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Acta Mathematica Sinica,Chinese Series
2018, Vol.34 Num.8
Online: 2018-08-15

1195 Ke Feng LIU, Xiao Kui YANG
Minimal Complex Surfaces with Levi-Civita Ricci-flat Metrics
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 Kähler Calabi-Yau surfaces and Hopf surfaces.
2018 Vol. 34 (8): 1195-1207 [Abstract] ( 2 ) [HTML 1KB] [PDF 239KB] ( 18 )
1208 Min RU
A Cartan's Second Main Theorem Approach in Nevanlinna Theory
In 2002, in the paper entitled "A subspace theorem approach to integral points on curves", Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt's subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt's subspace in Nevanlinna theory is H. Cartan's Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan's original theorem. We call such method "a Cartan's Second Main Theorem approach". In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.
2018 Vol. 34 (8): 1208-1224 [Abstract] ( 5 ) [HTML 1KB] [PDF 274KB] ( 18 )
1225 Hendrik HERRMANN, Chin Yu HSIAO, Xiao Shan LI
An Explicit Formula for Szegö Kernels on the Heisenberg Group
In this paper, we give an explicit formula for the Szegö kernel for (0, q) forms on the Heisenberg group Hn+1.
2018 Vol. 34 (8): 1225-1247 [Abstract] ( 2 ) [HTML 1KB] [PDF 318KB] ( 4 )
1248 Song Ying LI, Duong Ngoc SON
The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces
Let (M, θ) be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ1 of the Kohn-Laplacian □b on (M, θ). In the present paper, we give a sharp upper bound for λ1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ1 when the pseudohermitian structure θ is volume-normalized.
2018 Vol. 34 (8): 1248-1258 [Abstract] ( 2 ) [HTML 1KB] [PDF 215KB] ( 3 )
1259 Bo YANG, Fang Yang ZHENG
On Compact Hermitian Manifolds with Flat Gauduchon Connections
Given a Hermitian manifold (Mn, g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call ▽s=(1-s/2)▽c + s/2 ▽b the s-Gauduchon connection of M, where ▽c and ▽b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s=0 (a flat Chern connection) or s=2 (a flat Bismut connection) are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either s ≥ 4 + 2√3 ≈ 7.46 or s ≤ 4-2√3 ≈ 0.54 and s ≠=0, then g is Kähler. We also show that, when n=2, g is always Kähler unless s=2. Therefore non-Kähler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.
2018 Vol. 34 (8): 1259-1268 [Abstract] ( 2 ) [HTML 1KB] [PDF 204KB] ( 7 )
1269 Siqi FU, Mei-Chi SHAW
Bounded Plurisubharmonic Exhaustion Functions and Levi-flat Hypersurfaces
In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Fornæss exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.
2018 Vol. 34 (8): 1269-1277 [Abstract] ( 3 ) [HTML 1KB] [PDF 205KB] ( 4 )
1278 Fu Sheng DENG, Hui Ping ZHANG, Xiang Yu ZHOU
Minimum Principle for Plurisubharmonic Functions and Related Topics
This is a survey about some recent developments of the minimum principle for plurisubharmonic functions and related topics.
2018 Vol. 34 (8): 1278-1288 [Abstract] ( 2 ) [HTML 1KB] [PDF 214KB] ( 7 )
1289 Xiang Yu ZHOU, Lang Feng ZHU
Optimal L2 Extension and Siu's Lemma
In this paper, we discuss our most recent results on the optimal L2 extension problem and Siu's lemma as applications of the strong openness property of multiplier ideal sheaves obtained by Guan and Zhou.
2018 Vol. 34 (8): 1289-1296 [Abstract] ( 2 ) [HTML 1KB] [PDF 209KB] ( 13 )
1297 Miroslav ENGLIŠ, Hao XU
Higher Laplace-Beltrami Operators on Bounded Symmetric Domains
It was conjectured by the first author and Peetre that the higher Laplace-Beltrami operators generate the whole ring of invariant operators on bounded symmetric domains. We give a proof of the conjecture for domains of rank ≤ 6 by using a graph manipulation of Kähler curvature tensor. We also compute higher order terms in the asymptotic expansions of the Bergman kernels and the Berezin transform on bounded symmetric domain.
2018 Vol. 34 (8): 1297-1312 [Abstract] ( 3 ) [HTML 1KB] [PDF 331KB] ( 8 )
1313 Der Chen CHANG, Shu Cheng CHANG, Ying Bo HAN, Chien LIN
On the CR Poincaré-Lelong Equation, Yamabe Steady Solitons and Structures of Complete Noncompact Sasakian Manifolds
In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR (2n + 1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.
2018 Vol. 34 (8): 1313-1344 [Abstract] ( 3 ) [HTML 1KB] [PDF 354KB] ( 16 )
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