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Non-homothetic complete periodic contact forms with constant Tanaka--Webster scalar curvature

Jeffrey S. Case, Yuya Takeuchi

TL;DR

The work addresses the problem of finding complete contact forms with constant Tanaka–Webster scalar curvature $R_{\theta}$ on non-compact strictly pseudoconvex CR manifolds, and proves that, under mild hypotheses, the universal cover of a compact base with infinite profinite completion of its fundamental group carries infinitely many pairwise non-homothetic CSC contact forms.The authors develop a general mechanism based on infinite towers of finite connected coverings and on unit-volume CR Yamabe minimizers on the finite covers, lifting to a periodic CSC structure on the universal cover and invoking Schoen's rigidity to enforce infinitude of the moduli.They then instantiate the main result in concrete CR-geometric settings, including complements of $\mathbb{R}$- and $\mathbb{C}$-spheres in the CR sphere, circle bundles over Kähler manifolds, and boundaries of Reinhardt domains, thereby providing CR analogues of non-compact Yamabe-type multiplicity phenomena with broad geometric applications.

Abstract

We study the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature on non-compact strictly pseudoconvex CR manifolds. We prove that, under mild assumptions, the universal cover of a compact strictly pseudoconvex CR manifold admits infinitely many non-homothetic such contact forms whenever its fundamental group has infinite profinite completion. As applications, we treat complements of real or complex spheres in the standard CR sphere, as well as circle bundles over compact Kähler manifolds and the boundary of a Reinhardt domain.

Non-homothetic complete periodic contact forms with constant Tanaka--Webster scalar curvature

TL;DR

The work addresses the problem of finding complete contact forms with constant Tanaka–Webster scalar curvature $R_{\theta}$ on non-compact strictly pseudoconvex CR manifolds, and proves that, under mild hypotheses, the universal cover of a compact base with infinite profinite completion of its fundamental group carries infinitely many pairwise non-homothetic CSC contact forms.The authors develop a general mechanism based on infinite towers of finite connected coverings and on unit-volume CR Yamabe minimizers on the finite covers, lifting to a periodic CSC structure on the universal cover and invoking Schoen's rigidity to enforce infinitude of the moduli.They then instantiate the main result in concrete CR-geometric settings, including complements of $\mathbb{R}$- and $\mathbb{C}$-spheres in the CR sphere, circle bundles over Kähler manifolds, and boundaries of Reinhardt domains, thereby providing CR analogues of non-compact Yamabe-type multiplicity phenomena with broad geometric applications.

Abstract

We study the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature on non-compact strictly pseudoconvex CR manifolds. We prove that, under mild assumptions, the universal cover of a compact strictly pseudoconvex CR manifold admits infinitely many non-homothetic such contact forms whenever its fundamental group has infinite profinite completion. As applications, we treat complements of real or complex spheres in the standard CR sphere, as well as circle bundles over compact Kähler manifolds and the boundary of a Reinhardt domain.
Paper Structure (20 sections, 12 theorems, 61 equations)

This paper contains 20 sections, 12 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. CR geometry
  3. CR manifolds
  4. Tanaka--Webster connection
  5. Sasakian manifolds
  6. Schoen's Theorem for CR automorphism group
  7. The CR automorphism group of $S^{2 n + 1}$ and $\mathbb{H}^{2 n + 1}$
  8. CR Yamabe problem
  9. Complex hyperbolic geomerty
  10. Complex hyperbolic Kleinian group
  11. Infinite tower of finite connected coverings
  12. Profinite completion and residual finiteness
  13. Infinite tower of finite connected coverings
  14. Proof of \ref{['thm:infinite-geometric-moduli-general']}
  15. Complements of spheres
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$ such that keyassumption holds and $\pi_{1}(M)$ has infinite profinite completion. Then $\# \mathop{\mathrm{\mathcal{M}}}\nolimits(\widetilde{M}, T^{1, 0} \widetilde{M}) = \infty$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: Schoen1995*Theorems 3.3' and 3.4'
  • Lemma 2.3
  • proof
  • Theorem 4.1: Nayatani1999*Theorem 2.4(ii)
  • Definition 5.1
  • Lemma 5.2
  • ...and 15 more