Planar Wulff shape is unique equilibrium

Morgan, Frank

doi:10.1090/S0002-9939-04-07661-0

Planar Wulff shape is unique equilibrium
HTML articles powered by AMS MathViewer

by Frank Morgan PDF

Proc. Amer. Math. Soc. 133 (2005), 809-813

Abstract:

In $\mathbf {R}^2$, for any norm, an immersed closed rectifiable curve in equilibrium for fixed area must be the Wulff shape (possibly with multiplicity).

References

A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. I, Amer. Math. Soc. Transl. (2) 21 (1962), 341–354. MR 0150706, DOI 10.1090/trans2/021/09
Hiroshi Mori, On surfaces of right helicoid type in $H^{3}$, Bol. Soc. Brasil. Mat. 13 (1982), no. 2, 57–62. MR 735120, DOI 10.1007/BF02584676
Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. MR 1043269, DOI 10.2307/1971494
Nicolaos Kapouleas, Constant mean curvature surfaces in Euclidean three-space, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 318–320. MR 903742, DOI 10.1090/S0273-0979-1987-15575-3
Frank Morgan, Cylindrical surfaces of Delaunay, preprint (2003).
Frank Morgan, Hexagonal surfaces of Kapouleas, Pacific J. Math., to appear.
Frank Morgan, Riemannian geometry, 2nd ed., A K Peters, Ltd., Wellesley, MA, 1998. A beginner’s guide. MR 1600519
Bennett Palmer, Stability of the Wulff shape, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3661–3667. MR 1473676, DOI 10.1090/S0002-9939-98-04641-3
Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), no. 4, 568–588. MR 493671, DOI 10.1090/S0002-9904-1978-14499-1
Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243. MR 815044