Stuart James Hall, Thomas Murphy
On compact Riemannian manifolds with a large isometry group we investigate the invariant spectrum of the ordinary Laplacian. For either a toric Kaehler metric, or a rotationally-symmetric metric on the sphere, we produce upper bounds for all eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
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If a normalized Kähler-Ricci flow $g(t),t\in[0,\infty),$ on a compact Kähler $n$-manifold, $n\geq 3$, of positive first Chern class satisfies $g(t)\in 2πc_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will converge along a subsequence to a Kähler-Ricci soliton.
We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds of complex dimension three, and for projective manifolds of all dimensions if assuming the abundance conjecture.
For an ancient solution of the mean curvature flow, we show that each time slice M_t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family M_t that evolves under mean curvature flow for all negative time t.
Bounding the A-hat genus using scalar curvature lower bounds and isoperimetric constants
本文利用标量曲率下界、Neumann 等周常数和体积，通过 Dirac 算子谱分析证明了光滑闭 spin Riemannian 流形 $\widehat{A}$ 亏格的上界，并构造例子说明该常数必要性，部分回答了 Gromov 的问题。
Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature
该研究针对围长为3且β>α的充分正则图，通过关联离散Ricci曲率与局部匹配性质的新方法，证明了其Lin-Lu-Yau曲率的精确下界，并由此推导出该类图的精确直径与特征值界。
We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metrics on complex projective space $\mathbb{C}P^{n}$ is exhibited. We derive an analogous bound in the case when the metric is extremal and a detailed study is made of the accuracy of the bound in the case of Calabi's extremal metrics on $\mathbb{C}P^{2}\sharp -\mathbb{C}P^{2}$.
Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, t\in\mathbb{R}$. In particular, there are no topological obstructions for metrics with $\varepsilon$-pinched Weyl curvature and negative scalar curvature.
Bounding Curvature Measure on Manifolds with Singularities
研究Alexandrov空间中去掉奇点后的光滑流形，证明其标量曲率积分由仅依赖于维数和奇点参数的常数上界控制，推广了Petrunin的有界曲率积分结果。
We show that every closed symplectic four-dimensional manifold admits compatible almost Kaehler metrics of negative scalar curvature.
We study the behaviour of the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a Kähler-Einstein metric of negative scalar curvature, recovering a result of Wu and Yau.
We prove that any compact four-manifold admits a Riemannian metric with negative isotropic curvature in the sense of Micallef and Moore.
An updated version with a few corrections.
Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.
In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.
Motivated by the torus stability problem, in this work we study Kähler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of Kähler metrics with almost non-negative scalar curvature will converge to flat torus weakly.
A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact Kähler manifold admitting a Kähler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this paper we shall consider a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact Kähler manifolds of almost quasi-negative holomorphic sectional curvature. We also obtain a gap-type theorem for the inequality $\int_Xc_1(K_X)^n>0$ in terms of the holomorphic sectional curvature. In the discussions, w…
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.
The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional curvature only if either the manifold is diffeomorphic to $\mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.
In a recent work of Brendle-Hirsch-Johne, a notion of intermediate curvature was introduced to extend the classical non-existence theorem of positive scalar curvature on torus to product manifolds. In this work, we study the rigidity when the intermediate curvature is only non-negative when the ambient dimension is at most $5$.
We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.
Liouville property and non-negative Ollivier curvature on graphs
证明非负Ollivier曲率图的Liouville性质并改进函数集中性结果。
We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$
This is a significantly improved version with new applications. We show that there are many cohomogeneity one manifolds which do not admit an analytic invariant metric with non-negative sectional curvature, although they do have a smooth one. In particular, there are no invariant metric with positive curvature.
Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.
In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.
Remark on semi-positive holomorphic sectional curvature and quasi-negative $k$-Ricci curvature
本文记录了两个注记：对于紧致Kähler流形，半正全纯截面曲率时MRC纤维化的有理维数等于非真正平坦方向的数量；拟负$k$-Ricci曲率或更一般的拟负混合曲率$C_{a,b}$时，典范丛是丰富的。
Open $3$-manifolds with non negative Ricci curvature in a spectral or integral sense
该研究证明了在特定条件下，具有非负曲率的开放3流形与三维欧几里得空间形同构。
We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.
We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg.
We show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index.
This article is an application of the author's paper about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d'Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition. As an application, we draw figures of discrete constant negative Gaussian curvature surfaces given by this method.
Closed self-similar solutions to flows by negative powers of curvature
本文研究翘曲乘积流形中负曲率幂次流的闭自相似解，证明其必为切片，关键引入新辅助函数。
We give a geometric obstruction to the non-negativity of the sectional curvature in the total spaces of certain Riemannian submersions with totally geodesic fibers; applications of this obstruction to several examples are given.
Remarks on Kähler orbifolds of non-negative Ricci curvature
本文证明了凯勒轨道折面的非负曲率版本的 Kobayashi 定理，证明了非负曲率的紧凯勒轨道折面在满足一定条件时是单连通的。
We show that the results of Foulon (1997 an 2002) and Kim (2007) (independently, Deng and Hou (2007)) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local.
We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d complete Riemannian manifold with non-negative Ricci curvature, there exists a smooth Ricci flow starting from it.
We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly $\textnormal{PIC}_1$ cases, of which the results are new. We complete the discussion of the almost preservation problem by Bamler-Cabezas-Rivas-Wilking, and the 2-non-negative case generalizes a result in 3D by Simon-Topping to higher dimensions. As an application, we use the local …
In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
Quaternion-Kähler manifolds with non-negative quaternionic sectional curvature
本文引入四元数截面曲率，将 Gray 定理推广到四元数-凯勒流形：证明 Wolf 空间具有非负四元数截面曲率，反之亦然，关键利用近凯勒 twistor 空间。
Volume growth and integral curvature bound for non-negatively curved three-manifolds
本文研究三维非负曲率流形，证明欧氏体积增长等价于平均二次曲率衰减，并给出基于标量曲率积分阈值的判定及间隙定理。
Notes on conformal metrics of negative curvature on manifolds with boundary
本文通过Morse函数构造共形度量，使修正Schouten张量特征值落入给定锥，从而证明紧致带边3-流形上的任意黎曼度量都共形于负截面曲率度量。
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minim…
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetri…
Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree
本文研究具有负Gauduchon度的紧Hermitian流形上预定Chern数量曲率问题，利用几何流收敛性在非正及变号曲率函数情形下建立了存在性与不存在性结果。
We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations of some classical results concerning moduli spaces of positive scalar curvature metrics.
This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant.
In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.
In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L^2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line with an hypersurface.
In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypersurface.
In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds, that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case where both are equal to two, where existence of non-negatively curved metrics is known.
In this paper, we study certain compact 4-manifolds with non-negative sectional curvature $K$. If $s$ is the scalar curvature and $W_+$ is the self-dual part of Weyl tensor, then it will be shown that there is no metric $g$ on $S^2 \times S^2$ with both (i) $K > 0$ and (ii) $ {1/6} s - W_+ \ge 0$. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: ``If a simply-connected, closed 4-manifold $M^4$ admits a metric $g$ of non-negative curvature operator, then $M^4$ is one of $S^4$, $\Bbb CP^2$ and $S^2 \times S^…
We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, and discuss some open questions.
On quaternionic bisectional curvature
本文研究了四元Kähler流形的四元数双截面曲率，证明非负该曲率仅在四元数射影空间中实现，并指出非负截面曲率不蕴含非负四元数双截面曲率，同时给出了A. Gray紧Kähler流形分类结果的简短新证明。
We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denote the Hamilton's cigar soliton and some compact Kähler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kähler Ricci soliton with $Ric\geq 0$, together with subquadratic volume growth or $\limsup_{r\to \infty} rS<1$ must have u…
We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. We give conditions under which such an extension is possible, and also give obstructions to such an extension. The methods we use are based on a maximum principle and the Ahlfors--Schwarz Lemma. We apply these considerations to compactification of Riemann surfaces.
We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold and it is diffeomorphic to the product of $k$ copies of the 3-sphere.
K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.
Three-manifolds with non-negatively pinched Ricci curvature
研究非负曲率pinching的三维流形的Ricci流解，证明其具有标度不变曲率衰减和pinching保持，结合Lott和Deruelle-Schulze-Simon的工作，证明Hamilton的pinching猜想。
Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature
本文证明紧致 Kähler 流形若存在关于 Chern 联络的拟负全纯截面曲率的 Hermitian 度量，则其典范丛丰沛，且该度量无需为 Kähler 度量，从而验证了 Yang-Zheng 拟负猜想的 Kähler 情形。
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