Curvature of hyperkaehler quotients

Gauge theory and the division algebras

We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this formulation we prove that (in flat space) these equations can be understood in terms of moment maps on the space of connections and the moduli space of solutions is obtained via a generalised symplectic quotient: a Kaehler quotient in dimension 2, a hyperkaehler quotient in dimension…

Hyperholomorpic connections on coherent sheaves and stability

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for i…

Uniqueness of self-similar solutions to flows by quotient curvatures

In this paper, we consider a family of closed hypersurfaces which shrink self-similarly with speed of quotient curvatures. We show that the only such hypersurfaces are shrinking spheres.

总商曲率的比较

Comparison of total quotient curvature

本文研究了总商曲率的比较定理，证明了爱因斯坦度量在总商曲率上的严格界限，并推广了标量曲率的体积比较定理和σ_k-曲率的刚性结果。

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on Brieskorn quotients in dimension 5

We show that the moduli space of Riemannian metrics of nonnegative sectional curvature on certain quotients of Brieskorn varieties in dimension 5 has infinitely many path components. The same is true for the corresponding moduli space of positive Ricci curvature.

Curvature explosion in quotients and applications

We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.

Hypersurfaces with constant curvature quotients in warped product manifolds

In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients $\frac{\mathcal{H}_{2k+1}}{\mathcal{H}_{2k}}$ in the warped product manifolds. Here $\mathcal{H}_{2k}$ is the $k$-th Gauss-Bonnet curvature and $\mathcal{H}_{2k+1}$ arises from the first variation of the total integration of $\mathcal{H}_{2k}$. Hence the quotients considered here are in general different from $\frac{σ_{2k+1}}{σ_{2k}}$, where $σ_k$ are the usual mean curvatures. We prove several rigidity and Bernstein type results for compact or non-compact hypersurfaces corresponding to such quotients…

The sectional curvature remains positive when taking quotients by certain nonfree actions

We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.

Metrics of positive Ricci curvature on quotient spaces

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonnegative Ricci and almost nonnegative sectional curvature. Moreover, if N has finite fundamental group, then N admits also metrics of positive Ricci curvature. Particular examples include infinite families of simply connected manifolds with the rational cohomology rings and integr…

Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an inf…

On doubly periodic minimal surfaces in $\mathbb H^2 \times \mathbb R$ with finite total curvature in the quotient space

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in $\mathbb H^2$ or a hyperbolic translation along a geodesic in $\mathbb H^2.$ In fact, we prove that if a properly immersed minimal surface in $\mathbb H^2\times\mathbb R/G$ has finite total curvature then its total curvature is a multiple of $2π,$ and…

Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional assumptions in [13] by Wallach and the second author.

Periodic constant mean curvature surfaces in H^2 x R

In this paper we study minimal and constant mean curvature (cmc) periodic surfaces in H^2 x R. More precisely, we consider quotients of H^2 x R by discrete groups of isometries generated by horizontal hyperbolic translations f and/or a vertical translation T. In the quotient by the Z^2 subgroup of the isometry group generated by any f and T, we prove an Alexandrov-type theorem for cmc surfaces, i.e., an analysis of compact embedded cmc surfaces in such a quotient. The remainder of the paper is devoted to construct examples of periodic minimal surfaces in H^2 x R.

凸超曲面上曲率商方程的内部曲率估计

Interior curvature estimate for curvature quotient equations on convex hypersurfaces

本文针对满足σₙ/σₙ₋₂(λ)=f(X)>0的凸图，研究其曲率商方程的内部曲率估计问题

Deformations of Scalar-Flat Anti-Self-Dual metrics and Quotients of Enriques Surfaces

In this article, we prove that a quotient of a K3 surface by a free Z_2+Z_2 action does not admit any metric of positive scalar curvature. This shows that the scalar flat anti self-dual metrics (SF-ASD) on this manifold can not be obtained from a family of metrics for which the scalar curvature changes sign, contrary to the previously known constructions of this kind of metrics on manifolds of b+=0.

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

We prove that a four-dimensional gradient shrinking Ricci soliton with $δW^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of $M\times\mathbb{C}$, where $M$ is a Riemann surface. The main arguments are curvature decompositions, the Weitzenböck formula for half Weyl curvature, and the maximum principle.

轨道流形上正曲率稳态梯度Ricci孤立子的刚性

Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

本文研究轨道流形上正曲率稳态梯度Ricci孤立子的刚性，证明其数量曲率非负，并在正曲率算子或正截面曲率条件下，此类孤立子必须是Bryant孤立子的有限商。

Scalar curvature and deformations of complex structures

We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kaehler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kaehler reduction, which is a hyperkaehler reduction for a particular choice of the spectral function. The system can be formally complexified using a flat connection on the space of first-order deformations that are compatible with a Kaehler metric. We describe a variational characterization of the equations, a Futaki invariant for the system, and a gener…

通过中间胖度在球面乘积及其商空间上构造正 $\mathrm{Ric}_2$ 曲率

Positive $\mathrm{Ric}_2$ curvature on products of spheres and their quotients via intermediate fatness

本文推广胖度概念，在 10 至 14 维闭流形（含球面乘积及其商空间）上构造了正 $\mathrm{Ric}_2$ 曲率度量，提供了包括非单连通空间在内的无限多个新例子。

Orbifolds and manifold quotients with upper curvature bounds

We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak--Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.

Classification of expanding and steady Ricci solitons with integral curvature decay

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar curvature decay. As a corollary, we obtain that the only three dimensional steady solitons with less…

关于给定曲率商方程严格凸解的内部曲率估计的注记

A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotient

本文针对椭圆型Weingarten曲率商方程的严格凸解，借助方程算子的凹性不等式与新颖辅助函数，通过初等最大值原理证明其先验内部曲率界，还得到低维特殊拉格朗日曲率方程严格凸解的对应估计。

闵可夫斯基空间中Hessian商型曲率方程的Dirichlet问题

The Dirichlet problem for Hessian quotient type curvature equations in Minkowski space

针对非凸区域，在不假设下解和Serrin型条件下建立先验估计，证明闵可夫斯基空间中一类Hessian商型曲率方程Dirichlet问题的存在性。

Curvature, diameter, and quotient manifolds

Gromov showed that there is an upper bound on the Betti numbers of all closed Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether such manifolds (if simply connected) fall into only finitely many rational homotopy types. We give a negative answer, in fact in dimension 6, which is the smallest possible. We also give counterexamples to some related questions in dimensions 7 and 9, improving the original counterexamples by Fang and Rong which were in dimensions at least 22.

The Dirichlet problem for a class of Hessian quotient equations in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$

In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen as a prescribed curvature problem and a continuous work of [12].

Complete gradient shrinking Ricci solitons with pinched curvature

We prove that any $n$--dimensional complete gradient Ricci soliton with pinched Weyl curvature is a finite quotient of $\RR^{n}$, $\RR \times \SS^{n-1}$ or $\SS^{n}$. In particular, we do not need to assume the metric to be locally conformally flat.

Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire self-shrinking solutions on $\mathbb{C}^n$ to the Kähler-Ricci flow under certain conditions.

正常 Hessian 截面曲率的 Hessian 流形

Hessian manifolds of positive constant Hessian sectional curvature

本文分类了正常 Hessian 截面曲率的 Hessian 流形标准模型，通过引入 Kähler 流形在交换李群 Hamilton 作用下的商结构实现该分类，填补了正曲率情形的空白。

On quotients of spaces with Ricci curvature bounded below

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of $\mathsf{RCD}^{…

关于Q曲率与标量曲率之比的Yamabe问题

A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature

本文研究了在给定共形类中寻找使得Q曲率与标量曲率之比为常数的共形度量问题，通过新的Sobolev不等式定义了新的Yamabe常数，并证明了在特定条件下该问题的解存在性。

余齐性三中的正曲率与有理椭圆性

Positive curvature and rational ellipticity in cohomogeneity three

本文证明了商空间无边界的闭、单连通、正曲率余齐性三流形是有理椭圆的，推广了齐性、余齐性一及余齐性二流形的相关结果。

The Alexandrov problem in a quotient space of $\mathbb H^2\times \mathbb R$

We prove an Alexandrov type theorem for a quotient space of $\mathbb H^2\times \mathbb R$. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb H^2\times \mathbb R$ by a subgroup of isometries generated by a parabolic translation along horocycles of $\mathbb H^2$ and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in $\mathbb H^2\times\mathbb R$ and we prove a multi-valued Rado theorem for small perturbations of the helicoid in $\mathbb H^2\times\mathbb R$.

凸毛细超曲面的 prescribed $L_{p}$ 曲率问题

Prescribed $L_{p}$ curvature problem for convex capillary hypersurface

本文研究了欧几里得半空间中凸毛细超曲面的 prescribed $L_p$ 曲率问题，通过转化问题为 Hessian 商类型方程，证明了解的存在性和唯一性，并应用该方法解决了毛细 $L_p$ Christoffel-Minkowski 问题及相关特征值问题。

预定 $L_p$ 商曲率问题及相关特征值问题

Prescribed $L_p$ quotient curvature problem and related eigenvalue problem

本文研究预定 $L_p$ 商型曲率问题的可容许光滑解存在性，针对 $p$ 与 $k-l+1$ 的不同关系给出解的存在性结果，并在 $p<k-l+1$ 情形下证明偶函数条件是必要的。

Some sharp estimates for convex hypersurfaces of pinched normal curvature

For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient $R/r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.

球商的环面紧化的曲率性质与Shafarevich猜想

Curvature properties and Shafarevich conjecture for toroidal compactifications of ball quotients

本文研究有限体积复双曲流形的环面紧化，探讨了其上满足非正曲率性质的Kähler度量的存在性，并通过对足够深的非均匀算术格群商证明万有覆盖为Stein，验证了Shafarevich猜想。

Noncompact Shrinking 4-Solitons with Nonnegative Curvature

We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0.

Scalar Curvature Estimates by Parallel Alternating Torsion

We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric. We also correct a mistake in the treatment of odd-dimensional spaces in arXiv:math/0010199 and arXiv:0705.0500

Geometry of Cyclic Quotients, I: Knotted Totally Geodesic Submanifolds in Positively Curved Spheres

We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the technique borrows a lot from the Mostow-Siu-Gromov-Thurston constuction of exotic negatively curved manifolds.

Local curvature estimates of long-time solutions to the Kähler-Ricci flow

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.

关于双曲3维空间中常平均曲率为1的曲面方程的达布可积性

On the Darboux Integrability of the Constant Mean Curvature One Equation for Surfaces in Hyperbolic 3 space

研究双曲3维空间中常平均曲率为1的曲面方程，利用群论构造表明相关椭圆型偏微分方程组等同于椭圆型刘维尔方程，证明方程达布可积性及等价商表示，给出显式等价映射。

Four-manifolds of Pinched Sectional Curvature

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere or their quotients. Furthermore, we also classify Einstein manifolds …

负曲率中发散测地线的等分布

Equidistribution of divergent geodesics in negative curvature

本文研究非紧有限体积负曲率流形及几何有限树商空间中的发散测地线，证明其随复杂度增大分别趋向最大熵测度与 Bowen–Margulis 测度等分布。

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in $M$ centered at a point $o\in M$. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the …

Gursky-Malchiodi Q-曲率流的全局收敛性

Global Convergence of the Gursky-Malchiodi $Q$-curvature Flow

本文证明了Gursky-Malchiodi Q-曲率流在任意初始能量下的全局收敛性，通过建立非局部Lojasiewicz-Simon不等式并构造测试气泡来解决Q-曲率和Paneitz算子的复杂性。

具有秩一非阿贝尔对称性的正曲率六维流形的轨道分层与光滑拓扑

Orbit stratifications and smooth topology of positively curved six-manifolds with rank-one nonabelian symmetry

本文研究带SO(3)或SU(2)有效等距作用的正曲率六维流形，细化轨道分层并完成多类商空间的光滑拓扑分类，建立相关曲率障碍，部分结果由ChatGPT-6-astra发现并验证。

A positively curved manifold homeomorphic to T$_1$S$^4$

We construct a new 7-dimensional manifold with positive sectional curvature which is 2-connected with π_3=\Z_2 and admits an isometric group action with one dimensional quotient.

Inverse curvature flow in anti-de Sitter-Schwarzschild manifold

In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.

Classification of Cohomogeneity One Manifolds in Low Dimensions

A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

A comparison theorem for steady Ricci solitons

We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton $N^{n-1}\times\mathbb{R}$, where $N^{n-1}$ is Ricci flat, or isometric to the Bryant soliton (up to scalings), provided that a couple of geometric conditions inspired by the cigar soliton hold. As an application, we prove that any complete non-compact steady Ricci soliton with positive Ricci curvature controlled by the scalar curvature $R$, curvature tensor $Rm$ satisfying $|Rm|r\to o(1)$ and $R\to\infty$, as $r\to\infty$, must be the Bryant soliton.…

Gromoll-Meyer球面上的一个正曲率度量

A positively curved metric on the Gromoll-Meyer sphere

该文在Sp(2)上构造双不变度量的显式形变，使Gromoll-Meyer exotic 7-球面在充分小正参数下获得正截面曲率，证明结合均匀估计与代数分析，并给出临界集上三次系数的正下界。

Pinching estimates and motion of hypersurfaces by curvature functions

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives.

关于具有常标量曲率的收缩梯度Ricci溶剂刚性的一个注记

A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

本文研究了具有常标量曲率的收缩梯度Ricci溶剂的刚性性质，证明了其在特定条件下可视为有限商空间，扩展了Cheng-Zhou的结果到高维情况。

Integral pinched 3-manifolds are space forms

In this paper we prove that, under an explicit integral pinching assumption between the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein metric with positive curvature. In particular this implies that the manifold is diffeomorphic to a quotient of ${\Bbb S}^3$.