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Willmore surfaces in 4-dimensional conformal manifolds

Changping Wang, Zhenxiao Xie

TL;DR

The paper develops a conformal-geometry framework for Willmore surfaces in four-dimensional conformal manifolds, deriving the first and second variations of the conformal Willmore functional $\mathcal{W}$ and expressing the Euler–Lagrange equation in conformally invariant form. It introduces Möbius-invariant data, including the invariants $\phi$, $\psi$, and the Möbius metric, to rewrite Willmore functionals and prove index theorems for isotropic/anti-isotropic points. A central result is the strict Willmore stability of the Clifford torus in $\mathbb{C}P^2$, together with a lower bound $\lambda_1^{Jacobi}\ge 12$ for complex curves, supporting Montiel–Urbano’s conjecture that the Clifford torus minimizes Willmore energy among tori; the work also shows that minimal $2$-spheres in 4D locally symmetric spaces must be super-minimal to be Willmore. Together, these results illuminate the minimization landscape of the Willmore functional in 4D and provide tools (holomorphic differentials, conformal invariants) for classifying Willmore spheres and tori in curved ambient geometries.

Abstract

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.

Willmore surfaces in 4-dimensional conformal manifolds

TL;DR

The paper develops a conformal-geometry framework for Willmore surfaces in four-dimensional conformal manifolds, deriving the first and second variations of the conformal Willmore functional and expressing the Euler–Lagrange equation in conformally invariant form. It introduces Möbius-invariant data, including the invariants , , and the Möbius metric, to rewrite Willmore functionals and prove index theorems for isotropic/anti-isotropic points. A central result is the strict Willmore stability of the Clifford torus in , together with a lower bound for complex curves, supporting Montiel–Urbano’s conjecture that the Clifford torus minimizes Willmore energy among tori; the work also shows that minimal -spheres in 4D locally symmetric spaces must be super-minimal to be Willmore. Together, these results illuminate the minimization landscape of the Willmore functional in 4D and provide tools (holomorphic differentials, conformal invariants) for classifying Willmore spheres and tori in curved ambient geometries.

Abstract

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in , we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.
Paper Structure (10 sections, 19 theorems, 218 equations)

This paper contains 10 sections, 19 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Conformal geometry of surfaces in conformal 4-manifold
  3. Basic isometric invariants on $(M^2, x^*h)$
  4. Some formulas on complex covariant derivatives
  5. Some conformal invariants of $x(M^2)$
  6. An Index theorem for (anti-)isotropic points
  7. The first variation of the Willmore functional
  8. Willmore $2$-spheres in $4$-dimensional locally symmetric spaces
  9. The second variation of the Willmore functional
  10. The strictly Willmore stability of Clifford torus in $\mathbb{C}P^2$

Key Result

Theorem 1

In a $4$-dimensional locally symmetric space, to be Willmore, a minimal $2$-sphere must be super-minimal.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 36 more