Intrinsic Lipschitz sections of no-linear quotient maps

TL;DR

This paper explores intrinsic Lipschitz sections of nonlinear quotient maps, focusing on their properties in Carnot groups, including dilation invariance and conditions for sum closure, extending previous linear cases.

Contribution

It extends the theory of intrinsic Lipschitz sections to nonlinear quotient maps, deriving a Leibniz formula and analyzing their behavior in Carnot groups.

Findings

Derived a Leibniz formula for intrinsic slope under weaker conditions

Showed dilation invariance of Lipschitz sections in Carnot groups

Provided conditions for sum of sections to be intrinsically Lipschitz in step 2 Carnot groups

Abstract

Le Donne and the author introduced the so-called intrinsically Lipschitz sections of a fixed quotient map $\pi$ in the context of metric spaces. Moreover, the author introduced the concept of intrinsic Cheeger energy when the quotient map is also linear. In this note we investigate about the non linearity of $\pi$. In particular, we find a Leibniz formula for the intrinsic slope when $\pi$ satisfies a weaker condition. After that, we focus our attention on Carnot groups and using the properties of intrinsic dilations we show that the dilation of a Lipschitz section is so too. Finally, in Carnot groups of step 2, we give a suitable additional condition in order to get the sum of two intrinsically Lipschitz sections is so too.