Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Anciaux, Henri²  Guilfoyle, Brendan³  Romon, Pascal¹ 

[1] Univ Paris Est, F-77454 Champs Sur Marne 2, Marne La Valle, France

[2] Univ Sao Paulo, IME, BR-05508090 Sao Paulo, Brazil

[3] Inst Technol, Dept Math & Comp, Tralee, Co Kerry, Ireland

关键词: Lagrangian surfaces;    Minimal surfaces;    Hamiltonian stationary surfaces;    Pseudo-Kahler metric;   

DOI  :  10.1016/j.geomphys.2010.09.017

来源: Elsevier

【 摘 要 】

Given an oriented Riemannian surface (Sigma, g), its tangent bundle T Sigma enjoys a natural pseudo-Kahler structure, that is the combination of a complex structure 2, a pseudo-metric G with neutral signature and a symplectic structure Omega. We give a local classification of those surfaces of T Sigma which are both Lagrangian with respect to Omega and minimal with respect to G. We first show that if g is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in R-3 or R-1(3) induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS2 or TH2 respectively. We relate the area of the congruence to a second-order functional F = f root H-2 - K dA on the original surface. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free