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Classification of Willmore $2$-spheres in $S^n$

Xiang Ma, Franz Pedit, Peng Wang

TL;DR

The paper delivers a complete classification of Willmore 2-spheres in S^n by unveiling a tripartite structure: totally isotropic spheres arising from normal-horizontal twistor curves, Möbius-congruent minimal surfaces with embedded planar ends, and strictly k-isotropic spheres generated by an (m−k)-step adjoint-transform sequence culminating in a strictly m-isotropic minimal surface in $ ext{R}^n$. Central to the approach is the construction of a harmonic sequence in real Grassmannians over the Lorentz space, together with a universal Gauss map h_0 and two intertwined harmonic-flow sequences that link the Willmore data to Euclidean minimal surfaces via adjoint transforms. The authors overcome singularities in adjoint transformations by a Frenet-bundle framework and holomorphic-differential arguments, enabling a global, finite adjoint sequence on S^2 and providing a robust geometric mechanism for constructing and understanding all Willmore 2-spheres. The work significantly extends prior S^5 results to all ambient dimensions, enriching the interplay between Möbius geometry, harmonic maps, and twistor theory, and supplying explicit pathways to generate new examples through adjoint chains from strictly m-isotropic minimal surfaces.

Abstract

This paper resolves a long-standing open problem by providing a classification of Willmore $2$-spheres in $S^n$. We show that any such $2$-sphere is either totally isotropic--originating from the projection of a special twistor curve in the twistor bundle over an even-dimensional sphere--or strictly $k$-isotropic, obtained via $(m-k)$ steps of adjoint transforms of a strictly $m$-isotropic minimal surface in $\mathbb{R}^n$, where $0\leq k\leq m\leq[n/2]-2$. Our approach hinges on the construction of a harmonic sequence in the real Grassmannian over the Lorentz space, derived from the harmonic conformal Gauss map of the original Willmore sphere. This sequence terminates finitely and generalizes, in part, the classical theory of harmonic sequences for harmonic $2$-spheres in complex Grassmannians.

Classification of Willmore $2$-spheres in $S^n$

TL;DR

The paper delivers a complete classification of Willmore 2-spheres in S^n by unveiling a tripartite structure: totally isotropic spheres arising from normal-horizontal twistor curves, Möbius-congruent minimal surfaces with embedded planar ends, and strictly k-isotropic spheres generated by an (m−k)-step adjoint-transform sequence culminating in a strictly m-isotropic minimal surface in . Central to the approach is the construction of a harmonic sequence in real Grassmannians over the Lorentz space, together with a universal Gauss map h_0 and two intertwined harmonic-flow sequences that link the Willmore data to Euclidean minimal surfaces via adjoint transforms. The authors overcome singularities in adjoint transformations by a Frenet-bundle framework and holomorphic-differential arguments, enabling a global, finite adjoint sequence on S^2 and providing a robust geometric mechanism for constructing and understanding all Willmore 2-spheres. The work significantly extends prior S^5 results to all ambient dimensions, enriching the interplay between Möbius geometry, harmonic maps, and twistor theory, and supplying explicit pathways to generate new examples through adjoint chains from strictly m-isotropic minimal surfaces.

Abstract

This paper resolves a long-standing open problem by providing a classification of Willmore -spheres in . We show that any such -sphere is either totally isotropic--originating from the projection of a special twistor curve in the twistor bundle over an even-dimensional sphere--or strictly -isotropic, obtained via steps of adjoint transforms of a strictly -isotropic minimal surface in , where . Our approach hinges on the construction of a harmonic sequence in the real Grassmannian over the Lorentz space, derived from the harmonic conformal Gauss map of the original Willmore sphere. This sequence terminates finitely and generalizes, in part, the classical theory of harmonic sequences for harmonic -spheres in complex Grassmannians.

Paper Structure

This paper contains 32 sections, 39 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. The Willmore sequence and the real harmonic sequence
  3. Other related works
  4. Organization of the paper
  5. Willmore surfaces and adjoint transforms
  6. Surface theory in Möbius geometry
  7. Review of Willmore surfaces
  8. Dual surfaces and S-Willmore surfaces
  9. The adjoint transform
  10. Isotropic order of surfaces in $S^n$
  11. Harmonic sequence associated to a harmonic $2$-sphere
  12. Harmonic sequence in view of Chern-Wolfson
  13. Real harmonic sequence for harmonic maps with a Frenet-bundle structure
  14. Geometric structure of $k$-isotropic Willmore $2$-spheres
  15. Totally isotropic Willmore $2$-spheres & $\hbox{normal-horizontal}$ twistor curves
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $y:S^2\to S^n$ be a Willmore immersion. Then $y$ belongs to one of the following cases.

Theorems & Definitions (89)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • ...and 79 more