尖点奇点连杆上的切触结构

每个尖点奇点链环都拥有无穷多个强辛填充

Every cusp singularity link admits infinitely many strong symplectic fillings

本文证明孤立复曲面奇点的链环（若为Sol^3或SL(2;R)流形）具有无穷多个互不同胚且非爆破相关的强辛填充，进而推出所有尖点奇点链环都有无穷多个极小强辛填充。

Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k = N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $\mathbb{H}^N$ for any $N \geq 2…

Rational cuspidal curves and symplectic fillings

A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no sym…

Contact Structures on AR-singularity links

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Nemethi. We also show that the construction cannot always work in the reverse direction: in fact the U-filtration depth of contact Ozsvath-Szabo invariant obstructs the exist…

奇异杆上的正则切触结构

Canonical contact structures on some singularity links

我们确定的规范切触结构的链接的一个简单的椭圆形或尖点奇异性的一个斯坦填充的Legendrian的身体图。我们还表明，典型的切触结构的数字Gorenstein表面奇点的链接是平凡的被认为是一个真正的平面丛。

来自带尖点的双曲曲面上的圆周平稳测度不能被拟对称映射拉直

Stationary measures on the circle from hyperbolic surfaces with cusps cannot be straightened by quasi-symmetries

本文证明：由带尖点的有限面积双曲曲面基本群上随机游走产生的、满足尖点游走指数衰减的圆周平稳测度，其关于勒贝格测度的奇异性在拟对称映射下保持稳定，从而解决了McMullen猜想。