** 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 ,

* *

** 1.1. Cylinders **

Let be the intersection of the Koranyi -ball with the hyperplane . Let

** 1.2. Excess **

The excess of inside a horizontal cylinder in horizontal direction is

** 1.3. Comments **

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 .

**2. Proof **

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 .

Let .

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 , .

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