ON THE LAGRANGIAN ANGLE AND THE KÄHLER ANGLE OF IMMERSED SURFACES IN THE COMPLEX PLANE C2

doi:10.1007/s10473-019-0617-4

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (6): 1695-1712.doi: 10.1007/s10473-019-0617-4

Previous Articles Next Articles

ON THE LAGRANGIAN ANGLE AND THE KÄHLER ANGLE OF IMMERSED SURFACES IN THE COMPLEX PLANE C²

Xingxiao LI¹, Xiao LI²

1. School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, China;
2. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received:2018-04-09 Online:2019-12-25 Published:2019-12-30
Contact: Xiao LI,E-mail:lxlixiaolx@163.com E-mail:lxlixiaolx@163.com
Supported by:
The first author was supported by National Natural Science Foundation of China (11671121, 11871197).

Abstract

Abstract: In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in C². Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C² than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kähler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kähler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kähler angle which imply rigidity theorems of self-shrinkers with Kähler angle under the condition that ∫_M|h|²e^-|x|²/2 dV_M<∞, where h and x denote, respectively, the second fundamental form and the position vector of the surface.

Key words: Kähler angle, Lagrangian angle, self-shrinker

CLC Number:

53C24

Xingxiao LI, Xiao LI. ON THE LAGRANGIAN ANGLE AND THE KÄHLER ANGLE OF IMMERSED SURFACES IN THE COMPLEX PLANE C²[J].Acta mathematica scientia,Series B, 2019, 39(6): 1695-1712.

Trendmd

References

[1] Chern S S, Wolfson J. Minimal surfaces by moving frames. Amer J Math, 1983, 105:59-83
[2] Boton J, Jensen G R, Rigoli M, et al. On conformal minimal immersions of S² into CPⁿ. Math Ann, 1988, 279:599-620
[3] Li Z Q. Counterexamples to the conjecture on minimal S² in CPⁿ with constant Kähler angle. Manuscripta Math, 1995, 88(1):417-431
[4] Wood L M. Minimal 2-spheres in CPⁿ with constant Kähler angle. Manuscripta Math, 2000, 103(1):1-8
[5] Li X X, Li A M. Minimal surfaces in S⁶ with constant Kähler angles. Acta Math Sin (in Chinese), 1996, 39:196-203
[6] Li X X. A classification theorem for complete minimal surfaces in S⁶ with constant Kähler angles. Arch Math, 1999, 72:385-400
[7] Zhang Y. A note on conical Kähler-Ricci flow on minimal elliptic Kähler surface. Acta Math Sci, 2018, 38B(1):169-176
[8] Chong T, Dong Y, Lin H, et al. Rigidity theorems of complete Kähler-Einstein manifolds and complex space forms. Acta Math Sci, 2019, 39B(2):339-356
[9] Morvan J M. Classe de Maslov d'une immersion lagrangienne et minimalité. C R Acad Sci Paris, 1981292(13):633-636
[10] Cieliebak K, Goldstein E. A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions. J Symplectic Geom, 2004, 2(2):261-266
[11] Schoen R, Wolfson J. Minimizing area among Lagrangian surfaces:the mapping problem. J Differential Geom, 2001, 58(1):1-86
[12] Arsie A. Maslov class and minimality in Calabi-Yau manifolds. J Geom Phys, 2000, 35:145-156
[13] Castro I, Lerma A M. The clifford torus as a self-shrinker for the Lagrangian mean curvature flow. Int Math Res Not, 2014, 6:1515-1527
[14] Lotay J D, Pacini T. From Lagrangian to totally real geometry:coupled flows and calibrations. Comm Anal Geom, 2018, in press
[15] Chen J, Li J. Mean curvature flow of surface in 4-manifolds. Adv Math, 2001, 163(2):287-309
[16] Chen J, Tian G. Moving symplectic curves in Kähler-Einstein surfaces. Acta Math Sin, 2000, 16(4):541-548
[17] Arezzo C, Sun J. Self-shrinker for the mean curvature flow in arbitrary codimension. Math Z, 2013, 274:993-1027
[18] Han X, Sun J. Translating solitons to symplectic mean curvature flows. Ann Glob Anal Geom, 2010, 38:161-169
[19] Li H, Wang X F. New characterizations of the clifford torus as a Lagrangian self-shrinker. J Geom Anal, 2017, 27(2):1393-1412
[20] Chen J, Tian G. Minimal surfaces in Riemannian 4-manifolds. Geom Funct Anal, 1997, 7:873-916
[21] Colding T H, Minicozzi W P II. Generic mean curvature flow I:generic singularities. Ann Math, 2012, 175:755-833
[22] Cao H-D, Li H. A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc Var Partial Differential Equations, 2013, 46:879-889
[23] Li H, Wei Y. Classification and rigidity of self-shrinkers in the mean curvature flow. J Math Soc Japan, 2014, 66(3):709-734
[24] Gilbarg D, Trudinger N S. Elleptic Partial Differential Equations of Second Order. Berlin:Springer, 2001:32-36

ON THE LAGRANGIAN ANGLE AND THE KÄHLER ANGLE OF IMMERSED SURFACES IN THE COMPLEX PLANE C²

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 10