This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
U. Abresch, Constant mean curvature tori in terms of elliptic functions. Preprint Bonn, 1985.
A. Alexandrov, Uniqueness theorems for surfaces in the Large I, A.M.S. Translations Series 2, vol. 21, 341–354, 1962 (Russian original of 1956). See also Annali Mat. Pura Appl. 58, 303–315(1962).
J.L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z., 185, 339–353 (1984).
J.L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds, Math. Z. 197, 123–138 (1988).
J.L.M. Barbosa, J.M. Gomes and A.M. da Silveira, Foliation of 3-dimensional space forms by surfaces with constant mean curvature, Bol.Soc.Bras.Mat., 18, 1–12 (1987).
R. Bryant, Surfaces of mean curvature one in hyperbolic space, to appear in Asterisque.
S.S Chern, On the curvature of a piece of hypersurface in euclidean space of N-dimensions, Abh.Math.Sem. Hamburg, 29, 77–91, 1965.
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent.Math., 82, 121–132, 1985.
D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl.Math. 33, 199–211, 1980.
G. Darboux, Leçons sur la théorie générale des surfaces, Chelsea Publish. Co. New York, 3rd edition, 1972, vol. III, pp. 314 and 315.
M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey, 1976
M. do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tôhoku Math.J. 34, 425–435, 1982.
C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante. J.Math. Pure et App., 16, 309–321, 1841.
M. do Carmo and C.K. Peng, Stable complete minimal surfaces in R 3 are planes, Bull. A.M.S., 1, 685–709, 1979.
M. do Carmo and A.M. da Silveira, Index and total curvature of surfaces with constant mean curvature, Preprint, 1988.
A. El Soufi and S. Ilias, Sur l'existence de hypersurfaces minimales stables ou d'indice 1 e des hypersurfaces à courbure moyenne stables. Preprint Grenoble, 1987.
K. Frensel, Stable complete surfaces with constant mean curvature, to appear in Anais Acad.Bras.Cien. See also: Doctor Thesis at IMPA, Rio, 1988.
A.M. da Silveira, Stability of complete noncompact surfaces with constant mean curvature, Math.Ann. 277, 629–638, 1987.
R. Gulliver, Index and total curvature of complete minimal surfaces, Proc.Symp. Pure Math. 44, 207–211, 1986.
E. Heintze, Extrinsic upper bounds for λ1, Preprint, Augsburg, 1986.
H. Hopf, Über Flächen mit einer Relation Zwischen den Hauptkrumnungen, Math.Nachr., 4, 232–249, 1951.
Wu-Yi Hsiang, A symmetry theorem on isoperimetric regions, Preprint, Berkeley, 1988.
D.A. Hoffman and R. Osserman, The Gauss map of surfaces in R 3 and R 4. Proc. London Math.Soc. 3(50), 27–56, 1985.
J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math., 88, 62–105, 1968.
D.A. Hoffman, R. Osserman and R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in R 3 and R 4, Comm.Math.Helv., 57, 519–531 (1982).
K. Kapouleas, Constant mean curvature surfaces in euclidean three-spaces, Thesis, Berkeley, 1987, see also Bull. A.M.S., 17, 318–320, 1989.
K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math.Ann. 245, 89–99 (1979).
N. Korevaar, R. Kusner, B. Salomon, The structure of complete embedded surfaces with constant mean curvature, Preprint, Berkeley, 1988.
B. Lawson, Complete minimal surfaces in S 3, Ann. of Math. 92, 335–374 (1970).
F. Lopez and A. Ros, Complete minimal surfaces with index one and stable constant mean curvature surfaces. Preprint, Granada, 1987.
W.H. Meeks III, The topology and geometry of embedded surfaces of constant mean curvature. Preprint, Amherst, 1986.
R. Osserman, to appear.
B. Palmer, Ph.D. Thesis, Stanford 1986.
R. Pedrosa, On the uniqueness of isoperimetric regions in cylindric spaces, Thesis, Berkeley, 1988.
R. Reilly, On the first eigenvalue of the laplacian for compact submanifolds of euclidean spaces, Comm. Math. Helv., 52, 525–533, 1977.
G. Roussos, Ph.D. Thesis, Minnesota, 1986.
E. Schmidt, Beweis der isoperimetrischen Eigenschaft der Kugel in hyperbolischen und sphärischen Raum jeder Dimensionszahl, Math. Z., 49, 1–109, 1943–44.
W. Seaman, Helicoids of constant mean curvature and their Gauss maps, Pacific J. of Math., 110, 389–396 (1984).
B. Smyth, The generalization of Delaunay's Theorem to constant mean curvature surfaces with continuous internal symmetry, Preprint, 1988.
H. Wente, Counterexample to a conjecture of H. Hopf, Pacific J.Math., 121, 193–243 (1986).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this chapter
Cite this chapter
do Carmo, M.P. (1989). Hypersurfaces of constant mean curvature. In: Carreras, F.J., Gil-Medrano, O., Naveira, A.M. (eds) Differential Geometry. Lecture Notes in Mathematics, vol 1410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086419
Download citation
DOI: https://doi.org/10.1007/BFb0086419
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51885-3
Online ISBN: 978-3-540-46858-5
eBook Packages: Springer Book Archive