** A flux formula for area-stationary surfaces the Heisenberg group **

Joint with M. Galli.

**1. Rulings **

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.

** 2.2. Application **

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.

** 2.3. Consequences **

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. *

** 2.4. Questions **

The formula does not easily generalize to intrinsic graphs.

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