Height estimate and slicing formulas in the Heisenberg group

TL;DR

This paper establishes a height estimate for perimeter-minimizing sets in the Heisenberg group, linking geometric deviation to excess via a novel coarea formula, advancing understanding of sub-Riemannian geometry.

Contribution

It introduces a new height estimate for $ ext{Lambda}$-minima in the Heisenberg group using a novel coarea formula, connecting geometric oscillation to perimeter minimization.

Findings

Proves a height estimate in the Heisenberg group for perimeter minimizers.

Develops a new coarea formula for rectifiable sets in the Heisenberg group.

Links the height estimate to the excess measure of the set.

Abstract

We prove a height-estimate (distance from the tangent hyperplane) for $\Lambda$-minima of the perimeter in the sub-Riemannian Heisenberg group. The estimate is in terms of a power of the excess ($L^2$-mean oscillation of the normal) and its proof is based on a new coarea formula for rectifiable sets in the Heisenberg group.