Entropy and collapsing of compact complex surfaces

具有边界的三维 Alexandrov 空间的坍缩

Collapsing three-dimensional Alexandrov spaces with boundary

本文延续[MY]的工作，研究带非空边界的三维紧 Alexandrov 空间在坍缩情形下的拓扑分类问题，并给出其拓扑结构的确定结果。

Compact null hypersurfaces and collapsing Riemannian manifolds

Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three-dimensional Riemannian metrics which collapses with bounded curvature and applying known results on the topology of manifolds which collapse. The result is then applied to general relativity, where it implies a restriction on the topology of smooth compact Cauchy horizons in spacetimes with vario…

$ALH^*$-引力瞬子的坍缩

Collapsing of $ALH^*$-Gravitational Instantons

本文证明弱 del Pezzo 曲面配对产生的 $ALH^*$-引力瞬子在大复结构极限下坍缩为带特殊 Kähler 度量的穿孔平面，并给出模空间的部分紧化，局部为多面体复形。

A collapsing ancient solution of mean curvature flow in $\mathbb{R}^3$

We construct a compact, convex ancient solution of mean curvature flow in $\mathbb R^{n+1}$ with $O(1)\times O(n)$ symmetry that lies in a slab of width $π$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $π$ and in no smaller slab.

Finite time collapsing of the Kähler-Ricci flow on threefolds

We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first Chern class.

坍缩的不朽Kähler-Ricci流

Collapsing immortal Kähler-Ricci flows

本文研究带半丰沛典范丛的紧Kähler流形上的Kähler-Ricci流，通过演化度量的渐近展开证明其坍缩为饭高纤维化底的典范度量，验证了Song-Tian的两个猜想。

Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures

In this short note, exploits of constructions of $\mathcal{F}$-structures coupled with technology developed by Cheeger-Gromov and Paternain-Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature.…

紧致Kähler曲面上Kähler Ricci流的有限时间奇点为第一类

Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

本文证明紧致Kähler曲面上有限时间体积坍缩的Kähler–Ricci流满足Type I曲率界，无需对称性或有理类假设，方法结合辛拓扑与孤子分类，并利用Bamler紧致性理论。

Nash熵界紧凯勒流形的最优几何估计

Optimal geometric estimates for compact Kähler manifolds of a Nash entropy bound

本文针对具有一致有界q-Nash熵的紧凯勒流形，证明了具有最优指数的Sobolev型不等式和局部体积非坍塌性。

测地电流在紧致双曲面上的熵与自交数

Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces

研究测地电流在双曲面上的熵与自交数关系，建立熵与自交数及系统最小长度的定量上界，证明小自交数导致小熵。

紧致Kähler曲面上Ricci流的有限时间奇点是Type I型的

Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

本文证明紧致Kähler曲面上Kähler-Ricci流的体积坍缩有限时间奇点满足Type I曲率界并分类切流，结合已有结果得出所有有限时间奇点均为Type I型。

Orbifold regularity of weak Kahler-Einstein metrics

In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.

复曲面上Kähler-Ricci流的有限时间坍缩奇点属于Type I型

Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

本文证明紧致Kähler曲面上有限时间坍缩的Kähler-Ricci流奇点均为Type I型，结合已有结果，所有有限时间奇点均为Type I型。

Existence of Typical Scales for Manifolds with Lower Ricci Curvature Bound

For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds subconverge to a product of a Euclidean and a compact space. All Euclidean factors have the same dimension, all possible compact factors satisfy the same diameter bounds and their dimension does not depend on the choice of the base point (along a fixed subsequence).