The Singular CR Yamabe Problem and Hausdorff Dimension
Sagun Chanillo, Paul C. Yang
TL;DR
This work addresses the singular CR Yamabe problem on compact pseudo-hermitian manifolds and derives a sharp bound on the Hausdorff dimension of the singular set under a complete CR conformal deformation with controlled negative scalar curvature, proving $\dim_{\mathcal{H}}(S)\le \frac{Q-2}{2}$ for $Q=2n+2$. It develops a CR potential-theoretic framework, employs a CR version of Moser iteration, and uses capacity arguments to mirror the conformal Schoen–Yau program in the CR setting. In dimension three, it proves injectivity of the CR developing map for spherical manifolds with $Y(M)>0$ and nonnegative Paneitz operator by comparing Green functions and invoking the positive mass theorem, with the CR mass $A\ge0$ and $A>0$ under the cited results. Collectively, these results extend the Schoen–Yau program to CR geometry, clarifying the structure of singular sets and the global behavior of developing maps in CR manifolds.
Abstract
We consider a compact pseudo-hermitian manifold (M,θ, J), that is a manifold equipped with a contact form θand CR structure J. We consider a conformal deformation of the contact form to obtain a complete, singular contact form and a corresponding Yamabe problem. We estimate then the Hausdorff dimension of the singular set. The conformal geometry analog of this result is due to R. Schoen and S. -T. Yau. Results of this type have their origin in work by Huber for Riemann surfaces. In the second part of our paper we investigate the CR developing map for three dimensional CR manifolds. We establish the injectivity of the developing map essentially using the same strategy as Schoen and Yau for the conformal case which is based on the positive mass theorem. Higher dimensional analogs of Huber's theorem in the conformal case for Q curvature are due to Alice Chang, Jie Qing and P. Yang.