×

Level sets of Gauss curvature in surfaces of constant mean curvature. (English) Zbl 1097.53036

The paper characterizes the topological properties of the level set of the Gauss mean curvature of an embedded surface \(M\) with nonempty boundary in \(\mathbb R^3\). The main theorem classifies 3 possible cases and implies an interesting consequence concerning the convexity of \(M\). Indeed, this paper contributes to the discussion of convexity, which is largely studied in the current literature.

MSC:

53C40 Global submanifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] H.S. Brascamp - E. LIEB, On extention of the Brunn-Minkowski and Prekoja-Leindler theorems , J. Funct. Anal. , 11 ( 1976 ), pp. 366 - 389 . Zbl 0334.26009 · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[2] L.A. Caffarelli - A. FRIEDMAN, Convexity of solutions of semilinear elliptic equations , Duke. Math. J. , 52 ( 1985 ), pp. 31 - 456 . Article | MR 792181 | Zbl 0599.35065 · Zbl 0599.35065 · doi:10.1215/S0012-7094-85-05221-4
[3] J.T. Chen - W.H. Huang , Convexity of capillary surfaces in outer space , Invent. Math. , 67 ( 1982 ), pp. 253 - 259 . MR 665156 | Zbl 0496.76005 · Zbl 0496.76005 · doi:10.1007/BF01393817
[4] R. Finn , Existence criteria for capillary free surfaces without gravity , Indiana Univ. Math. J. , 32 ( 1982 ), pp. 439 - 460 . MR 697648 | Zbl 0487.76012 · Zbl 0487.76012 · doi:10.1512/iumj.1983.32.32032
[5] R. Finn , Comparison Principles in Capillarity, Calculus of Variations , Lecture Notes in Mathematics , vol. 1357 , Springer-Verlag , 1988 . MR 976235 | Zbl 0692.35006 · Zbl 0692.35006
[6] H. Hopf , Lectures on Differential Geometry in the Large , Lecture Notes in Mathematics , vol. 1000 , Springer-Verlag , 1983 . MR 707850 | Zbl 0526.53002 · Zbl 0526.53002
[7] W.H. Huang , Superhamonicity of curvatures for surfaces of constant mean curvature , Pacific J. of Math. , 152 , no. 2 ( 1992 ), pp. 291 - 318 . Article | MR 1141797 | Zbl 0767.53040 · Zbl 0767.53040 · doi:10.2140/pjm.1992.152.291
[8] W.H. Huang - CHUN-CHI LIN, Negatively Curved Sets in Surfaces of Constant Mean Curvature in R3 , Arch. Rat. Mech. Anal. , 141 , no. 2 ( 1998 ), pp. 105 - 116 . MR 1615516 | Zbl 0941.53011 · Zbl 0941.53011 · doi:10.1007/s002050050074
[9] N.J. Korevarr , Convex solutions to nonlinear elliptic and parabolic boundary value problems , Indiana Univ. Math. J. , 32 ( 1983 ), pp. 603 - 614 . MR 703287 | Zbl 0481.35024 · Zbl 0481.35024 · doi:10.1512/iumj.1983.32.32042
[10] N.J. Korevarr - J.L. Lewis , Convex solutions to nonlinear elliptic equations having constant rank Hessians , Arch. Rat. Mech. Anal. , 32 ( 1987 ), pp. 19 - 32 . Zbl 0624.35031 · Zbl 0624.35031 · doi:10.1007/BF00279844
[11] H. Wente , Counterexample to a conjecture of H. Hopf , Pacific J. Math. , 121 ( 1986 ), pp. 193 - 243 . Article | MR 815044 | Zbl 0586.53003 · Zbl 0586.53003 · doi:10.2140/pjm.1986.121.193
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.