Height estimate for minimal surfaces in Heisenberg groups
With R. Monti.
1. The main result
Theorem 1 In , . There exist constats , , such that if is a local minimizer of perimeter containing the origin, contained in horizontal cylinder , and has small excess in direction ,
then the intersection of with a slightly smaller cylinder is nearly contained in a strip in direction ,
Let be the intersection of the Koranyi -ball with the hyperplane . Let
The excess of inside a horizontal cylinder in horizontal direction is
The inequality we prove can be thought of as a non linear version of Poincaré inequality.
It fails for : there, there are sets with constant horizontal normal, and therefore vanishing excess, which are not vertical planes.
It is a first step in a regularity theory. We expect a reverse Poincaré inequality of the form
This would imply an estimate on excess,
and one knows that this implies that the normal is , and the boundary is .
Follows Schoen-Simon 1980.
2.1. Step 0
If is small enough, is contained in a bounded strip. This follows from the fact that vertical hyperplanes are minimizing.
2.2. Step 1
Let be such that hyperplane divides into two pieces of equal perimeters.
Let be such that hyperplane divides into a piece of perimeter .
We prove that . Indeed, the density estimate states that
2.3. Step 2
This is the serious step. We show that . By Hölder,
Here, we have used the coarea formula, , and is the perimeter measure for the sub-Riemannian metric of hyperplane . This is new.
Using the isoperimetric inequality (this is the place where the argument collapses for ), this is estimated below by
2.4. Step 3
Let . Then . For this, one uses the isometry , .