Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

TL;DR

This paper characterizes when intrinsic graphs in Carnot groups are $C^1_H$ regular through uniform intrinsic differentiability, linking it to H"older properties and providing an area formula for such maps.

Contribution

It offers a new characterization of uniform intrinsic differentiability in Carnot groups and derives an explicit area formula for these maps.

Findings

Characterization of $C^1_H$ regularity via H"older properties.

Strengthened results for step-2 Carnot groups with horizontal regularity.

An explicit area formula for uniformly intrinsically differentiable maps.

Abstract

In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of H\"older properties along the projections of left-invariant vector fields on the graph. We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group. As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.