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
When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em&...When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-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
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics.We prove that a compact locally conformal Kähler manifold with the constant nonpositive holomorphi...We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics.We prove that a compact locally conformal Kähler manifold with the constant nonpositive holomorphic sectional curvature is K?hler.We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature,and complete non-Kähler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensors.展开更多
We utilize homology and co-homology of a K3-Kähler manifold as a model for spacetime to derive the cosmic energy density of our universe and subdivide it into its three fundamental constituents, namely: 1) or...We utilize homology and co-homology of a K3-Kähler manifold as a model for spacetime to derive the cosmic energy density of our universe and subdivide it into its three fundamental constituents, namely: 1) ordinary energy;2) pure dark energy and 3) dark matter. In addition, the fundamental coupling of dark matter to pure dark energy is analyzed in detail for the first time. Finally, the so-obtained results are shown to be in astounding agreement with all previous theoretical analysis as well as with actual accurate cosmic measurements.展开更多
Let(M,g)be a Kähler surface andΣbe aβ-symplectic critical surface in M.If L_(q)(Σ)is bounded for some q>3,then we give a uniform upper bound for the Kähler angle onΣ.This bound only depends on M,q,βa...Let(M,g)be a Kähler surface andΣbe aβ-symplectic critical surface in M.If L_(q)(Σ)is bounded for some q>3,then we give a uniform upper bound for the Kähler angle onΣ.This bound only depends on M,q,βand the Lq functional ofΣ.For q>4,this estimate is known and we extend the scope of q.展开更多
Let M be a complete Kähler surface andbe a symplectic surface which is smoothly immersed in M.Letαbe the Kähler angle ofin M.In the previous paper Han and Li(JEMS 12:505–527,2010)2010,we study the symple...Let M be a complete Kähler surface andbe a symplectic surface which is smoothly immersed in M.Letαbe the Kähler angle ofin M.In the previous paper Han and Li(JEMS 12:505–527,2010)2010,we study the symplectic critical surfaces,which are critical points of the functional L=1 cosαdμin the class of symplectic surfaces.In this paper,we calculate the second variation of the functional L and derive some consequences.In particular,we show that,if the scalar curvature of M is positive,is a stable symplectic critical surface with cosα≥δ>0,whose normal bundle admits a holomorphic section X∈L2(),thenis holomorphic.We construct symplectic critical surfaces in C2.We also prove a Liouville theorem for symplectic critical surfaces in C2.展开更多
In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form assoc...In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form associated with the r-Jacobi algebra A.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
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
Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariant...Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariants of their equivariant normal R-test configurations in terms of the combinatory data.Based onHan and Li“Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties”,we compute the semistable limit of aK-unstable FanoG-compactification.As an application,we show that for the two smooth K-unstable Fano SO4(C)-compactifications,the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.展开更多
Let E be a holomophic vector bundle over a compact Astheno-Kähler manifold(M,ω).The authors would prove that E is a numerically flat vector bundle if E is pseudoeffective and the first Chern class c_(1)^(BC)(E)i...Let E be a holomophic vector bundle over a compact Astheno-Kähler manifold(M,ω).The authors would prove that E is a numerically flat vector bundle if E is pseudoeffective and the first Chern class c_(1)^(BC)(E)is zero.展开更多
In this short note,we give a proof of our partial C^0-estimate for Kähler-Einstein metrics.Our proof uses a compactness theorem of Cheeger-Colding-Tian and L^2-estimate for∂^--operator.
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When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-
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 Natural Science Foundation of China(Grant No.11801516)Zhejiang Provincial Natural Science Foundation(Grant No.LY19A010017)。
文摘We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics.We prove that a compact locally conformal Kähler manifold with the constant nonpositive holomorphic sectional curvature is K?hler.We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature,and complete non-Kähler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensors.
文摘We utilize homology and co-homology of a K3-Kähler manifold as a model for spacetime to derive the cosmic energy density of our universe and subdivide it into its three fundamental constituents, namely: 1) ordinary energy;2) pure dark energy and 3) dark matter. In addition, the fundamental coupling of dark matter to pure dark energy is analyzed in detail for the first time. Finally, the so-obtained results are shown to be in astounding agreement with all previous theoretical analysis as well as with actual accurate cosmic measurements.
基金supported by the National Natural Science Foundation of China(Grant No.11871436).
文摘Let(M,g)be a Kähler surface andΣbe aβ-symplectic critical surface in M.If L_(q)(Σ)is bounded for some q>3,then we give a uniform upper bound for the Kähler angle onΣ.This bound only depends on M,q,βand the Lq functional ofΣ.For q>4,this estimate is known and we extend the scope of q.
基金The research was supported the National Natural Science Foundation of China,No.11131007,No.11471014The research was also supported by the Doctoral Programme Foundation of Institution of Higher Education of China,No.20110002110064.
文摘Let M be a complete Kähler surface andbe a symplectic surface which is smoothly immersed in M.Letαbe the Kähler angle ofin M.In the previous paper Han and Li(JEMS 12:505–527,2010)2010,we study the symplectic critical surfaces,which are critical points of the functional L=1 cosαdμin the class of symplectic surfaces.In this paper,we calculate the second variation of the functional L and derive some consequences.In particular,we show that,if the scalar curvature of M is positive,is a stable symplectic critical surface with cosα≥δ>0,whose normal bundle admits a holomorphic section X∈L2(),thenis holomorphic.We construct symplectic critical surfaces in C2.We also prove a Liouville theorem for symplectic critical surfaces in C2.
文摘In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form associated with the r-Jacobi algebra A.
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.
文摘Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariants of their equivariant normal R-test configurations in terms of the combinatory data.Based onHan and Li“Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties”,we compute the semistable limit of aK-unstable FanoG-compactification.As an application,we show that for the two smooth K-unstable Fano SO4(C)-compactifications,the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.
基金supported by the National key R&D Program of China(No.2020YFA0713100)the National Natural Science Foundation of China(No.12141104)the Jiangsu Funding Program for Excellent Postdoctoral Talent(No.2023ZB491).
文摘Let E be a holomophic vector bundle over a compact Astheno-Kähler manifold(M,ω).The authors would prove that E is a numerically flat vector bundle if E is pseudoeffective and the first Chern class c_(1)^(BC)(E)is zero.
文摘In this short note,we give a proof of our partial C^0-estimate for Kähler-Einstein metrics.Our proof uses a compactness theorem of Cheeger-Colding-Tian and L^2-estimate for∂^--operator.
