Constant mean curvature surfaces in 𝑀²×𝐑

Hoffman, David; de Lira, Jorge; Rosenberg, Harold

doi:10.1090/S0002-9947-05-04084-5

Constant mean curvature surfaces in $M^2\times \mathbf {R}$
HTML articles powered by AMS MathViewer

by David Hoffman, Jorge H. S. de Lira and Harold Rosenberg PDF

Trans. Amer. Math. Soc. 358 (2006), 491-507 Request permission

Abstract:

The subject of this paper is properly embedded $H-$surfaces in Riemannian three manifolds of the form $M^2\times \mathbf {R}$, where $M^2$ is a complete Riemannian surface. When $M^2={\mathbf R}^2$, we are in the classical domain of $H-$surfaces in ${\mathbf R}^3$. In general, we will make some assumptions about $M^2$ in order to prove stronger results, or to show the effects of curvature bounds in $M^2$ on the behavior of $H-$surfaces in $M^2\times \mathbf {R}$.

References

J. Lucas M. Barbosa and Ricardo Sa Earp, Prescribed mean curvature hypersurfaces in $H^{n+1}(-1)$ with convex planar boundary. I, Geom. Dedicata 71 (1998), no. 1, 61–74. MR 1624726, DOI 10.1023/A:1005046021921
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
Pascal Collin, Topologie et courbure des surfaces minimales proprement plongées de $\mathbf R^3$, Ann. of Math. (2) 145 (1997), no. 1, 1–31 (French). MR 1432035, DOI 10.2307/2951822
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Karsten Große-Brauckmann and Meinhard Wohlgemuth, The gyroid is embedded and has constant mean curvature companions, Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499–523. MR 1415998, DOI 10.1007/BF01261761
David Hoffman and Hermann Karcher, Complete embedded minimal surfaces of finite total curvature, Geometry, V, Encyclopaedia Math. Sci., vol. 90, Springer, Berlin, 1997, pp. 5–93. MR 1490038, DOI 10.1007/978-3-662-03484-2_{2}
David Hoffman and William H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), no. 1, 1–34. MR 1038356, DOI 10.2307/1971506
Wu-teh Hsiang and Wu-Yi Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math. 98 (1989), no. 1, 39–58. MR 1010154, DOI 10.1007/BF01388843
H. Karcher, Embedded minimal surfaces derived from Scherk’s examples, Manuscripta Math. 62 (1988), no. 1, 83–114. MR 958255, DOI 10.1007/BF01258269
Hermann Karcher, The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64 (1989), no. 3, 291–357. MR 1003093, DOI 10.1007/BF01165824
Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
H. Blaine Lawson Jr., Lectures on minimal submanifolds. Vol. I, Monografías de Matemática [Mathematical Monographs], vol. 14, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1977. MR 527121
Rafe Mazzeo and Frank Pacard, Bifurcating nodoids, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 169–186. MR 1941630, DOI 10.1090/conm/314/05430
William H. Meeks III and Harold Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), no. 2, 727–758. MR 2153399, DOI 10.4007/annals.2005.161.727
W. H. Meeks III and H. Rosenberg The theory of minimal surfaces in $M\times \mathbf {R}.$ To appear in Comment. Math. Helv.
Barbara Nelli and Harold Rosenberg, Minimal surfaces in ${\Bbb H}^2\times \Bbb R$, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263–292. MR 1940353, DOI 10.1007/s005740200013
B. Nelli and H. Rosenberg. Global Properties of Constant Mean Curvature Surfaces in $\mathbf {H}^2\times \mathbf {R}$. To appear in Pacific J. of Mathematics.
Pál-Andrej Nitsche, Existence of prescribed mean curvature graphs in hyperbolic space, Manuscripta Math. 108 (2002), no. 3, 349–367. MR 1918082, DOI 10.1007/s002290200267
Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), no. 4, 1357–1394. MR 1757077, DOI 10.1512/iumj.1999.48.1614
Harold Rosenberg, Minimal surfaces in ${\Bbb M}^2\times \Bbb R$, Illinois J. Math. 46 (2002), no. 4, 1177–1195. MR 1988257
Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211–239. MR 1216008
Richard M. Schoen, Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 321–358. MR 765241, DOI 10.1007/978-1-4612-1110-5_{1}7
J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413–496. MR 282058, DOI 10.1098/rsta.1969.0033
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146, DOI 10.1007/978-1-4684-0152-3
Matthias Weber and Michael Wolf, Teichmüller theory and handle addition for minimal surfaces, Ann. of Math. (2) 156 (2002), no. 3, 713–795. MR 1954234, DOI 10.2307/3597281
M. Weber and M. Wolf, Minimal surfaces of least total curvature and moduli spaces of plane polygonal arcs, Geom. Funct. Anal. 8 (1998), no. 6, 1129–1170. MR 1664793, DOI 10.1007/s000390050125
Meinhard Wohlgemuth, Higher genus minimal surfaces by growing handles out of a catenoid, Manuscripta Math. 70 (1991), no. 4, 397–428. MR 1092145, DOI 10.1007/BF02568387