Akito Futaki
We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold constructed from a toric diagram with a constant height.
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Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.
On the Ricci curvature of Kahler-Ricci Flow
本文研究Kähler-Ricci流中Kodaira维数为一的流形的Ricci曲率收敛性，揭示其在局部拓扑下的收敛行为。
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.
In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $E^{n}$.
K{ä}hler Einstein manifolds and the Calabi curvature operator
本研究针对卡拉比流形上的卡拉比曲率算子，证明其满足不同正性条件时对应标量曲率、里奇曲率等的正性结论，还得出满足特定条件的紧卡拉比爱因斯坦流形必具有非负常全纯截面曲率的贡献。
By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.
For any complete noncompact K$\ddot{a}$hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.
We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.
Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with initial data $(M^3,g_0)$ has a long time solution. This extends a recent result of Ma and Zhu. We also have a higher dimensional version, and we reprove a K$\ddot{a}$hler analogy due to Chau, Tam and Yu.
One-sided type-D Ricci-flat multi-centre metrics
本文基于LeBrun–Tod ansatz的Tod ansatz简化形式构造单侧D型里奇平坦多中心度量，系统研究其结构并结合多孤子解方案得出相关孤子数结论，推测结论对一般中心数成立。
Twins in K{ä}hler and Sasaki geometry
本文提出加权极值 Kähler 孪生与极值 Sasaki 孪生概念，将 Hirzebruch 曲面上的孪生现象推广到加权极值度量，并重点研究环面 Sasaki 情形下 Sasaki 锥中多条极值射线的存在性。
We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang.
We show that an expanding gradient Ricci solitons which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.
In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other examples of Ricci-flat cones over 4-manifolds are unstable, and that Ricci-flat cones over products of Einstein manifolds and over Kahler-Einstein manifolds with h^(1,1)>1 are unstable in dimension less than 10. As results of independent interest, our computations indicate that the Page…
We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n,n+1).
We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expanders is connected.
We consider Ricci flow on a closed surface with cone points. The main result is: given a (nonsmooth) cone metric g_0 over a closed surface there is a smooth Ricci flow g(t) defined for (0,T], with curvature unbounded above, such that g(t) tends to g_0 as t tends to 0. This result means that Ricci flow provides a way for instantaneously smoothening cone points. We follow an argument of P. Topping modifying his reasoning for cusps of negative curvature; in that sense we can consider cusps as a limiting zero-angle cone, and we generalize to any angle between 0 and 2π.
We prove that a shrinking gradient Ricci soliton which is asymptotic to a Kähler cone along some end is itself Kähler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally Kähler.
Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 Φ$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.
Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
本文证明了 Kähler–Ricci 收缩子的标量曲率仅依赖维数的一致界，并建立了非紧收缩子无穷远处唯一切锥与其极化 Fano 纤维化基的同胚关系。
It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension 4 with non-Euclidean volume growth who has at least two distinct tangent cones at infinity and one of them has a smooth cross section.
Expanding Ricci solitons coming out of weakly PIC1 metric cones
本文研究从弱PIC1度量锥中涌现的完备里奇流，证明其必为扩张梯度里奇孤立子，并由此推出完备弱PIC1凯勒流形的无穷远度量锥典范双全纯等价于复欧氏空间。
We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone.
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity
研究构造了一个非Kähler扩张梯度Ricci流形的例子，其在无限远处具有Kähler切锥，展示了Kähler锥可通过光滑AC扩张梯度Ricci流形去奇异化，但无法通过光滑AC扩张梯度Kähler-Ricci流形实现。
Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle
本文证明当锥角趋于零时，锥形凯勒-里奇流收敛到唯一的凯勒-里奇流，该流在除子外光滑并沿除子具有尖点奇异性。
Ricci flows which terminate in cones
本文证明在完备流形上具有二次曲率衰减且终止于锥端的 Ricci 流必为梯度收缩孤子。
We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point and Gromov-Hausdorff converges to a two dimensional projective orbifold. This gives the first example of the conical Kähler-Ricci flow contracting the cone divisor to a single point. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing sing…
Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $β\in (0,1)$. We show that in any positive co-homology class on $X$ there is a Kähler metric with cone angle $2πβ$ along $D$ which has bounded Ricci curvature. We use this result together with the Aubin-Yau continuity method to give an alternative proof of a well-known existence theorem for Kahler-Einstein metrics with cone singularities.
A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic volume ratio of gradient Ricci solitons.
We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $γ$ is a limit geodesic in $X$ then along the interior of $γ$ same scale measure metric tangent cones $T_{γ(t)}X$ are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.
We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat Kähler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.
This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for $g$-weighted Kähler-Ricci solitons ($g$-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular $g$-soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sa…
In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's $λ$ and $ν$ functionals of cones and characterize their finiteness in terms of the $λ$-functional of the link. As an application, we characterize manifolds with conical singularities on which a $λ$-functional can be defined and get upper bounds on the $ν$-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman's pseudolocality theo…
We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0; n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n; n+1). More generally, for each N > 1 we prove that the condition CD(N-1;N) for a wei…
Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones
研究凯勒锥面分解上收缩凯勒 - 里奇孤子的唯一性，通过证明其渐近锥形，结合埃斯帕扎结果得出在双全纯映射拉回下至多有一个此类孤子，证实宋 - 张猜想特殊情况并讨论其他结果。
This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature con-ditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation:
Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the action is locally free on the cross section. This generalizes the metric cone theorem of Cheeger-Colding to the Kähler case. We also discuss some applications to complete Kähler manifolds with nonnegative bisectional curvature.
The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds, introduced by R. Hamilton in 1982. It is from the achievements made by G. Perelman with this technique in 2002 when the Ricci flow has been established in a discipline itself, generating a great interest in the community. This thesis contains four original results. First result…
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…
We show that the only metric measure space with the structure of an $N$-cone and with two-sided synthetic Ricci bounds is the Euclidean space ${\mathbb R}^{N+1}$ for $N$ integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymtotic of the heat kernel in the $L^2$-transport distance. Moreover, we establish a beautiful rigidity results of independent interest which characterize the $N$-dimensional standard sphere ${\mathbb S}^N$ as the unique minimizer of $$\int_X\int_X \cos d (x,y)\, m(d y)\,m(d x)$$ among all metri…
This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegat…
Nonnegative Ricci curvature, metric cones, and virtual abelianness
本文研究非负 Ricci 曲率开流形的基本群，在逃逸率非 1/2 且万有覆盖渐近为度量锥时证明其虚拟阿贝尔性，并在欧氏体积增长常数至少为 L 时给出指数界 C(n,L)。
Examples of tangent cones of non-collapsed Ricci limit spaces
本研究拓展了Colding-Naber关于非坍缩Ricci极限空间切锥同胚不唯一的结果，构造了5维及以上极限空间，使任意容许核心度量的有限流形族都可作为同一点切锥的截面。
We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and for any negative line bundle $L$ over a compact Kähler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_{1}(K…
We describe a framework for constructing the Ricci-flat metrics on the total space of the canonical bundle over $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ (the del Pezzo surface of rank one). We construct explicitly the first-order deformation of the so-called `orthotoric metric' on this manifold. We also show that the deformation of the corresponding conformal Killing-Yano form does not exist.
Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.
We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $π_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.
On an invariant curvature cone along 4-dimensional Ricci flow
本文研究满足曲率锥约束的四维非紧致完备流形，证明了拓扑与几何间隙定理，并研究了具有曲率下界流形的正则性结果。
Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class. We prove the sharp rate of convergence of the metric to the cone metric. For compact Kähler classes this is the same as for the Ricci-flat ALE metrics of P. Kronheimer and D. Joyce.
We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.
From Kähler Ricci solitons to Calabi-Yau Kähler cones
研究带有Kähler Ricci soliton的光滑Fano流形，通过动量多面体上的权函数方法，证明其与高维复射影空间乘积的规范锥是Calabi-Yau锥，并探讨了相关分支应用。
The author has proved that a crepant resolution Y of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,\R). These manifolds are generalizations of the Ricci-flat ALE Kähler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric Kähler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. …
Limits of manifolds with a Kato bound on the Ricci curvature
该研究针对Ricci曲率满足一致Kato界的黎曼流形序列的Gromov-Hausdorff极限，证明了能量收敛、切锥的RCD条件与度量锥性质、分层存在性及体积收敛到Hausdorff测度，推进了带曲率积分界流形的极限理论。
We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold $(M,g)$ which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold $(M^n,g)$ is of order $n$ and thereby close a small gap in a paper by Cheeger and Tian.
In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the time derivatives of the conformal factor are bounded and for each fixed time, the conformal factor has an explicit asymptotic expansion near the cone points.
A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone
本文证明了对任意在 $\mathbb{S}^3$ 上自由作用的有限子群 $Γ$，存在具有非负 Ricci 曲率且渐近锥为 $C(\mathbb{S}_δ^3 /Γ)$ 的 4 维完备 Riemannian 流形，从而回答了 4 维非塌缩切锥的拓扑障碍问题并实现了其拓扑分类。
Kähler-Ricci Tangent Flows are Infinitesimally Algebraic
本研究证明奇异收缩Kähler-Ricci孤立子的切锥为正规仿射代数簇且两类正则集一致，同时给出Hörmander的$L^2$估计的抛物型证明，可用于求解相关孤立子上的$\bar{\theta}$方程。
Given closed Riemannian manifold $(M^n, g)$ of positive Ricci curvature $Ricci(g) \geq (n-1)g$ we study isoperimetric regions on the spherical cone over $M$. When $g$ is Einstein we use this to compute the Yamabe constant of $(M \times {\bf R}, g + dt^2)$ and so to obtain lower bounds for the Yamabe invariant of $M\times S^1$.
Consider a limit space $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α^n$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of $Y$ at a point $p\in Y$ are known to be metric cones $C(X)$, however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the closed subset of compact metric spaces $X$ which arise as cross sections for the tangents cones of $Y$ at $p$. In this paper we study the properties of $\barΩ_{Y,p}$. In particular, we give necessary and sufficient conditions for an open smooth family $Ω\equiv (X_s,g_s)$ of closed man…
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