Surfaces with parallel mean curvature vector in and
Authors:
Francisco Torralbo and Francisco Urbano
Journal:
Trans. Amer. Math. Soc. 364 (2012), 785813
MSC (2010):
Primary 53A10; Secondary 53B35
Published electronically:
October 3, 2011
MathSciNet review:
2846353
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Two holomorphic Hopf differentials for surfaces of nonnull parallel mean curvature vector in and are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of and is established. Using this, surfaces with vanishing Hopf differentials (in particular, spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of and is proved.
 1.
Uwe
Abresch and Harold
Rosenberg, A Hopf differential for constant mean curvature surfaces
in 𝑆²×𝑅 and
𝐻²×𝑅, Acta Math. 193
(2004), no. 2, 141–174. MR 2134864
(2006h:53003), http://dx.doi.org/10.1007/BF02392562
 2.
Benoît
Daniel, Isometric immersions into 3dimensional homogeneous
manifolds, Comment. Math. Helv. 82 (2007),
no. 1, 87–131. MR 2296059
(2008a:53058), http://dx.doi.org/10.4171/CMH/86
 3.
Paul
F. Byrd and Morris
D. Friedman, Handbook of elliptic integrals for engineers and
scientists, Die Grundlehren der mathematischen Wissenschaften, Band
67, SpringerVerlag, New York, 1971. Second edition, revised. MR 0277773
(43 #3506)
 4.
Ildefonso
Castro and Francisco
Urbano, Minimal Lagrangian surfaces in
𝕊²×𝕊², Comm. Anal. Geom.
15 (2007), no. 2, 217–248. MR 2344322
(2008j:53107)
 5.
Bangyen
Chen, On the surface with parallel mean curvature vector,
Indiana Univ. Math. J. 22 (1972/73), 655–666. MR 0315606
(47 #4155)
 6.
Shiu
Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math.
Helv. 51 (1976), no. 1, 43–55. MR 0397805
(53 #1661)
 7.
Isabel
Fernández and Pablo
Mira, A characterization of constant mean curvature surfaces in
homogeneous 3manifolds, Differential Geom. Appl. 25
(2007), no. 3, 281–289. MR 2330457
(2008e:53012), http://dx.doi.org/10.1016/j.difgeo.2006.11.006
 8.
Dirk
Ferus, The torsion form of submanifolds in
𝐸^{𝑁}, Math. Ann. 193 (1971),
114–120. MR 0287493
(44 #4697)
 9.
Katsuei
Kenmotsu and Detang
Zhou, The classification of the surfaces with parallel mean
curvature vector in twodimensional complex space forms, Amer. J.
Math. 122 (2000), no. 2, 295–317. MR 1749050
(2001a:53094)
 10.
Maria
Luiza Leite, An elementary proof of the AbreschRosenberg theorem
on constant mean curvature immersed surfaces in
𝕊²×ℝ and ℍ²×ℝ, Q.
J. Math. 58 (2007), no. 4, 479–487. MR 2371467
(2008j:53013), http://dx.doi.org/10.1093/qmath/ham020
 11.
Jorge
H. S. De Lira and Feliciano
A. Vitório, Surfaces with constant mean curvature in
Riemannian products, Q. J. Math. 61 (2010),
no. 1, 33–41. MR 2592022
(2011b:53146), http://dx.doi.org/10.1093/qmath/han030
 12.
Takashi
Ogata, Surfaces with parallel mean curvature vector in
𝑃²(𝐂), Kodai Math. J. 18 (1995),
no. 3, 397–407. MR 1362916
(96i:53061), http://dx.doi.org/10.2996/kmj/1138043479
 13.
Shing
Tung Yau, Submanifolds with constant mean curvature. I, II,
Amer. J. Math. 96 (1974), 346–366; ibid. 97 (1975),
76–100. MR
0370443 (51 #6670)
 1.
 U. Abresch and H. Rosenberg.
A Hopf differential for constant mean curvature surfaces in and . Acta Math. 193 (2004) 141174. MR 2134864 (2006h:53003)
 2.
 D. Benôit.
Isometric immersions into 3dimensional homogeneous manifolds. Comment. Math. Helv. 82 (2007) 87131. MR 2296059 (2008a:53058)
 3.
 P.F. Byrd and M.D. Friedman.
Handbook of Elliptic Integral for Engineers and Scientists. SpringerVerlag, New York (1971). MR 0277773 (43:3506)
 4.
 I. Castro and F. Urbano.
Minimal Lagrangian surfaces in . Comm. Anal. and Geom. 15 (2007) 217248. MR 2344322 (2008j:53107)
 5.
 B.Y. Chen.
On the surface with parallel mean curvature vector. Indiana Univ. Math. J. 22 (1973) 655666. MR 0315606 (47:4155)
 6.
 S.Y. Cheng.
Eigenfunctions and nodal sets. Comment. Math. Helv. 51 (1976) 4355. MR 0397805 (53:1661)
 7.
 I. Fernández and P. Mira.
A characterization of constant mean curvature surfaces in homogeneous 3manifolds. Differential Geom. Appl. 25 (2007) 281289. MR 2330457 (2008e:53012)
 8.
 D. Ferus.
The torsion form of submanifolds in . Math. Ann. 193 (1971) 114120. MR 0287493 (44:4697)
 9.
 K. Kenmotzu and D. Zhou.
The classification of the surfaces with parallel mean curvature vector in twodimensional complex space forms. Amer. J. Math. 122 (2000) 295317. MR 1749050 (2001a:53094)
 10.
 M.L. Leite.
An elementary proof of the AbreschRosenberg theorem on constant mean curvature immersed surfaces in and . Quart.J. Math. 58 (2007) 479487. MR 2371467 (2008j:53013)
 11.
 J.H.S. de Lira and A. Vitorio.
Surfaces with constant mean curvature in Riemannian products. Quart.J. Math. 61 (2010) no 1, 3341. MR 2592022 (2011b:53146)
 12.
 T. Ogata,
Surfaces with parallel mean curvature in . Kodai Math. J. 18 (1995) 397407. MR 1362916 (96i:53061)
 13.
 S.T. Yau.
Submanifolds with constant mean curvature. I. Amer. J. Math. 96 (1974) 346366. MR 0370443 (51:6670)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
53A10,
53B35
Retrieve articles in all journals
with MSC (2010):
53A10,
53B35
Additional Information
Francisco Torralbo
Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email:
ftorralbo@ugr.es
Francisco Urbano
Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email:
furbano@ugr.es
DOI:
http://dx.doi.org/10.1090/S000299472011053468
PII:
S 00029947(2011)053468
Received by editor(s):
December 30, 2008
Received by editor(s) in revised form:
October 27, 2009, and February 15, 2010
Published electronically:
October 3, 2011
Additional Notes:
This research was partially supported by an MCyTFeder research project MTM200761775 and a Junta Andalucĭa Grant P06FQM01642.
Article copyright:
© Copyright 2011 American Mathematical Society
