Notes of Manuel Ritore’s lecture

A flux formula for {C^1} area-stationary surfaces the Heisenberg group

Joint with M. Galli.

1. Rulings

Our main result is

Theorem 1 A complete immersed {C^1} 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 {t}-graphs. It is more natural to consider intrinsic graphs (any {C^1} surface, away from characteristic points, is locally an intrinsic graph).

Cheng, Hwang, Malchiodi and Yang show that, for {C^2} minimal {t}-graphs, characteristic points are either isolated or come in continuous curves. In the {C^1} 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 {U} multiplied by a cut-off near the boundary. This gives

\displaystyle  \begin{array}{rcl}  \int_{\partial\Sigma}\langle\langle U,T \rangle S-|N_h|U,\xi\rangle=0, \end{array}

where {\xi} is the unit normal to the boundary, {N_h} is the horizontal projection of the Riemannian unit normal {N}, and {N=N_h+S}. This can be viewed as a first integral of the equations.

2.2. Application

I prove the theorem above. Assume {\Sigma} is an intrinsic graph. Use contours {\partial\Sigma} 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 {C^1} 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 {C^1_h} intrinsic graphs.


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