Dan Knopf
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
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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。
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.
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.
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.
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 establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.
We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, K{ä}hler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curv…
Refined behavior description of the normalized Ricci flow on homogeneous spaces
本文研究规范化Ricci流下齐性空间Ricci曲率正性的保持问题，发现无限多个广义Wallach空间能保持或丧失该正性，并改进了参数重合情形的先前结果。
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.
In this note we show that, under certain curvature positivity conditions (the weak $\operatorname{PIC}-2$ condition or the nonnegative bisectional curvature condition), a complete and noncompact expanding breather of the Ricci flow is also an expanding gradient Ricci soliton. This is the first no expanding breather theorem in the noncompact category.
We show that closed manifolds supporting a nonpositively curved metric with negative $([\frac{n}{4}]+1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov.
We show how to associate with each graph with a certain property (positivity) a family of simply connected solvable Lie groups endowed with left-invariant Riemannian metrics that are Ricci solitons (called solsolitons). We classify them up to isometry, obtaining families depending on many parameters of explicit examples of Ricci solitons. A classification of graphs with up to 3 coherent components according to positivity is also given.
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…
Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.
In this note we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.
The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{g}) plays an important role in both geometric measure theory and the study of Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and with bounded Ricci curvature we show the underlying space has finite f-volume. As a consequence such manifolds, including shrinking Ricci solitons, have finite fundamental group. The analysis can be extended to classify shrinking solitons under convexity or concavity assumptions on the measure function.
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.
We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: The half-$PIC$ condition. It is a slight weakening of the positive isotropic curvature ($PIC$) condition introduced by M. Micallef and J. Moore. We observe that the half-$PIC$ condition is preserved by the Ricci flow and satisfies a maximality property among all Ricci flow invariant positivity conditions on the curvature of oriented 4-manifolds. We also study some geometric and topological aspects of half-$PIC$ manifolds.
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduced $hh$-curvature on finite time. Next, it is shown that the evolution of Ricci scalar is a parabolic-type equation and if the initial Finsler metric is of positive flag curvature, then the flag curvature and the Ricci scalar remain positive as long as the solution exists. Finall…
We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform $C^1$ bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler-Ricci flow to the Sasakian setting. We also show that the positivity of transverse of holomorphic bisectional curvature is preserved along the flow, using the methods and the results proved by Bando and Mok in Kahler setting. In particular, we show that the Sasaki-Ricci flow wou…
Transverse Rigidity of Shrinking Sasaki-Ricci Solitons
本文研究了Sasaki-Ricci溶iton作为Sasaki-Ricci流奇点模型的性质，建立了基本方程并证明了标量曲率正性，提出两个横向刚性判据，指出常标量曲率低维溶iton必为Sasaki-Einstein，且调和Weyl张量溶iton为球面有限商。
We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that every such manifold has first Betti number $b_1 =1$ and Hodge numbers $h^{p,0} =0$ for $p>0$, $h^{0,1} =1$, $h^{0,q} =0$ for $q>1$.
RC-positivity of complex surfaces
本文证明非平坦Ricci-flat Kähler复曲面的全纯切丛具有RC正Hermite度量，并揭示RC正性在张量运算下不保持，且弱于一致RC正性，不蕴含有理连通性。
Complex Monge-Amp{è}re equations on Hermitian Manifolds: From bounded to smooth solutions
研究紧致Hermitian流形上复Monge-Ampère方程有界弱解的正则性，通过支配原理和先验估计证明有界解在数据正则点集光滑，并恢复已有结果。
Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow
本文研究带孤立锥形奇点的空间上的Ricci de Turck流，证明其保持Ricci曲率初始正则性与附加假设下的数量曲率正性，为奇异Ricci流的曲率算子正性研究奠定基础。
In this paper we construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a gluing construction for Ricci-positive manifolds with corners generalizing a gluing construction of Perelman for Ricci-positive manifolds with boundaries. Our construction gives a sufficient condition for gluing together two Ricci-positive manifolds with corners along isometric …
It is observed that for complex surfaces, the positivity of the Ricci curvature is preserved by the Kähler-Ricci flow, under the additional assumption that the sum of the two lowest eigenvalues of the traceless curvature operator is non-negative.
Harmonic Gauge on the Space of Riemannian Metrics and Its Role in the Ricci-DeTurck Flow
本文研究了黎曼度量空间上的调和度量，证明其在Hilbert-Einstein函数的变分中消除散度项，并诱导出自然的椭圆结构，从而建立 gauge fixing、椭圆算子理论与几何刚性的联系。
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。
Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.
Ricci Curvature and Betti Numbers of Hessian Manifolds
研究Hessian度量在由第一Koszul形式生成的codimension-one foliation叶上的Ricci曲率性质，发现单叶非负Ricci曲率迫使Hessian度量为平坦，且给出Betti数的精确界。
We extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $β(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0 \leq β(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq β(M^4,[g]) < 4(1 + ε)$ for $ε> 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to…
In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the …
In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasi-negative Chern-Ricci curvature is ample.
In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the Kähler-Ricci flow and the Ricci flow.
We consider a periodic problem for the motion of a charged particle in a magnetic field. Introducing a notion of Ricci curvature for such Lagrangian systems and using the methods of the calculus of variations in the large, we prove the existence of periodic motions for such particles under a condition of positivity of the Ricci curvature.
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.
Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow
本文研究非紧复流形上瞬时完备Chern-Ricci流的强唯一性，在弱假设下建立初始度量的唯一性，并证明欧几里得度量下流的平稳性及复曲面Ricci流的Kähler性质保持。
In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these exampl…
We establish a new version of the CR almost Schur Lemma which gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold to be a constant in terms of the norm of the traceless Webster Ricci tensor and the pseudohermitian torsion under a certain positivity condition. In the torsion-free case, i.e. for a compact Sasakian manifold, our positivity condition coincides with the known one and we obtain a better estimate
We prove that a compact Riemannian manifold of dimension $m \geq 3$ with harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of $\lfloor\frac{m-1}{2}\rfloor$-positivity originates from recent work of Petersen and Wink, who proved a similar Tachibana-type theorem under the stronger condition that the manifold be Einstein. We show that the same rigidity property holds for complete manifolds assuming either parabolicity, an integral…
In this paper, we study any Kähler manifold where the positive orthogonal bisectional curvature is preserved on the Kähler Ricci flow. Naturally, we always assume that the first Chern class $C_1$ is positive. In particular, we prove that any irreducible Kähler manifold with such property must be biholomorphic to $\mathbb{C}\mathbb{P}^n. $ This can be viewed as a generalization of Siu-Yau\cite{Siuy80}, Morri's solution \cite{Mori79} of the Frankel conjecture. According to [8], note that any Kähler manifold with 2-positive traceless bisectional curvature operator is preserved under Kahler Ricci …
PIC1 pinched manifolds are flat or compact
本文证明了所有维度下，缩陷里奇曲率的流形要么平坦要么紧致，发展了处理无限远坍缩流形的提升技术，并改进了里奇流的曲率估计和间隙定理。
Let $M^n$ be a complete noncompact K$\ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}$$_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $$dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $\mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or…
First we confirm a conjecture asserting that any compact Kähler manifold $N$ with $\Ric^\perp>0$ must be simply-connected by applying a new viscosity consideration to Whitney's comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension $n$ under the condition $\Ric_k>0$ (for some $k\in \{1, \cdots, n\}$, with $\Ric_n$ being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár-Miyaoka-Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a secon…
The structure and rigidiy of FBCMC hypersurfaces
本文研究FBCMC超曲面的结构与刚性，利用k-tri-Ricci和3-中间Ricci曲率条件证明5维流形中稳定自由边界极小超曲面的刚性，并刻画其单端性。
Volume comparison and rigidity for positive holomorphic sectional curvature
该研究证明了正全纯截面曲率紧Kähler流形的体积上界，回答了Xiong与Yang的问题，还给出了Zhang体积估计和Liu刚性定理的不同证明，其证明由ChatGPT 5.6 Sol Pro完成。
Rigidity results with curvature conditions from Lichnerowicz Laplacian and applications
该研究将Bochner技巧推广到完备非紧黎曼流形，研究Lichnerowicz拉普拉斯算子支配的L^Q-调和张量，得到新的刚性与消没定理，并应用于超曲面、ALE空间及ADM质量刚性问题。
For Einstein four-manifolds with positive scalar curvature, we derive relations among various positivity conditions on the curvature tensor, some of which are of great importance in the study of the Ricci flow. These relations suggest possible new ideas to study the well-known rigidity conjecture for positively curved Einstein four-manifolds.
Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic
研究具有非零欧拉特征的闭非正曲率四维流形，运用高斯 - 博内定理证明其单纯体积为正，部分解决相关猜想，还证实了格罗莫夫关于负里奇曲率时单纯体积正性的猜想。
Ricci curvature and normalized Ricci flow on generalized Wallach spaces
本文研究广义Wallach空间上规范化Ricci流保持Ricci曲率正性的条件，证明在$a_1+a_2+a_3\le 1/2$时正性不被保持，并给出$>1/2$时保持正性的充分条件及$\operatorname{SO}$型空间的进一步结果。
Free Boundary Stable Minimal Hypersurfaces in Positively Curved 4-Manifolds
该研究证明在正曲率4流形中，自由边界稳定极小超曲面的刚性由非负2中间Ricci曲率和标量曲率严格正性共同决定。
We define the orthogonal Bakry-Émery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on Kähler and quaternionic Kähler manifolds under positivity assumption on the orthogonal Bakry-Émery tensor. Moreover, under such assumptions on the orthogonal Bakry-Émery tensor and the holomorphic or quaternionic sectional curvature on a Kähler manifold or a quaternionic Kähler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.
In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for submanifolds in hyperbolic spaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial submanifold is positive, but does not imply positivity of the sectional curvature of $M_0$.
In this paper we consider Cartan-Hadamard manifolds (i.e. simply connected of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold when the curvature is bounded, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Caldéron-Zygmund inequalities and the $L^p$-positivity preserving property, i.e. $u\in L^p\ \&\ (-Δ+ 1)u\ge 0 \Rightarrow u\ge 0$. The main tool is a new class of first and second order Hardy-type inequalities on Ca…
The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds $\operatorname{SO}(n)/\operatorname{SO}(n-2)$
本文研究 Stiefel 流形上的归一化 Ricci 流，证明其保持正 Ricci 曲率、有限时间改善混合曲率，并从动力系统角度刻画正曲率参数域中的不变集。
We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem under the weakly pinching condition. In particular, we obtain a classification theorem for weakly pinched hypersurfaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial hypersurface is positive, but does not imply positivity of the…
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