Jeffrey Streets
We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on $T^k$ bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.
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Dynamical stability of Pluriclosed and Generalized Ricci solitons
本研究探讨 pluriclosed 流与广义 Ricci 流的稳定性，证明了广义爱因斯坦 - 希尔伯特泛函二阶变分非正且无穷小形变可积时流动力学稳定，以及第一陈类消失时 pluriclosed 稳态孤子动力学稳定的结论。
Long-time existence of the pluriclosed flow on some fibrations
本文证明在非不变初始数据下，某些紧致李群商空间以及非正曲率Kähler流形上的全纯主环面丛上，多闭流的长时存在性，并给出复曲面上该流长时存在性的新证明。
Derivative estimates of pluriclosed flow
本文为pluriclosed流提供导数估计，控制高阶导数，并在二维及Hermitian-symplectic情形下导出挠率估计和单调量，证明所有Hermitian-symplectic孤子为Kähler Ricci孤子。
We show pluriclosed flow preserves the Hermitian-symplectic structures. And we observe that it can actually become a flow of Hermitian-symplectic forms when an extra evolution equation determined by the Bismut-Ricci form is considered. Moreover, we get a topological obstruction to the long-time existence in arbitrary dimension.
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex man…
Chern-Ricci flow and t-Gauduchon Ricci-flat condition
本文研究Chern-Ricci流下的t-Gauduchon Ricci平坦条件，提供了不保持该条件的例子、非平凡新解及几何流的Gromov-Hausdorff极限行为，明确描述了特定主丛上pluriclosed流的极限。
Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not ad…
Complete pluriclosed metrics with vanishing Bismut Ricci form on the affine quadric
研究仿射二次曲面上SO(4)-不变且Bismut Ricci形式消失的pluriclosed度量族，证明其局部与全局存在性、完备性及退化条件，并给出曲率公式与和乐群性质。
Shrinking Kähler-Ricci solitons on toric fano fibrations
本文证明了光滑环面Fano纤维化上存在完备的梯度收缩Kähler-Ricci孤立子，这是该领域的重要存在性结果。
Pluriclosed flow on Oeljeklaus-Toma manifolds
本文研究了 pluriclosed 流在 Oeljeklaus-Toma 流形上的全局存在性及收敛性，同时探讨了广义 Kähler-Ricci 流的先验估计。
Pluriclosed 3-folds with vanishing Bismut Ricci form: General theory in the quasi-regular case
研究非Kähler BHE 3 折的通用理论，通过维度减少到特殊Kähler几何，揭示其与cscK度量的相似性及构造方法。
Pluriclosed flow and the Hull-Strominger system
本文提出Pluriclosed流的自然扩展以构造Hull-Strominger系统的解，给出多种几何表述并证明先验估计，推广Yau的C^3估计，解决一致椭圆解的光滑性，并证明特殊背景上的全局存在与收敛。
Homogeneous Generalized Ricci flows II
该研究关联李群左不变广义度量的广义里奇曲率与度量李代数的里奇曲率，证明相关流的子收敛性，建立标准 Bismut 平坦度量的动力学稳定性并构造非标准不稳定例子。
A note on the pluriclosed flow on balanced manifolds with $c_1=0$
本文研究 $c_1=0$ 的紧致平衡流形上 pluriclosed 流长时间解的存在性，并证明在齐性背景下该流光滑收敛到 Kähler 度量，为 Fino-Vezzoni 猜想提供新证据。
On the shrinking solitons of generalized Ricci flow
本文研究广义 Ricci 流的收缩孤子，证明紧流形上的梯度收缩孤子必为 Ricci 孤子，并在宽泛上同调条件下刻画 pluriclosed 孤子，同时构造首个非平凡例子作为奇点模型。
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long time solution to the pluriclosed flow in 2-step nilmanifolds.
We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifolds by the volume growth. As an application, we give the uniqueness of tangent cones of collapsing manifolds with some conditions.
In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability for isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973), to manifolds with local bounded Ricci covering geometry. Our two generalized results have been used in a recent work of Xiaochun Rong in generalizing Gromov's almost flat manifolds theorem to maximally collapsed manifolds with local bounded Ricci covering geometry.
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics' in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricc…
Kähler-Ricci shrinkers and Fano fibrations
本文用双有理代数几何证明Kähler-Ricci收缩孤立子是拟射影簇，并提出将其存在性与极化Fano纤维化K-稳定性及奇点切流相联系的统一猜想。
Cohomogeneity one Ricci Solitons from Hopf Fibrations
本文针对主轨道迷向表示含两个不可约子和的情形，证明非平凡丛上存在非相似完备稳态与膨胀余齐次性一 Ricci 孤立子，并用于重构 Böhm 的正标量曲率 Einstein 度量。
We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. …
Singular Kähler--Ricci shrinkers and polarized Fano fibrations
本文证明任何作为 Kähler-Ricci 流非塌缩极限的奇异 Kähler-Ricci 收缩子具有局部代数复解析簇结构，并推导出单连通、唯一端、唯一切触锥等几何性质。
Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration
本文研究了基于四元数霍普夫纤维的非收缩里齐流形，发现两类新的流形家族并探讨其几何性质。
Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on non-standard Hopf surfaces, completing the existence question for such structures. Moreover this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex su…
In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)<n. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kahler metric along the normalized Kahler-Ricci flow converge to -m outside the singular set of this fibration.
Let $X$ be a Kähler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $ω$ be a fixed Kähler form on $X$. By Yau's theorem, there exists a unique Ricci-flat Kähler form $ρ\vert_{X_y}$ for each fiber, which is cohomologous to $ω\vert_{X_y}$. This family of Ricci-flat Kähler forms $ρ\vert_{X_y}$ induces a smooth $(1,1)$-form $ρ$ on $X$ with a normalization condition. In this paper, we prove that the direct image of $ρ^{n+1}$ is positive on the base $Y$. We also discuss several byproducts, among them the local triviality of families of Calabi…
The homogeneous generalized Ricci flow
本文基于括号流框架研究Lie群离散商上的广义Ricci流，证明了任意维solvmanifolds上的整体存在性，定义了exact Courant algebroids上的广义Ricci孤立子并分析了其在nilmanifolds上的极限行为，同时给出了不变广义度量的曲率新公式。
We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a Weil-Petersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimens…
Compactness and Rigidity of Complete Kähler Ricci Shrinkers
本文借助极化法诺纤维化结构，探究完备凯勒里奇收缩子的紧性、刚性及相关几何性质。
We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when the regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.
Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow
本文研究了凯勒-里茨流中Fano纤维的几何结构，推导了直径界和标量曲率估计，并给出了复蒙日-安培流势的精确估计。
Fibrations, the First Betti Number, and Almost Nonnegative Ricci Curvature
本文在几乎非负Ricci曲率和额外正则性假设下，证明了流形纤维化为b1-环面的定理，去除了Yamaguchi结果中所需的截面曲率上界，并推广到非光滑RCD空间。
Collapsing immortal Kähler-Ricci flows
本文研究带半丰沛典范丛的紧Kähler流形上的Kähler-Ricci流，通过演化度量的渐近展开证明其坍缩为饭高纤维化底的典范度量，验证了Song-Tian的两个猜想。
Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kahler manifold. Differing from Fateev's examples most of our examples are non-collapsed. The construction of this paper, different from the ad hoc anastz of Fateev, is systematic, generalizing (as well as unifying) the previous constructions of Einstein metrics by Bourguignon-Karcher, Jensen, and Ziller in…
On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.
We extend Cheeger-Gromov's and Anderson's convergence theorems to regular limit spaces of manifolds with bounded Ricci curvature and local covering geometry, by establishing the $C^{1,α}$-regularities that are the best one may expect on those Ricci-limit spaces. As an application we prove an optimal generalization of Fukaya's fibration theorem on collapsed manifolds with bounded Ricci curvature, which also improves the original version to $C^{1,α}$ limit spaces.
Shrinking gradient Kähler-Ricci solitons are simply-connected
本文通过证明光滑极化Fano纤维化不存在非平凡有限覆盖，并结合Sun-Zhang与Wylie的已有结果，得出收缩凯勒-里奇孤立子必为单连通的结论。
Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers
研究非塌缩凯勒 - 里奇流有限时间奇点及里奇收缩子无穷远处几何，证明凯勒 - 里奇流时间切片收敛等结论，还将其与法诺纤维化联系，应用于证明特定收缩子无穷远处切空间唯一性。
We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with …
K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers
本文证明了渐近锥形Kähler-里奇收缩者度量的存在性与K-极稳定性之间的等价关系。
We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov-Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a fam…
In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics, that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges locally smoothly away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fib…
We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first Chern class.
In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We show the existence of new invariant Einstein metrics on homogeneous bisymmetric fibrations…
Almost Hermitian geometry and generalized Ricci flow
本文研究全斜对称Nijenhuis张量且Bismut挠率闭合的殆埃尔米特流形（almost-SKT），引入超定椭圆性条件保证受约束Bismut–Ricci流短时存在，并在六维紧流形上构造非可积例子，揭示其与广义Ricci流的联系。
In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge-Ampère equation, which is called a Monge-Ampère fibration. There exist two canonical types of generalized Weil-Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil-Petersson metric, we obtain an explicit curvature formula and prove that the holomorphic bisectional curvature is non-positive, the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature are all bounded from above by a negative constant. For a holomorphic vector bundle o…
J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost Kähler manifolds generalising Kähler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-Kähler structures on several nilmanifolds and on twistor fibrations over hyperbolic space studied by J. Fine and D. Panov. The latter lead to examples of non-Kähler static solutions of symplectic curvature flow which can be seen as analogues of Kähler-Einstein manifolds in Kähler-Ricci flow.
We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of generalized Kahler-Einstein metrics as the limits of the Kahler-Ricci flow for some holomorphic fibrations (in the spirit of Song and Tian "The Kahler-Ricci flow on surfaces of positive Kodaira dimension", arXiv:math/0602150).
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
Let X be a complex manifold fibered over the base S and let L be a relatively ample line bundle over X. We define relative Kahler-Ricci flows on the space of all Hermitian metrics on L with relatively positive curvature. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds and the case when L is the relative (anti-) canonical line bundle. The main theme studied is whether posivity in families is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibres. The quantization of this setting is als…
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibration associated with the large complex structure limit, namely an n-sphere, …
Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits
本文研究紧凯勒流形上有限时间非坍缩凯勒-里奇流的热核估计，构造其格罗莫夫-豪斯多夫极限，证明固定基点I型爆破的子序列度量流极限为奇异收缩凯勒-里奇孤子空间，移动基点的极限度量流在熵挤压条件下为度量孤子。
The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers
该研究针对满足Ricci曲率下界条件且局部万有覆盖具有Reifenberg控制的完备黎曼流形，证明了Ricci流的存在性与唯一性，揭示了坍缩闭流形的奇异幂零纤维化性质，明确了Reifenberg假设的必要性。
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