Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

TL;DR

This paper explores invariant surface area functionals in 3D CR geometry, deriving their Euler-Lagrange equations, analyzing solutions, and linking one functional to volume renormalization in the singular CR Yamabe problem.

Contribution

It introduces explicit CR invariant surface area elements, derives their energy functional equations, and connects these to volume renormalization in the singular Yamabe problem.

Findings

Derived explicit formulas for CR invariant surface area elements.

Obtained Euler-Lagrange equations for these functionals.

Linked one energy functional to volume renormalization in the singular Yamabe problem.

Abstract

We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization.