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A Liouville theorem for CR Yamabe type equation on Sasakian manifolds

Biqiang Zhao

TL;DR

This work proves Liouville-type rigidity theorems for a CR Yamabe-type semilinear equation on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature. Using Jerison–Lee's differential identity together with refined integral estimates for an energy-like quantity $\mathcal{M}$ and a subcritical nonlinearity, it is shown that $\mathcal{M}=0$, which forces $H'(f)=2H(f)$ and hence $F(u)=C u^{1+\frac{2}{n}}$. Consequently, the manifold must be CR isometric to the Heisenberg group $\mathbb{H}^n$ and the positive solution $u$ takes the Jerison–Lee model form; results are established for $n=1$, $n=2$, and $n\ge 3$ under appropriate volume-growth conditions. This delivers a unifying rigidity framework across dimensions in the Sasakian setting with nonnegative curvature.

Abstract

In this paper, we study the CR Yamabe type equation \begin{align} Δ_b u+F(u)=0 \nonumber \end{align} on complete noncompact $(2n+1)$-dimensional Sasakian manifolds with nonnegative curvature. Under some assumptions, we prove a rigidity result, that is, the manifold is CR isometric to Heisenberg group $\mathbb{H}^n$. The proofs are based on the Jerison-Lee's differential identity combining with integral estimates.

A Liouville theorem for CR Yamabe type equation on Sasakian manifolds

TL;DR

This work proves Liouville-type rigidity theorems for a CR Yamabe-type semilinear equation on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature. Using Jerison–Lee's differential identity together with refined integral estimates for an energy-like quantity and a subcritical nonlinearity, it is shown that , which forces and hence . Consequently, the manifold must be CR isometric to the Heisenberg group and the positive solution takes the Jerison–Lee model form; results are established for , , and under appropriate volume-growth conditions. This delivers a unifying rigidity framework across dimensions in the Sasakian setting with nonnegative curvature.

Abstract

In this paper, we study the CR Yamabe type equation \begin{align} Δ_b u+F(u)=0 \nonumber \end{align} on complete noncompact -dimensional Sasakian manifolds with nonnegative curvature. Under some assumptions, we prove a rigidity result, that is, the manifold is CR isometric to Heisenberg group . The proofs are based on the Jerison-Lee's differential identity combining with integral estimates.

Paper Structure

This paper contains 6 sections, 13 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem 1.1
  4. Proof of Theorem 1.2
  5. Proof of Theorem 1.3
  6. Acknowledgments

Key Result

Theorem 1.1

Let $(M^3,J,\theta)$ be a 3-dimensional complete noncompact Sasakian manifold with nonnegative Tanaka-Webster scalar curvature and assume that $F$ is a subcritical function with exponent $3$. Let $u$ be a positive solution to (1.1) such that $u$ tends to zero at infinity. If there exists a constant Then $(M^3,J,\theta)$ is isometric to $\mathbb{H}^1$ with its standard structures and $u$ is the fo

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1
  • ...and 9 more