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.
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 3 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 improves earlier results by Garofalo et al. where the distance to the boundary appeared with a negative power.
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 5 Let 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 ,
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
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.