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The CR-Volume of Horizontal Submanifolds of Spheres

Jacob Bernstein, Arunima Bhattacharya

Abstract

We study an analog in CR-geometry of the conformal volume of Li-Yau. In particular, to submanifolds of odd-dimensional spheres that are Legendrian or, more generally, horizontal with respect to the sphere's standard CR-structure we associate a quantity that is invariant under the CR-automorphisms of the sphere. We apply this concept to a corresponding notion of Willmore energy.

The CR-Volume of Horizontal Submanifolds of Spheres

Abstract

We study an analog in CR-geometry of the conformal volume of Li-Yau. In particular, to submanifolds of odd-dimensional spheres that are Legendrian or, more generally, horizontal with respect to the sphere's standard CR-structure we associate a quantity that is invariant under the CR-automorphisms of the sphere. We apply this concept to a corresponding notion of Willmore energy.
Paper Structure (20 sections, 24 theorems, 265 equations)

This paper contains 20 sections, 24 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Geometric background
  3. Basic constructs on Euclidean space
  4. Contact, Sasaki and CR-geometry of $\mathbb{S}^{2n+1}$
  5. Complex automorphisms of unit ball in $\mathbb{C}^{n+1}$ and CR-automorphisms of $\mathbb{S}^{2n+1}$
  6. Contact gauge transformations and their Levi-Civita connections
  7. Horizontal and Legendrian submanifolds of $\mathbb{S}^{2n+1}$
  8. Properties of horizontal submanifolds
  9. Transformation of extrinsic curvatures for horizontal submanifolds
  10. Characterization of CR-umbilical horizontal submanifolds
  11. Lower bounds for CR-volume and rigidity
  12. Alternate form of CR-volume
  13. Rigidity of $\lambda_{CR}$-minimizers
  14. CR-volume as conformal invariant
  15. CR-Willmore Energy
  16. ...and 5 more sections

Key Result

Theorem 1.1

If $\Sigma\subset \mathbb{S}^{2n+1}$ is an $m$-dimensional closed horizontal submanifold and $\mathbb{S}^m$ is an $m$-dimensional totally geodesic horizontal sphere in $\mathbb{S}^{2n+1}$, then Equality holds if and only if $\Sigma$ is a contact Whitney sphere, i.e., $\Sigma=\Psi(\mathbb{S}^m)$ for some $\Psi\in Aut_{CR}(\mathbb{S}^{2n+1})$. Moreover, there is a $\Psi\in Aut_{CR}(\mathbb{S}^{2n+1

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Proposition 1.7
  • Remark 1.8
  • Lemma 2.1
  • Proposition 2.2
  • ...and 44 more