Stability, Finiteness and Dimension Four

Wasserstein空间的稳定性与有限性

Stability and Finiteness of Wasserstein Spaces

本文在Gromov--Hausdorff收敛及其等变形式下，证明Wasserstein空间在奇异与非奇异空间类上的稳定性，并给出Perelman稳定性定理的Wasserstein空间版本。

On a conformal gap and finiteness theorem for a class of four manifolds

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound for a family of conformal metricssatisfying a suitable curvature equation.

Orbifold homeomorphism finiteness based on geometric constraints

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps…

Length of a shortest closed geodesic in manifolds of dimension four

In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric|\leq 3$, volume $vol(M)>v>0$, and diameter $diam(M)<D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$. The proofs of our result are based on a recent theorem of diffeomorphism finiteness of the manifolds satisfying the above conditions proven by J. Cheeger and A. Naber.

3维切触次黎曼流形中亏格$g\geq 1$的曲面

Surfaces of genus $g\geq 1$ in 3D contact sub-Riemannian manifolds

该研究针对3维切触次黎曼流形中亏格大于1的闭曲面，证明其可通过不同嵌入方式获得有限或无限的诱导距离，并进一步探究了该距离有限性与非有限性的结构稳定性。

K-stability of cubic fourfolds

We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler-Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu-Xu, and Ambro-Kawamata's non-vanishing theorem for Fano fourfolds.