Transactions of the American Mathematical Society

Author(s): David Hoffman; Jorge H. S. de Lira; Harold Rosenberg
Journal: Trans. Amer. Math. Soc. 358 (2006), 491-507.
MSC (2000): Primary 53C27, 58J60
Posted: September 26, 2005
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Abstract: The subject of this paper is properly embedded

surfaces in Riemannian three manifolds of the form $M^2\times \mathbf{R}$ , where

is a complete Riemannian surface. When $M^2={\mathbf R}^2$ , we are in the classical domain of

surfaces in ${\mathbf R}^3$ . In general, we will make some assumptions about

in order to prove stronger results, or to show the effects of curvature bounds in

on the behavior of

surfaces in $M^2\times \mathbf{R}$ .

1.: J. L. Barbosa and R.S. Earp.
Prescribed mean curvature in $H\sp{n+1}(-1)$ with convex planar boundary, I.
Geom. Dedicata, 71(1):61-74, 1998. MR 1624726 (99d:53064)
2.: M. Crandall and P. Rabinowitz.
Bifurcation, perturbation of simple eigenvalues and linearized stability.
Arch. Rational Mech. Anal., 52:161--180, 1973. MR 0341212 (49:5962)
3.: P. Collin.
Topologie et courboure des surfaces minimales proprement plongées de ${\mathbf {R} ^3}$ .
Ann. of Mathematics, 145:1-31, 1997. MR 1432035 (98d:53010)
4.: D. Gilbarg and N. S. Trudinger.
Elliptic partial differential equations of second order.
Springer-Verlag, New York, 2nd edition, 1983. MR 0737190 (86c:35035)
5.: Karsten Grosse-Braukmann and Meinhard Wohlgemuth.
The gyroid is embedded and has constant mean curvature companions.
Calc. Var., 4:499-523, 1996. MR 1415998 (97k:53011)
6.: D. Hoffman and H. Karcher.
Complete embedded minimal surfaces of finite total curvature.
In Encyclopedia of Mathematics, pages 5-93, 1997.
R. Osserman, editor, Springer Verlag. MR 1490038 (98m:53012)
7.: D. Hoffman and W. H. Meeks III.
Embedded minimal surfaces of finite topology.
Annals of Mathematics, 131:1-34, 1990. MR 1038356 (91i:53010)
8.: W.-T. Hsiang and W.-Y. Hsiang.
On the uniqueness of isoperimetric solutions and imbedded soap bubbles in non-compact symmetric spaces.
Inventiones Math. 98:39-58, 1989. MR 1010154 (90h:53078)
9.: H. Karcher.
Embedded minimal surfaces derived from Scherk's examples.
Manuscripta Math., 62:83-114, 1988. MR 0958255 (89i:53009)
10.: H. Karcher.
The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions.
Manuscripta Math., 64:291-357, 1989. MR 1003093 (90g:53010)
11.: N. Korevaar, R. Kusner, and B. Solomon.
The structure of complete embedded surfaces with constant mean curvature.
Journal of Differential Geometry, 30:465-503, 1989. MR 1010168 (90g:53011)
12.: H. B. Lawson.
Complete minimal surfaces in .
Annals of Mathematics, 92:335-374, 1970. MR 0270280 (42:5170)
13.: H. B. Lawson, Jr.
Lectures on Minimal Submanifolds.
Publish or Perish Press, Berkeley, 1971. MR 0576752 (82d:53035b)
14.: R. Mazzeo and F. Pacard.
Bifurcating nodoids.
Contemp. Math., 314:169-186, 2002. MR 1941630 (2004b:53015)
15.: W. H. Meeks III and H. Rosenberg.
The uniqueness of the helicoid.
Annals of Mathematics 161(2):723-754, 2005. MR 2153399
16.: W. H. Meeks III and H. Rosenberg
The theory of minimal surfaces in $M\times \mathbf{R}.$
To appear in Comment. Math. Helv.
17.: B. Nelli and H. Rosenberg.
Minimal surfaces in ${\mathbf H}^2\times \mathbf{R}.$
Bull. Bras. Math. Soc., New Series, 33(2):263-292, 2002. MR 1940353 (2004d:53014)
18.: 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.
19.: P. A. Nitsche.
Existence of prescribed mean curvature graphs in hyperbolic space.
Manuscripta Math., 108(3):349-367, 2002. MR 1918082 (2003f:53015)
20.: R. Pedrosa and M. Ritoré.
Isoperimetric domains in the Riemannian product of a circle with simply connected space forms and applications to free boundary problems.
Indiana Univ. J. of Mathematics, 48:1357-1394, 1999. MR 1757077 (2001k:53120)
21.: H. Rosenberg.
Minimal surfaces in ${M}^2\times {\mathbf {R}}$ .
Illinois J. Math, 46(4):1177-1195, 2002. MR 1988257 (2004d:53015)
22.: H. Rosenberg.
Hypersurfaces of constant curvature in space forms.
Bull Sc. Math, 117:211-239, 1993. MR 1216008 (94b:53097)
23.: R. Schoen.
Analytic aspects of the harmonic map problem.
In Seminar on Nonlinear Partial Differential Equations, 1983. MR 0765241 (86b:58032)
24.: J. Serrin.
The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables.
Philos. Trans. Roy. Soc. London Ser. A, 264:413-496, 1969. MR 0282058 (43:7772)
25.: J. Smoller.
Shock waves and reaction-diffusion equations.
Springer-Verlag, New York, 1983. MR 0688146 (84d:35002)
26.: M. Weber and M. Wolf.
Teichmüller theory and handle addition for minimal surfaces.
Annals of Math. 156:713-795, 2002. MR 1954234 (2005j:53012)
27.: M. Weber and M. Wolf.
Minimal surfaces of least total curvature and moduli spaces of plane polygonal arcs.
GAFA, 8:1129-1170, 1998. MR 1664793 (99m:53020)
28.: M. Wohlgemuth.
Higher genus minimal surfaces by growing handles out of a catenoid.
Manuscripta Math., 70:397-428, 1991. MR 1092145 (91k:53021)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C27, 58J60

David Hoffman
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: hoffman@math.stanford.edu

Jorge H. S. de Lira
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Fortaleza - Ceará - Brasil
Email: jherbert@mat.ufc.br

Harold Rosenberg
Affiliation: Institut de Mathematiques de Jussieu, Paris XIII, France
Email: rosen@math.jussieu.fr

DOI: 10.1090/S0002-9947-05-04084-5
PII: S 0002-9947(05)04084-5
Received by editor(s): December 1, 2003
Posted: September 26, 2005
Additional Notes: The first author was partially supported by research grant DE-FG03-95ER25250/A007 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation, Division of Mathematical Sciences research grant DMS-0139410.
The second author was partially supported by Cooperaçao Brasil-França - Ministère des Affaires ètrangères (France) and CAPES (Brasil)
Copyright of article: Copyright 2005, American Mathematical Society

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