Generalized asymptotic expansions of Tian-Yau-Zelditch (Announcements)
This an announcement for the generalized asymptotic expansion of Tian-Yau-Zeldtich.
math.DG↗
arXiv subjects
Zhiqin Lu
Publications and source records attributed to Zhiqin Lu.
Proof of the normal scalar curvature conjecture
In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.
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On the DDVV Conjecture and the Comass in Calibrated Geometry (I)
In this paper, we proved a special case of the DDVV Conjecture.
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Recent developments of the DDVV Conjecture
In this paper, we give a survey of the recent develpoments of the DDVV conjecture.
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On the DDVV Conjecture and the Comass in Calibrated Geometry (II)
In this paper, we proved P(n,3), which is an important part of the DDVV conjecture. The general case will be treated in the next version of the paper.
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On the ground state of quantum layers
We provide some new results of the ground state of quantum layers.
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Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space
In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose second fundamental form tends to zero at infinity must have bound states.
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Quantum Layers over Surfaces Ruled Outside a Compact Set
In this paper, we proved the quantum layer over a surface which is ruled outside a compact set, asymptotically flat but not totally geodesic admits ground states.
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On the Geometry of Moduli Space of Polarized Calabi-Yau manifolds
In this paper, we study the Chern classes on the moduli space of polarized Calabi-Yau manifolds. We prove that the integrations of the invariants of the curvature of the Weil-Petersson metric are finite. In some special cases, they are even rational numbers.
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Analytic torsion for Calabi-Yau threefolds
After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa.
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Discrete spectrum of quantum tubes
This short paper is based on the talk of the second author given at the 2005 Hayama Symposium on Complex Analysis in Several Variables. The recent results of the authors on the spectrum of quantum tubes are summarized.
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On the Discrete Spectrum of Generalized Quantum Tubes
In this paper, we study the spectrum of quantum tubes. Under certain intrinsic assumptions of the asymptotically flat submanifold of the Euclidean space, we prove the existence of the ground state of the quantum tube. The work is a generalization of Duclos, Exner and Krejcirik (CMP, 223(1), 13-28, 2001) and ourselves(math.DG/0402252).
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Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds
In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric for Calabi-Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.
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On the Weil-Petersson volume and the first Chern Class of the moduli space of Calabi-Yau manifolds
In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.
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Finiteness of volume of moduli spaces
We give a ``physics proof'' of a conjecture made by the first author at Strings 2005, that the moduli spaces of certain conformal field theories are finite volume in the Zamolodchikov metric, using an RG flow argument.
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On the Weil-Petersson Geometry of Calabi-Yau Moduli
In this paper we proved that the Weil-Petersson volume of the Chern class of any order over the moduli space of Calabi-Yau manifolds is a rational number. We also found the necessary and sufficient condition of the incompleteness of Weil-Petersson metric in several variables case.
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On the Curvature Tensor of the Hodge Metric of Moduli Space of Polarized Calabi-Yau Threefolds
In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degeneration of Calabi-Yau threefolds is given.
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The log term of Szego Kernel
In this paper, we study the relations between the log term of the Szegö kernel of the unit circle bundle of the dual line bundle of an ample line bundle over a compact Kählermanifold. We proved a local rigidity theorem. The result is related to the classical Ramadanov Conjecture.
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