A flux formula for area-stationary surfaces the Heisenberg group
Joint with M. Galli.
Our main result is
Theorem 1 A complete immersed surface in Heisenberg group which is area stationary is foliated by horizontal lines (up to characteristic points).
First proved by S. Pauls under the assumption that horizontal Gauss map has Sobolev components. Then Cheng, Hwang and Yang proved it for -graphs. It is more natural to consider intrinsic graphs (any surface, away from characteristic points, is locally an intrinsic graph).
Cheng, Hwang, Malchiodi and Yang show that, for minimal -graphs, characteristic points are either isolated or come in continuous curves. In the case, they may form trees.
2. Flux formula
2.1. The formula
View sub-Riemannian area as a Riemannian functional. Apply the first variation formula to special variations: right-invariant vectorfields multiplied by a cut-off near the boundary. This gives
where is the unit normal to the boundary, is the horizontal projection of the Riemannian unit normal , and . This can be viewed as a first integral of the equations.
I prove the theorem above. Assume is an intrinsic graph. Use contours formed of 2 pieces of horizontal curves and two images (under the graph parametrization) of vertical segments.
For a ruled surface, one can make sense of second variation.
Theorem 2 A complete immersed surface in Heisenberg group, without characteristic points, which is area stationary and stable, must be a vertical plane.
The formula does not easily generalize to intrinsic graphs.