage, M. E.: Evolving plane curves by curvature in relative geometries. Duke Math. J. 72 (1993), 441-466.
22. [G] Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differential Geometry 32 (1990), 199-314.
23. [GH] Gage, M. E., Hamilton, R. S.: The heat equation shrinking convex plane curves. J. Differential Geometry 23 (1986), 69-96.
24. [GL] Gage, M., Li, Y.: Evolving plane curves by curvature in relative geometries II. Duke Math. J. 75 (1994), 79-98.
25. [Gr] Grayson, M.: The heat equation shrinks embedded plane curves to points. J. Differential Geometry 26 (1987), 285-314.
26. [Ha1] Hamilton, R. S.: Three-manifolds with positive Ricci curvature. J. Differential Geometry 17 (1982), 255-306.
27. [Ha2] Hamilton, R. S.: Four-manifolds with positive curvature operator. J. Differential Geometry 24 (1986), 153-179.
28. [Hu1] Huisken, G.: Flow by mean curvature of convex hypersurfaces into spheres. J. Differential Geometry 20 (1984), 237-268.
29. [Hu2] Huisken, G.: On the expansion of convex hypersurfaces by the inverse of symmetric curvatures functions. To appear.
30. [K] Krylov, N. V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. D. Reidel (1987).
31. [Sc] Schneider, R.: Convex bodies: The Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications Vol. 44, Cambridge University Press (1993).
32. [ST] Sapiro, G., Tannenbaum, A.: On affine plane curve evolution. J. Functional Analysis 119 (1994), 79-120.
33. [T] Tso, K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Comm. Pure Appl. Math. 38 (1985), 867-882.
34. [Ts] Tsai, D. H.: Geometric expansion of star-shaped plane curves. Comm. Analysis and Geometry 4 (1996), 459-480.
35. [U1] Urbas, J. I. E.: An expansion of convex surfaces. J. Differential Geometry 33 (1991), 91-125.
36. [U2] Urbas, J. I. E.: On the expansion of star-shaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205 (1990), 355-372.