Turkish Journal

of

Mathematics

math@tubitak.gov.tr

# Turkish Journal of Mathematics

Extremal Lagrangian submanifolds in a complex space form Nn(4c)

Shichang SHU
Department of Mathematics
Xianyang Normal University
Xianyang 712000 Shaanxi
P. R. CHINA
e-mail: shushichang@126.com
Annie Yi HAN
Department of Mathematics
Borough of Manhattan Community College
CUNY 10007 N.Y. USA
e-mail: DrHan@nyc.rr.com

Abstract: Let Nn(4c) be the complex space form of constant holomorphic sectional curvature 4c, j: M \to Nn(4c) be an immersion of an n-dimensional Lagrangian manifold M in Nn(4c). Denote by S and H the square of the length of the second fundamental form and the mean curvature of M. Let r be the non-negative function on M defined by r2=S-nH2, Q be the function which assigns to each point of M the infimum of the Ricci curvature at the point. In this paper, we consider the variational problem for non-negative functional U(j)=\intMr2dv=\intM(S-nH2)dv. We call the critical points of U(j) the Extremal submanifold in complex space form Nn(4c). We shall get the new Euler-Lagrange equation of U(j) and prove some integral inequalities of Simons' type for n-dimensional compact Extremal Lagrangian submanifolds j: M \to Nn(4c) in the complex space form Nn(4c) in terms of r2, Q, H and give some rigidity and characterization Theorems.

Key Words: Willmore Lagrangian submanifold, complex hyperbolic space, curvature, totally umbilical.

Turk. J. Math., 34, (2010), 129-144.
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Other articles published in the same issue: Turk. J. Math.,vol.34,iss.1.