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On minimizing surfaces of the CR invariant energy $E_1$

Jih-Hsin Cheng, Hung-Lin Chiu, Paul Yang, Yongbing Zhang

TL;DR

This work analyzes CR-invariant energy surfaces for the vanishing-energy condition $E_1=0$ in the Heisenberg group. It proves the governing equation is a hyperbolic PDE with CR-invariant surface measure $dA_1$, and establishes local uniqueness of absolute $E_1$-minimizers from non-characteristic initial data, including both smooth and real-analytic categories. It then classifies complete rotationally symmetric absolute $E_1$-minimizers, identifying four families up to CR transformations: two quadratic graphs and two shifted Heisenberg spheres, with the quadratic cases arising as dilation limits. Finally, it computes the second variation of $E_1$ for the Clifford torus in the CR 3-sphere and shows the torus is not a local minimizer, refining prior expectations about CR-minimizing torus-type surfaces. Together, these results illuminate rigidity, stability, and explicit minimizers for CR-invariant energies in low-dimensional CR geometry.

Abstract

We study a CR-invariant equation for vanishing $E_1$ surfaces in the 3-dimensional Heisenberg group. This is shown to be a hyperbolic equation. We prove the local uniqueness theorem for an initial value problem and classify all such global surfaces with rotational symmetry. We also show that the Clifford torus in the CR 3-sphere is not a local minimizer of $E_1$ by computing the second variation.

On minimizing surfaces of the CR invariant energy $E_1$

TL;DR

This work analyzes CR-invariant energy surfaces for the vanishing-energy condition in the Heisenberg group. It proves the governing equation is a hyperbolic PDE with CR-invariant surface measure , and establishes local uniqueness of absolute -minimizers from non-characteristic initial data, including both smooth and real-analytic categories. It then classifies complete rotationally symmetric absolute -minimizers, identifying four families up to CR transformations: two quadratic graphs and two shifted Heisenberg spheres, with the quadratic cases arising as dilation limits. Finally, it computes the second variation of for the Clifford torus in the CR 3-sphere and shows the torus is not a local minimizer, refining prior expectations about CR-minimizing torus-type surfaces. Together, these results illuminate rigidity, stability, and explicit minimizers for CR-invariant energies in low-dimensional CR geometry.

Abstract

We study a CR-invariant equation for vanishing surfaces in the 3-dimensional Heisenberg group. This is shown to be a hyperbolic equation. We prove the local uniqueness theorem for an initial value problem and classify all such global surfaces with rotational symmetry. We also show that the Clifford torus in the CR 3-sphere is not a local minimizer of by computing the second variation.

Paper Structure

This paper contains 11 sections, 20 theorems, 180 equations, 3 figures.

Table of Contents

  1. Hyperbolicity and CR invariance
  2. Proof of Theorem \ref{['mainthm2']}: the local uniqueness
  3. The smooth category
  4. Hyperbolic system
  5. The real analytic category
  6. Proof of Theorem \ref{['T-class']}: the classification
  7. Rotationally symmetric surfaces
  8. Proof of Theorem \ref{['T-2nd-var']}: second variation of $E_{1}$ for the Clifford torus
  9. First variation of $E_{1}:$review
  10. Second variation of $E_{1}$
  11. Second variation of $E_{1}$ for the Clifford torus: proof of Theorem \ref{['T-2nd-var']}

Key Result

Theorem 1

Suppose that on a nonsingular domain $u=u(r,{ \if@compatibility \mathchar"011E {} \mathchar"011E } )$ and $v=v(r,{ \if@compatibility \mathchar"011E {} \mathchar"011E } )$ are two (smooth) absolute $E_{1}$-minimizers having the same initial values $H,{ \if@compatibility \mathchar"010B

Figures (3)

  • Figure 1: (\ref{['soleq131']}) and (\ref{['soleq132']})
  • Figure 2: (\ref{['3-20']})
  • Figure 3: closed pictures

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 2.1
  • proof
  • ...and 27 more