** Horizontal convexity in the Heisenberg group **

Joint with Andrea Calogero and Alexandru Kristály.

**1. Alexandrov’s theorem in Euclidean space **

I think of a convex function as a function which stays above its supporting affine function at every point. The set of slopes (covectors) of these affine functions at is called the *subdifferential* of at . The union of all these sets over the domain is denoted by . Its measure is sometimes called the *Monge-Ampère measure* of .

Theorem 1 (Alexandrov)There is a dimension dependant constant with the following effect. Let be an open, bounded convex domain in . Let be a convex function on the closure of , which vanishes on the boundary. Then, for all ,

This is used in PDE (Caffarelli,…).

**2. Convexity in Heisenberg group **

Since left translations ar affine is exponential coordinates, Heisenberg group carries an affine structure. Therefore convex domains will simply be Euclidean convex.

Definition 2 (Several competing groups)Say a function on a convex domain of Heisenberg group is -convex if its restriction to every horizontal line of is convex.

The subdifferential of at is a subset of .

Note that there are H-convex functions in which are very irregular (e.g. Weierstrass) in the vertical direction.

** 2.1. Results **

We define a horizontal slicing diameter : this is the maximal diameter of the intersection of with horizontal planes , . We also define a horizontal slicing Monge-Ampère measure

Theorem 3There is a dimension dependant constant with the following effect. Let be an open, bounded convex domain in . Let be a convex function on the closure of , which vanishes on the boundary. Then, for all ,

This improves earlier results by Garofalo et al. where the distance to the boundary appeared with a negative power.

**3. Proof **

** 3.1. Back to the Euclidean case **

Lemma 4 (Comparison principle)Let , be continuous functions on the closure of . Assume that . Then

Indeed, any supporting hyperplane of the graph of , when raised, will touch the graph of .

Alexandrov compares the graph of with the cone on with vertex at . Its subdifferential is concentrated at the vertex. Let be the nearest point in the boundary. In the subdifferential , there is a covector of size . All othe covectors in satisfy

** 3.2. Failure of comparison principle in Heisenberg group **

There exists functions on a cyclinder , which are equal n the boundary and , but .

Indeed, set . Check that , so that contains the origin. Modify in an annulus,

where has support in an annulus. Assume that . Then , and achieves its minimum on at point . One can achieve that this never happens.

** 3.3. Comparison for convex functions **

What saves us is that comparison holds for convex functions.

Theorem 5Let be a convex domain in , let be convex functions on that are equal on the boundary. Assume that for some , there exists such that, for al different from ,

Then .

The proof uses degree theory for set valued maps. For simplicity, let us assume that is smooth, and . Let projected to . We view as a mapping of to . To show that belongs to its image, it suffices to show that the degree of on at is non zero. We check that this is the case when is replaced with . Then a linear homotopy allows to conclude. Indeed, assume by contradiction that the homotopy hits along , i.e. there exists a point and such that

Along the horizontal line from to ,

Take the convex combination of these two inequations, get and inequality that contradicts the assumption .

Computation of the index for .

** 3.4. End of the proof **

One gets

There remains to replace with . This relies of an Harnack inequality, which allows to replace with a nearby point where the horizontal plane is tilted and hits the boundary at a distance comparable to the distance of to the boundary.

Next sessions : April 4th and May 14th.