On the invariant surface area functionals in 3-dimensional CR geometry

TL;DR

This work extends Cheng–Yang–Zhang’s CR invariant surface area functionals $E_1$ and $E_2$ from constant Webster curvature and vanishing torsion to general 3D CR manifolds, deriving the corresponding Euler–Lagrange equations and applying them to concrete CR models. It computes explicit EL expressions on the disk bundle $B^1\times\mathbb{R}$ and identifies multiple zero-energy and nonzero-energy critical surfaces, including planar and radially defined ones, with detailed conditions for $E_1$ and $E_2$ criticality. In the Rossi sphere, the Clifford torus emerges as a zero-energy minimizer for $E_1$ at $t_0=4-\sqrt{15}$ and as a critical point for $E_2$ only at the same parameter; the paper also analyzes the unboundedness of $E_2$ in certain ranges and clarifies how the torus geometry influences criticality. Finally, the study of 3D tori shows how nonvanishing torsion precludes CHMY’s EL equations, motivating the general EL framework and demonstrating zero-energy and nonzero-energy critical tori via circle and ellipse generating curves.

Abstract

Cheng, Yang, and Zhang have studied two invariant surface area functionals in 3-dimensional CR manifolds. They deduced the Euler-Lagrange equations of the associated energy functionals when the 3-dimensional CR manifold has constant Webster curvature and vanishing torsion. In this paper, we deduce the the Euler-Lagrange equations of the energy functionals in a more general 3-dimensional CR manifold. Moreover, we study the invariant area functionals on the disk bundle, on the Rossi sphere, and on 3-dimensional tori.

Theorems & Definitions (28)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 4.1
• Lemma 4.2
• Theorem 5.1
• Lemma 6.1
• proof : Proof of Lemma \ref{['lem7.1']}
• Lemma 6.2