Minimal Lagrangian submanifolds via the geodesic Gauss map

TL;DR

This paper investigates the properties of the geodesic Gauss map associated with isometric immersions into spheres, establishing conditions under which it is minimal Lagrangian and exploring its deformation behavior.

Contribution

It provides explicit formulas for mean curvature vectors of the Gauss maps and shows minimality of the geodesic Gauss map for minimal surfaces, extending Palmer's work.

Findings

Geodesic Gauss map of minimal surfaces is minimal Lagrangian.

Deformations of the original immersion induce Hamiltonian deformations of the Gauss map.

Mean curvature vector of the Gauss map is always Hamiltonian.

Abstract

For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $\gamma$ projects this into the manifold of oriented geodesics in $S^n$ (the Grassmannian of oriented 2-planes in $\mathbb{R}^{n+1}$), giving a Lagrangian immersion of $UM^\perp$ into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of $f$, and show that when $f$ has conformal shape form this depends only on the mean curvature of $f$. In particular we deduce that the geodesic Gauss map of every minimal surface in $S^n$ is minimal Lagrangian. We also give simple proofs that: deformations of $f$ always correspond to Hamiltonian deformations of $\gamma$; the mean curvature vector of $\gamma$ is always a Hamiltonian vector field. This extends work of Palmer on the case when $M$ is a hypersurface.