On Three-Dimensional CR Yamabe Solitons

Abstract

In this paper, we investigate the geometry and classification of three-dimensional CR Yamabe solitons and pseudo-gradient CR Yamabe solitons. In the compact case, we obtain a classification result of three-dimensional CR Yamabe solitons under the assumption that their potential functions are in the kernel of the CR Paneitz operator. In addition, we obtain a structure theorem on the diffeomorphism types of complete three-dimensional pseudo-gradient CR Yamabe solitons (shrinking, steady, or expanding) of vanishing torsion.

Appendix

We include some basic facts about isoparametric functions mentioned in Remark 5.1. Recall that our isoparametric potential function $\varphi$ satisfies $|\nabla \varphi |= b(\varphi )$ for some function $b: \mathcal {R}\rightarrow \mathbb {R}$, where $\mathcal {R}$ denotes the range of $\varphi$.

Lemma 6.1

(cf. Lemma 3 in [34]) The only possible singular level sets of $\varphi$ are the smooth focal submanifolds

$$\begin{aligned} \Sigma _+:= \{ \varphi = \max _{x\in M} \varphi (x)\}\ \text{ and } \ \Sigma _-:= \{\varphi = \min _{x\in M} \varphi (x)\}. \end{aligned}$$

Proof

We follow the argument in [34]. Suppose that $b=|\nabla \varphi |=0$ on the level set $\Sigma _{c}:= \{x\in M |\varphi (x) = c\}$ for some critical value $c\in (\inf _{M}\varphi ,\sup _{M}\varphi )$. Then, there exists some number $\epsilon >0$ sufficiently small so that c is the only critical value in interval $[c, c+\epsilon ]$. In particular, we have

$$\begin{aligned} \int _c^{c+\epsilon } \frac{1}{|\nabla \varphi |} \text {d}\varphi =\lim _{a\rightarrow c^{+}} \int _a^{c+\epsilon } \frac{1}{|\nabla \varphi |} \text {d}\varphi = \epsilon . \end{aligned}$$

(5.6)

On the other hand, since $b\ge 0$ on $\mathcal {R}$ and c is an interior point in $\mathcal {R}$, one has $b'=0$ at c so the function b is locally bounded by its Taylor’s reminder around c, i.e., $b(x)\le A(x-c)^2$ for $x\in [c, c+\epsilon ]$ and some constant $A>0$. However, this would imply that

$$\begin{aligned} \int _c^{c+\epsilon } \frac{1}{|\nabla \varphi |} \text {d}\varphi \ge \int _c^{c+\epsilon } \frac{1}{A(x-c)^2} \text {d}\varphi =\infty , \end{aligned}$$

which contradicts (5.6). $\square$

Suppose that $\varphi$ is locally a constant function on some open set U of M, then b and $b'$ will vanish at the boundary of U, which is impossible as shown in the proof of Lemma 6.1 Therefore, we have the following lemma (which is used in the proof of Theorem 1.3).

Lemma 6.2

The non-constant potential function $\varphi$ cannot be locally constant. In particular, its critical set is at most two-dimensional.