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The Singular CR Yamabe Problem and Hausdorff Dimension

Year: 2025

Type: article

Abstract: We consider a compact pseudo-hermitian manifold (M,\theta, J), that is a manifold equipped with a contact form \theta and 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. These results go back to 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.

Source: Rutgers University Community Repository (Rutgers University)

DOI: 10.7282/00000493

Authors: Chanillo, Sagun, Yang, Paul C.

Institution: Rutgers, The State University of New Jersey

Language: English

Topic: Geometry and complex manifolds

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W6977856450

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