Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3

Chin-Yu Hsiao Institute of Mathematics, Academia Sinica, Taiwan Po Lam Yung Department of Mathematics, the Chinese University of Hong Kong

Complex Variables and Complex Analysis mathscidoc:1803.08004

Adv. Math. , 281, 734-822, 2015
Let $(\hat X,T^{1,0}\hat X)$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0}\hat X$ is a CR structure on $\hat X$. Fix a point $p\in\hat X$ and take a global contact form $\hat\theta$ so that $\hat\theta$ is asymptotically flat near $p$. Then $(\hat X,T^{1,0}\hat X,\hat\theta )$ is a pseudohermitian $3$-manifold. Let $G_p\in C^\infty(\hat X\setminus\set{p})$, $G_p > 0$, with $G_p(x)\sim\vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat X$. Consider the new pseudohermitian $3$-manifold with a blow-up of contact form $(\hat X\setminus\set{p},T^{1,0}\hat X,G^2_p\hat\theta)$ and let $\Box_{b}$ denote the corresponding Kohn Laplacian on $\hat X\setminus\set{p}$. In this paper, we prove that the weighted Kohn Laplacian $G^2_p\Box_b$ has closed range in $L^2$ with respect to the weighted volume form $G^2_p\hat\theta\wedge d\hat\theta$, and that the associated partial inverse and the Szeg\"{o} projection enjoy some regularity properties near $p$. As an application, we prove the existence of some special functions in the kernel of $\Box_{b}$ that grow at a specific rate at $p$. The existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3-dimensional CR geometry by Cheng-Malchiodi-Yang \cite{CMY}.
Kohn Laplacian, CR manifolds, CR positive mass theorem
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@inproceedings{chin-yu2015solving,
  title={Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3},
  author={Chin-Yu Hsiao, and Po Lam Yung},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180329233815707638997},
  booktitle={ Adv. Math. },
  volume={281},
  pages={734-822},
  year={2015},
}
Chin-Yu Hsiao, and Po Lam Yung. Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3. 2015. Vol. 281. In Adv. Math. . pp.734-822. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180329233815707638997.
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