In this paper,we give a survey of our recent results on extension theorems on Kähler manifolds for holomorphic sections or cohomology classes of(pluri)canonical line bundles twisted with holomorphic line bundles ...In this paper,we give a survey of our recent results on extension theorems on Kähler manifolds for holomorphic sections or cohomology classes of(pluri)canonical line bundles twisted with holomorphic line bundles equipped with singular metrics,and also discuss their applications and the ideas contained in the proofs.展开更多
In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the ...In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.展开更多
The formula for the quantum amplitude of the Veneziano dual resonance model is shown to be formally analogous to the dimensionality of a K-theoretical fractal quotient manifold of the non-commutative geometrical type....The formula for the quantum amplitude of the Veneziano dual resonance model is shown to be formally analogous to the dimensionality of a K-theoretical fractal quotient manifold of the non-commutative geometrical type. Subsequently this analogy is used to deduce the ordinary energy of the quantum particle and the dark energy of the quantum wave. The results agree completely with cosmological measurements. Even more surprisingly the sum of both energy expressions turned out to be exactly equal to Einstein’s iconic formula E = mc2. Consequently Einstein’s formula makes no distinction between ordinary and dark energy.展开更多
Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below.Suppose that N is a str...Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below.Suppose that N is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant.In this paper,we establish a Schwarz lemma for holomorphic mappings f from M into N.As applications,we obtain a Liouville type rigidity result for holomorphic mappings f from M into N,as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold.展开更多
In this short note,we compare our previous work on the off-diagonal expansion of the Bergman kernel and the preprint of Lu-Shiffman(arXiv:1301.2166).In particular,we note that the vanishing of the coefficient of p−1/2...In this short note,we compare our previous work on the off-diagonal expansion of the Bergman kernel and the preprint of Lu-Shiffman(arXiv:1301.2166).In particular,we note that the vanishing of the coefficient of p−1/2 is implicitly contained in Dai-Liu-Ma’s work(J.Differ.Geom.72(1),1-41,2006)and was explicitly stated in our book(Holomorphic Morse inequalities and Bergman kernels.Progress in Math.,vol.254,2007).展开更多
The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacet...The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.展开更多
In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As...In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.展开更多
The definition of Schrdinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrdinger flow of maps into S 2, and that there exists a unique local smooth solution for the initial va...The definition of Schrdinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrdinger flow of maps into S 2, and that there exists a unique local smooth solution for the initial value problem of one dimensional Schrdinger flow of maps into Khler manifolds. In case the targets are Khler manifolds with constant curvature, it is proved that one dimensional Schrdinger flow admits a unique global smooth solution.展开更多
基金the National Natural Science Foundation of China(11688101 and 11431013)the National Natural Science Foundation of China(12022110,11201347 and 11671306).
文摘In this paper,we give a survey of our recent results on extension theorems on Kähler manifolds for holomorphic sections or cohomology classes of(pluri)canonical line bundles twisted with holomorphic line bundles equipped with singular metrics,and also discuss their applications and the ideas contained in the proofs.
基金supported by National Key R&D Program of China (Grant No. 2021YFA1003100)supported by National Natural Science Foundation of China (Grant Nos. 11825101, 11522101, and 11431013)+1 种基金supported by the Talent Fund of Beijing Jiaotong Universitysupported by China Postdoctoral Science Foundation (Grant Nos. BX20230402 and 2023M743719)。
文摘In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.
文摘The formula for the quantum amplitude of the Veneziano dual resonance model is shown to be formally analogous to the dimensionality of a K-theoretical fractal quotient manifold of the non-commutative geometrical type. Subsequently this analogy is used to deduce the ordinary energy of the quantum particle and the dark energy of the quantum wave. The results agree completely with cosmological measurements. Even more surprisingly the sum of both energy expressions turned out to be exactly equal to Einstein’s iconic formula E = mc2. Consequently Einstein’s formula makes no distinction between ordinary and dark energy.
基金supported by National Natural Science Foundation of China(Grant Nos.12071386,11671330 and 11971401)。
文摘Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below.Suppose that N is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant.In this paper,we establish a Schwarz lemma for holomorphic mappings f from M into N.As applications,we obtain a Liouville type rigidity result for holomorphic mappings f from M into N,as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold.
基金X.Ma partially supported by Institut Universitaire de France.G.Marinescu partially supported by DFG funded projects SFB/TR 12 and MA 2469/2-2.
文摘In this short note,we compare our previous work on the off-diagonal expansion of the Bergman kernel and the preprint of Lu-Shiffman(arXiv:1301.2166).In particular,we note that the vanishing of the coefficient of p−1/2 is implicitly contained in Dai-Liu-Ma’s work(J.Differ.Geom.72(1),1-41,2006)and was explicitly stated in our book(Holomorphic Morse inequalities and Bergman kernels.Progress in Math.,vol.254,2007).
基金partially supported by the NSFC grant (12071485)partially supported by the Beijing Natural Science Foundation (1202012,Z190003)+1 种基金the NSFC grant (11701031,12071035)partially supported by the NSFC grant (11688101)。
文摘The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.
基金supported by National Key R&D Program of China(Grant No.2021YFA1003100)supported by NSFC(Grant Nos.11825101,11522101 and 11431013)+1 种基金supported by the Talent Fund of Beijing Jiaotong Universitysupported by China Postdoctoral Science Foundation(Grant Nos.BX20230402 and 2023M743719)。
文摘In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.
文摘The definition of Schrdinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrdinger flow of maps into S 2, and that there exists a unique local smooth solution for the initial value problem of one dimensional Schrdinger flow of maps into Khler manifolds. In case the targets are Khler manifolds with constant curvature, it is proved that one dimensional Schrdinger flow admits a unique global smooth solution.