Every quasitoric manifold admits an invariant metric of positive scalar curvature

Nonexistence of NNSC fill-ins with large mean curvature

In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower bound.

穿孔四流形上的自旋与数量曲率

Spin and scalar curvature on punctured four manifolds

本文证明非零符号差的闭自旋四流形在穿孔后仍不允许一致正数量曲率的完备度量，并发现Gromov的μ-气泡可作为APS边界问题的便捷边界。

Topological obstructions to nonnegative curvature

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.

Four-dimensional manifolds with positive biorthogonal curvature

We classify, up to homeomorphisms, the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive.

Positive scalar curvature and $10/8$-type inequalities on $4$-manifolds with periodic ends

We show $ 10/8$-type inequalities for some end-periodic $4$-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed $4$-manifolds which do not admit positive scalar curvature metrics.

正标量曲率的倍增猜想

The doubling conjecture for positive scalar curvature

本文研究正标量曲率 manifold 的倍增猜想，证明在满足特定基本群分裂条件时，该猜想成立，结合手术技术和面积最小超曲面理论，探讨如何调整 psc-metric 使给定超曲面成为极小、稳定极小或全测地。

广义肥皂泡与正数量曲率流形的拓扑

Generalized soap bubbles and the topology of manifolds with positive scalar curvature

本文利用广义肥皂泡（μ-气泡）方法，证明了低维闭非球面流形不承认正数量曲率度量，并由此确立 Schoen--Yau 刘维尔定理对所有非负数量曲率的局部共形平坦流形成立。

具有无限多个不同正Ricci曲率几何的奇数维流形

Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvature

针对n≥5的奇数维闭流形，证明其存在无限多个不同的正Ricci曲率几何，即正Ricci曲率度量模空间有无限多个连通分支，丰富了奇数维流形正Ricci曲率几何的研究。

Torus manifolds and non-negative curvature

A torus manifold $M$ is a $2n$-dimensional orientable manifold with an effective action of an $n$-dimensional torus such that $M^T\neq \emptyset$. In this paper we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If $M$ is a simply connected torus manifold which admits such a metric, then $M$ is diffeomorphic to a quotient of a free linear torus action on a product of spheres. We also classify rationally elliptic torus manifolds $M$ with $H^{\text{odd}}(M;\mathbb{Z})=0$ up homeomorphism.

维数为 $6k$ 的单连通流形上的正 Ricci 曲率度量

Metrics of Positive Ricci Curvature on Simply-Connected Manifolds of Dimension $6k$

本文通过标记二部图描述某些 $6k$ 维单连通流形，并利用作者先前结果构造正 Ricci 曲率度量，从而获得了大量新的 spin 和 non-spin 型正 Ricci 曲率流形实例。

Dissolving four-manifolds and positive scalar curvature

We prove that many simply connected symplectic four-manifolds dissolve after connected sum with only one copy of $S^{2}\times S^{2}$. For any finite group G that acts freely on the three-sphere we construct closed smooth four-manifolds with fundamental group G which do not admit metrics of positive scalar curvature, but whose universal covers do admit such metrics.

正标量曲率度量存在性的覆盖不稳定性

Covering instability for the existence of positive scalar curvature metrics

本文研究非可定向流形上正标量曲率度量的存在性，证明闭非可定向3-流形与其定向二重覆盖在PSC度量存在性上等价，并构造4至7维中定向二重覆盖允许PSC度量但自身不允许的同伦等价类无穷多反例，同时将Schoen-Yau归纳下降方法推广至非可定向情形。

Small Covers, infra-nilmanifolds and positive curvature

We show that all the small covers which are infra-nilmanifolds are exactly real Bott manifolds. This implies that any small cover which admits a flat Riemannian metric must be a real Bott manifold. In addition, we will study small covers which admit Riemannian metrics with positive or nonnegative Ricci curvature or sectional curvature. We will see that these geometric conditions put very strong restrictions on the topology of the small covers and the combinatorial structure of the underlying simple polytopes. Similar geometric problems are also studied for the real moment-angle complex of an a…

广义非正曲率中的非紧凸包

A non-compact convex hull in generalized non-positive curvature

本文研究容许锥形测地双梳的弱非正曲率度量空间，通过构造反例证明Gromov关于凸包紧性的问题在此设定下答案为否，即存在有限子集的闭σ-凸包非紧的完备度量空间。

Spaces of positive intermediate curvature metrics

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

具有一致正数量曲率的完备黎曼4-流形

Complete Riemannian 4-manifolds with uniformly positive scalar curvature

本文研究非紧4-流形上具有一致正数量曲率的完备黎曼度量的拓扑阻碍，证明该度量可唯一区分标准4-球，并给出大量奇异$\mathbb{R}^4$及驯服4-流形不允许此类度量的结论。

Obstructions to nonpositive curvature for open manifolds

We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood of a subcomplex of the same codimension. In this setting we show that each tangential homotopy type contains infinitely many open K(G,1) manifolds that admit no complete nonpositively curved metric even though their u…

Diffeomorphisms and positive curvature

We prove the existence of elements of infinite order in the homotopy groups of the spaces $\mathcal{R}_{Ric>0}(M)$ and $\mathcal{R}_{sec>0}(M)$ of positive Ricci and positive sectional curvature, provided that $M$ is high-dimensional and Spin, admits such a metric and has a non-vanishing rational Pontryagin class.