In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Forn^ess exponent and its relations with existence of domains with Levi-flat b...In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Forn^ess exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.展开更多
In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems tha...In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C^3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).展开更多
This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli- Chern class on compact complex manifolds, and proved that the （1, 1） curvature form of the Levi-Civita connection represent...This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli- Chern class on compact complex manifolds, and proved that the （1, 1） curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi- Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are K/ihler Calabi-Yau surfaces and Hopf surfaces.展开更多
基金supported by NSF(Grant No.DMS 1500952)supported by NSF(Grant No.DMS 1700003)
文摘In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Forn^ess exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.
基金supported by National Science Foundation of USA(Grant No.NSF-1363418)
文摘In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C^3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).
基金supported in part by NSFC(Grant No.11531012),NSFC(Grant No.11688101)supported in part by China’s Recruitment Program
文摘This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli- Chern class on compact complex manifolds, and proved that the （1, 1） curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi- Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are K/ihler Calabi-Yau surfaces and Hopf surfaces.
