Surfaces with Constant Mean Curvature
In this chapter we shall review some basic aspects of the theory of surfaces with constant mean curvature. Surfaces with constant mean curvature will arise as solutions of a variational problem associated to the area functional and related with the classical isoperimetric problem. We state the first and second variation formula for the area and we give the notion of stability of a cmc surface. Next, we introduce the complex analysis as a basic tool in the theory and this will allow to prove the Hopf theorem. Also, we compute the Laplacians of some functions that contain geometric information of a cmc surface. As the Laplacian operator is elliptic, the maximum principle yields height estimates of a graph of constant mean curvature. Finally, and with the aid of the expression of these Laplacians, we will derive the Barbosa-do Carmo theorem that characterizes a round sphere in the family of closed stable cmc surfaces of Euclidean space.
KeywordsMinimal Surface Compact Surface Jacobi Operator Round Sphere Isoperimetric Problem
- [Ale56]Alexandrov, A.D.: Uniqueness theorems for surfaces in the large I–V. Vestn. Leningr. Univ. 11 #19, 5–17 (1956); 12 #7, 15–44, 1957; 13 #7, 14–26, 1958; 13 #13, 27–34, 1958; 13 #19, 5–8, (1958). English transl. in Amer. Math. Soc. Transl. 21, 341–354, 354–388, 389–403, 403–411, 412–416 (1962) Google Scholar
- [Chv94]Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, New York (1994) Google Scholar
- [Jel53]Jellet, J.H.: Sur la surface dont la courbure moyenne est constant. J. Math. Pures Appl. 18, 163–167 (1853) Google Scholar
- [Lop07]López, R.: On uniqueness of graphs with constant mean curvature. J. Math. Kyoto Univ. 46, 771–787 (2007) Google Scholar