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ISSN 1088-6826(e) ISSN 0002-9939(p)

Lagrangian Bonnet pairs in $\mathbb{C}P^2$

Author(s): Huixia He; Hui Ma
Journal: Proc. Amer. Math. Soc. 137 (2009), 2725-2731.
MSC (2000): Primary 53C40; Secondary 53C42, 53D12
Posted: March 31, 2009
MathSciNet review: 2497485
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Abstract: In this paper we introduce Lagrangian Bonnet pairs in the complex projective plane $\mathbb{C}P^2$ and derive a Lawson-Tribuzy type theorem. We also present examples of compact Lagrangian Bonnet pairs with genus one in $\mathbb{C}P^2$ .

References:

1.: A.I. Bobenko, Exploring surfaces through methods from the theory of integrable systems. Lectures on the Bonnet problem. SFB288 Preprint 403 (1999), math.DG/9909003.
2.: O. Bonnet, Mémoire sur la théorie des surfaces applicables, J. École Polytechnique 42 (1867), 72-92.
3.: I. Castro and F. Urbano, On twistor harmonic surfaces in the complex projective plane, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 115-129. MR 1443591 (98d:53090)
4.: E. Cartan, Sur les couples de surfaces applicables avec conservation des courbures principales, Bull. Sci. Math. (2) 66 (1942), 55-72, 74-85. MR 0009876 (5:216e)
5.: S.S. Chern, Deformation of surfaces preserving principal curvatures, Differential Geometry and Complex Analysis, 155-163, Springer-Verlag, Berlin, 1985. MR 780041 (86h:53005)
6.: S.S. Chern and J.G. Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), 59-83. MR 692106 (84i:53056)
7.: J. Eells and J.C. Wood, Harmonic maps from surfaces to complex projective spaces, Adv. in Math. 49 (1983), 217-263. MR 716372 (85f:58029)
8.: J.A. Gálvez, A. Martínez and P. Mira, The Bonnet problem for surfaces in homogeneous -manifolds, math.DG/0612766.
9.: G. Kamberov, F. Pedit and U. Pinkall, Bonnet pairs and isothermic surfaces, Duke Math. J. 92(3) (1998), 637-644. MR 1620534 (99h:53009)
10.: H.B. Lawson Jr. and R. Tribuzy, On the mean curvature function for compact surfaces, J. Diff. Geom. 16 (1981), 179-183. MR 638784 (83e:53060)
11.: H. Ma, Hamiltonian stationary Lagrangian surfaces in $\mathbb{C}P^2$ , Ann. Global Anal. Geom. 27 (2005), 1-16. MR 2130529 (2005m:53152)
12.: H. Ma and M. Schmies, Examples of Hamiltonian stationary Lagrangian tori in $\mathbb{C}P^2$ , Geom. Ded. 118 (2006), 173-183. MR 2239455 (2007b:53124)
13.: Y.G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), 501-519. MR 1062973 (91f:58022)
14.: R. Schoen and J. Wolfson, Minimizing volume among Lagrangian submanifolds, in `Differential Equations: La Pietra 1996' (ed.: Giaquinta, Shatah and Varadhan), 181-199, Proc. Symp. Pure Math., 65, Amer. Math. Soc., Providence, RI, 1999. MR 1662755 (99k:53130)
15.: B.A. Springborn, Bonnet pairs in the -sphere, Differential geometry and integrable systems (Tokyo, 2000), 297-303, Contemp. Math., 308, Amer. Math. Soc., Providence, RI, 2002. MR 1955643 (2004b:53007)
16.: R. Tribuzy, A characterization of tori with constant mean curvature in space form, Bol. Soc. Bras. Math. 11 (1980), 259-274. MR 671469 (84d:53066)
17.: C. Wang, The classification of homogeneous surfaces in $\mathbb{C}P^2$ , Geometry and topology of submanifolds, X (Beijing/Berlin, 1999), 303-314, World Sci. Publ., River Edge, NJ, 2000. MR 1801923 (2001j:53082)

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Additional Information:

Huixia He
Affiliation: Department of Mathematics, Beijing University of Aeronautics and Astronautics (Beihang University), Beijing 100083, People's Republic of China
Email: hehx@buaa.edu.cn

Hui Ma
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China
Email: hma@math.tsinghua.edu.cn

DOI: 10.1090/S0002-9939-09-09890-6
PII: S 0002-9939(09)09890-6
Received by editor(s): December 24, 2007
Posted: March 31, 2009
Additional Notes: The first author is partially supported by NSFC grant No. 10701007.
The second author is partially supported by NSFC grant No. 10501028 and NKBRPC No. 2006CB805905.
Communicated by: Chuu-Lian Terng
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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