In this paper,we show the relation between the existence of twisted conical K?hler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds.This is based on our proofs of some openness theor...In this paper,we show the relation between the existence of twisted conical K?hler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds.This is based on our proofs of some openness theorems on the existence of twisted conical Kahler-Ricci solitons,which generalize Donaldson’s existence conjecture and the openness theorem of the conical K?hler-Einstein metrics to the conical soliton case.展开更多
In this paper,we study the uniformly strong convergence of the Kahler-Ricci flow on a Fano manifold with varied initial metrics and smoothly deformed complex structures.As an application,we prove the uniqueness of Kah...In this paper,we study the uniformly strong convergence of the Kahler-Ricci flow on a Fano manifold with varied initial metrics and smoothly deformed complex structures.As an application,we prove the uniqueness of Kahler-Ricci solitons in the sense of diffeomorphism orbits.The result generalizes Tian-Zhu’s theorem for the uniqueness of of Kahler-Ricci solitons on a compact complex manifold,and it is also a generalization of Chen-Sun’s result of the uniqueness of Kahler-Einstein metric orbits.展开更多
This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow ...This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.展开更多
We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kahler-Ricci flow on a minimal elliptic Kahler surface converges in the sense of currents to a generalized conica...We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kahler-Ricci flow on a minimal elliptic Kahler surface converges in the sense of currents to a generalized conical Kahler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.展开更多
This paper presents the complex dynamics synthesis of the combat dy-namics series called tensor-centric warfare (TCW;for the first three parts of the series, see [1] [2] [3]), which includes tensor generalization of c...This paper presents the complex dynamics synthesis of the combat dy-namics series called tensor-centric warfare (TCW;for the first three parts of the series, see [1] [2] [3]), which includes tensor generalization of classical Lanchester-type combat equations, entropic Lie-dragging and commutators for modeling warfare uncertainty and symmetry, and various delta-strikes and missiles (both deterministic and random). The present paper gives a unique synthesis of the Red vs. Blue vectorfields into a single complex battle-vectorfield, using dynamics on Kähler manifolds as a rigorous framework for extending the TCW concept. The global Kähler dynamics framework, with its rigorous underpinning called the Kähler-Ricci flow, provides not only a new insight into the “geometry of warfare”, but also into the “physics of warfare”, in terms of Lagrangian and Hamiltonian structures of the battlefields. It also provides a convenient and efficient computational framework for entropic wargaming.展开更多
In this paper, we explicitly construct some rotationally symmetric gradient pseudo- Kahler-Ricci solitons which depend on some parameters, on some line bundles and other bundles over projective spaces. We also discuss...In this paper, we explicitly construct some rotationally symmetric gradient pseudo- Kahler-Ricci solitons which depend on some parameters, on some line bundles and other bundles over projective spaces. We also discuss the ＂phase change＂ phenomenon caused by the variation of parameters.展开更多
In this article, we study the steady, shrinking, and expanding Kahler-Ricci solitons with vanishing Bochner-Weyl tensor and prove that, under this condition, the Ricci solitons must have constant holomorphic sectional...In this article, we study the steady, shrinking, and expanding Kahler-Ricci solitons with vanishing Bochner-Weyl tensor and prove that, under this condition, the Ricci solitons must have constant holomorphic sectional curvature.展开更多
In this note, we present a connection between equivariant Bott–Chern classesand K?hler–Ricci solitons. We also propose a generalized version the of the K–energy.
The authors show that the 2-non-negative traceless bisectional curvature is preserved along the Kahler-Ricci flow. The positivity of Ricci curvature is also preserved along the Kahler-Ricci flow with 2-non-negative tr...The authors show that the 2-non-negative traceless bisectional curvature is preserved along the Kahler-Ricci flow. The positivity of Ricci curvature is also preserved along the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature. As a corol- lary, the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature will converge to a Kahler-Ricci soliton in the sense of Cheeger-Cromov-Hausdorff topology if complex dimension n ≥ 3.展开更多
We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Qm = SOm＋2/SOmSO2. It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes -princ...We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Qm = SOm＋2/SOmSO2. It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes -principal or -isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in Qm = SOm＋2/SOmSO2 with commuting Ricci tensor.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12001532)supported by the Special Priority Program SPP 2026“Geometry at Infinity”from the German Research Foundation(DFG)。
文摘In this paper,we show the relation between the existence of twisted conical K?hler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds.This is based on our proofs of some openness theorems on the existence of twisted conical Kahler-Ricci solitons,which generalize Donaldson’s existence conjecture and the openness theorem of the conical K?hler-Einstein metrics to the conical soliton case.
基金supported by National Natural Science Foundation of China(Grant No.11971423)the Fundamental Research Funds for the Central Universities+2 种基金supported by National Natural Science Foundation of China(Grant No.11771019)Beijing Science Foundation(Grant No.Z180004)National Key R&D Program of China(Grant No.SQ2020YFA070059).
文摘In this paper,we study the uniformly strong convergence of the Kahler-Ricci flow on a Fano manifold with varied initial metrics and smoothly deformed complex structures.As an application,we prove the uniqueness of Kahler-Ricci solitons in the sense of diffeomorphism orbits.The result generalizes Tian-Zhu’s theorem for the uniqueness of of Kahler-Ricci solitons on a compact complex manifold,and it is also a generalization of Chen-Sun’s result of the uniqueness of Kahler-Einstein metric orbits.
基金Partially supported by NSF grants and a Simons fund.
文摘This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.
基金supported by the Science and Technology Development Fund(Macao S.A.R.),Grant FDCT/016/2013/A1the Project MYRG2015-00235-FST of the University of Macao
文摘We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kahler-Ricci flow on a minimal elliptic Kahler surface converges in the sense of currents to a generalized conical Kahler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.
文摘This paper presents the complex dynamics synthesis of the combat dy-namics series called tensor-centric warfare (TCW;for the first three parts of the series, see [1] [2] [3]), which includes tensor generalization of classical Lanchester-type combat equations, entropic Lie-dragging and commutators for modeling warfare uncertainty and symmetry, and various delta-strikes and missiles (both deterministic and random). The present paper gives a unique synthesis of the Red vs. Blue vectorfields into a single complex battle-vectorfield, using dynamics on Kähler manifolds as a rigorous framework for extending the TCW concept. The global Kähler dynamics framework, with its rigorous underpinning called the Kähler-Ricci flow, provides not only a new insight into the “geometry of warfare”, but also into the “physics of warfare”, in terms of Lagrangian and Hamiltonian structures of the battlefields. It also provides a convenient and efficient computational framework for entropic wargaming.
基金supported by the Natural Science Foundation of Fujian Province(2013J01027)
文摘In this paper, we explicitly construct some rotationally symmetric gradient pseudo- Kahler-Ricci solitons which depend on some parameters, on some line bundles and other bundles over projective spaces. We also discuss the ＂phase change＂ phenomenon caused by the variation of parameters.
基金supported by the National Natural Science Foundation of China under the grant numbers 11126073the Fundamental Research Funds for the Central Universities of SCUT under the grant number 2012ZB0017
文摘In this article, we study the steady, shrinking, and expanding Kahler-Ricci solitons with vanishing Bochner-Weyl tensor and prove that, under this condition, the Ricci solitons must have constant holomorphic sectional curvature.
基金Supported partially by NSF grants and a Simons fund
文摘In this note, we present a connection between equivariant Bott–Chern classesand K?hler–Ricci solitons. We also propose a generalized version the of the K–energy.
基金the National Science Foundation (No. DMS-0406346)
文摘The authors show that the 2-non-negative traceless bisectional curvature is preserved along the Kahler-Ricci flow. The positivity of Ricci curvature is also preserved along the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature. As a corol- lary, the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature will converge to a Kahler-Ricci soliton in the sense of Cheeger-Cromov-Hausdorff topology if complex dimension n ≥ 3.
基金supported by National Research Foundation of Korea (Grant No. NRF2015-R1A2A1A-01002459)
文摘We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Qm = SOm＋2/SOmSO2. It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes -principal or -isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in Qm = SOm＋2/SOmSO2 with commuting Ricci tensor.
