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CR Yamabe constant and inequivalent CR structures

Chanyoung Sung, Yuya Takeuchi

TL;DR

This work investigates the CR Yamabe constant $\\lambda(X)$ for compact strongly pseudoconvex CR manifolds, establishing integral reformulations and continuity under CR deformations. It constructs an infinite-dimensional family of CR structures on a fixed circle bundle over a Hodge manifold, showing the CR Yamabe constant varies continuously and can assume values strictly below a baseline in many cases. Moreover, the authors prove the existence of a compact simply-connected $(2n+1)$-manifold carrying two inequivalent strongly pseudoconvex CR structures with opposite signs of $\\lambda$, using sophisticated cohomological and diffeomorphism arguments to distinguish the structures. Together, these results highlight the richness of CR geometry beyond a single contact structure and provide tools to generate and distinguish inequivalent CR geometries with varying Yamabe behavior.

Abstract

The CR Yamabe constant is an invariant of a compact strongly pseudoconvex CR manifold and plays an important role in CR geometry. We show some integral formulae of the CR Yamabe constant. We also construct an infinite-dimensional family of strongly pseudoconvex CR structures with varying CR Yamabe constants and a compact simply-connected manifold admitting two strongly pseudoconvex CR structures with different signs of the CR Yamabe constant.

CR Yamabe constant and inequivalent CR structures

TL;DR

This work investigates the CR Yamabe constant for compact strongly pseudoconvex CR manifolds, establishing integral reformulations and continuity under CR deformations. It constructs an infinite-dimensional family of CR structures on a fixed circle bundle over a Hodge manifold, showing the CR Yamabe constant varies continuously and can assume values strictly below a baseline in many cases. Moreover, the authors prove the existence of a compact simply-connected -manifold carrying two inequivalent strongly pseudoconvex CR structures with opposite signs of , using sophisticated cohomological and diffeomorphism arguments to distinguish the structures. Together, these results highlight the richness of CR geometry beyond a single contact structure and provide tools to generate and distinguish inequivalent CR geometries with varying Yamabe behavior.

Abstract

The CR Yamabe constant is an invariant of a compact strongly pseudoconvex CR manifold and plays an important role in CR geometry. We show some integral formulae of the CR Yamabe constant. We also construct an infinite-dimensional family of strongly pseudoconvex CR structures with varying CR Yamabe constants and a compact simply-connected manifold admitting two strongly pseudoconvex CR structures with different signs of the CR Yamabe constant.
Paper Structure (6 sections, 9 theorems, 78 equations)

This paper contains 6 sections, 9 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. CR manifolds
  3. Formulae for CR Yamabe constant
  4. Continuity of the CR Yamabe constant
  5. Deformations of CR structures with varying CR Yamabe constants
  6. Construction of a manifold with different signs of CR Yamabe constants

Key Result

Theorem 1.1

Let $(M, J, \omega)$ be an $n$-dimensional compact Hodge manifold. Then the map is continuous. Moreover if $\theta_{0} / 2 \pi$ is a CR Yamabe minimizer, then for any $\varphi \in \mathcal{K} \setminus \mathcal{F}$. The assumption holds if $\omega$ has constant non-positive scalar curvature or it defines a Kähler-Einstein metric.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 11 more