Classification of Almost Quarter-Pinched Manifolds

Almost quarter-pinched Kähler metrics and Chern numbers

We prove that compact Kähler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit Kähler metrics with pinching close to 1/4.

Manifolds with 1/4-pinched flag curvature

We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain results for the case that the flag curvature is pinc…

以偶维球面空间形为纤维的丛及纤维向四分之一夹挤黎曼度量

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

本文证明：若光滑丛的每个纤维（实射影空间）带有连续变化的正曲率逐点强1/4夹挤度量，则丛的结构群可约化为标准圆度量的等距群。

Quarter-pinched Einstein metrics interpolating between real and complex hyperbolic metrics

We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete $1/4$-pinched negatively curved quaternionic Kähler (i.e. half conformally flat Einstein) metrics $g^c$, $c\ge 0$, on $\mathbb R^4$. The metric $g^0$ is the complex hyperbolic metric whereas the family $(g^c)_{c>0}$ is equivalent to a family of metrics $(h^b)_{b>0}$ depending on $b=1/c$ and smoothly extending to $b=0$ for which $h^0$ is the real hyperbolic metric. In this sense the one-loop deformation interpolates between the real and the complex hyperbolic metrics. We also determine th…

Einstein四维流形上的挤压自双曲Weyl曲率

Pinched self-dual Weyl curvature on Einstein four-manifolds

研究Einstein四维流形中挤压自双曲Weyl曲率的性质，证明其在特定条件下具有反自双曲结构。

具有近1/4夹挤度量的Kähler流形

Kähler manifolds with an almost $1/4$-pinched metric

本研究在Stover和Toledo发现的一类非局部对称紧Kähler流形上构造出近负1/4夹挤黎曼度量，证明其无法达到严格负1/4夹挤，且是Gromov-Thurston夹挤度量在复双曲场景的推广，填补了相关实例空白。