Optimal length estimates for stable CMC surfaces in 3-space forms

Mazet, Laurent

doi:10.1090/S0002-9939-09-09885-2

Optimal length estimates for stable CMC surfaces in $3$-space forms
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Proc. Amer. Math. Soc. 137 (2009), 2761-2765 Request permission

Abstract:

In this paper, we study stable constant mean curvature $H$ surfaces in $\mathbb {R}^3$. We prove that, in such a surface, the distance from a point to the boundary is less than or equal to $\pi /(2H)$. This upper bound is optimal and is extended to stable constant mean curvature surfaces in space forms.

References

Philippe Castillon, An inverse spectral problem on surfaces, Comment. Math. Helv. 81 (2006), no. 2, 271–286 (English, with English and French summaries). MR 2225628, DOI 10.4171/CMH/52
Tobias H. Colding and William P. Minicozzi II, Estimates for parametric elliptic integrands, Int. Math. Res. Not. 6 (2002), 291–297. MR 1877004, DOI 10.1155/S1073792802106106
Jose M. Espinar and Harold Rosenberg. A Colding-Minicozzi stability inequality and its applications, to appear in Trans. Amer. Math. Soc.
Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. MR 562550, DOI 10.1002/cpa.3160330206
H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
William H. Meeks III, Joaquín Pérez, and Antonio Ros. Stable constant mean curvature surfaces, preprint.
Antonio Ros and Harold Rosenberg. Properly embedded surfaces with constant mean curvature, preprint.
Harold Rosenberg, Constant mean curvature surfaces in homogeneously regular 3-manifolds, Bull. Austral. Math. Soc. 74 (2006), no. 2, 227–238. MR 2260491, DOI 10.1017/S000497270003567X
Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR 795231