On the behavior of quasi-local mass at the infinity along nearly round surfaces

Sasakian Geometry, Holonomy, and Supersymmetry

In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nearly Kaehler 6-manifolds. We then discuss the relations between the latter two and Sasaki-Einstein geometry.

四维无穷远处的内蕴Brown-York型质量

Intrinsic Brown--York Type Mass at Infinity in Four Dimensions

针对四维渐近平坦流形中的闭超曲面，定义内蕴参考平均曲率，导出边界项收敛至ADM质量并含形状修正的展开式，对近球面修正项在自然衰减条件下消失。

二维空间上稳态梯度Ricci-Yang-Mills孤立子的分类

Classification of Steady Gradient Ricci-Yang-Mills Solitons on Surfaces

本文构造了连接Hamilton cigar与球面的二维弦背景，提出了一族旋转对称的稳态梯度Ricci-Yang-Mills孤立子，并证明所有二维完整稳态梯度Ricci-Yang-Mills孤立子均来自该族。

从 $\mathbb{S}^2$ 到具有正 Ricci 曲率的近球面的收缩传输映射

Contractive transport maps from $\mathbb{S}^2$ to nearly spherical surfaces with positive Ricci curvature

本文证明每个近球面正曲率曲面都可表示为标准球面在收缩保体积映射下的像，方法结合了 Ricci 流、Kim-Milman 构造与多尺度 Bakry-Émery 判据。

将Caffarelli收缩定理推广到近球面流形

A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds

本文将Caffarelli收缩定理推广到近球面流形，并通过最优传输映射证明了Milman猜想的扰动形式。

球面中任意亏格极小曲面的高维族

High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres

本文在偶数维球面中构造任意亏格与共形结构的极小曲面的任意高维参数族，其维数随次数趋于无穷。

非负 Ricci 曲率且万有覆盖在无穷远处几乎 $k$-极的流形的基本群的有限生成性

Finite generation of fundamental groups for manifolds with nonnegative Ricci curvature whose universal cover is almost $k$-polar at infinity

本文证明非负 Ricci 曲率完备开流形在万有覆盖满足几乎 $k$-极条件时基本群有限生成且几乎阿贝尔，并在几乎最大体积增长时形变收缩到平坦流形。

近$G_2$-流形与$G_2$-拉普拉斯共流

Nearly $G_2$-manifolds and $G_2$-Laplacian co-flows

该研究探讨了近$G_2$-结构在修正$G_2$-拉普拉斯共流下的稳定性，发现许多自然来源于3-萨西亚几何的结构不稳定，特别是标准的近$G_2$-结构在7维球面上的高索引不稳定临界点。

Minimal Hypersurfaces in nearly $\mathrm{G}_2$ Manifolds

We study hypersurfaces in a nearly $\mathrm{G}_2$ manifold. We define various quantities associated to such a hypersurface using the $\mathrm{G}_2$ structure of the ambient manifold and prove several relationships between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the $\mathrm{G}_2$ structure of the ambient manifold, to be nearly K$\ddot{\text{a}}$hler. Then using the nearly $\mathrm{G}_2$ structure on the round sphere $S^7$, we prove that for a compact minimal hypersurface $M^6$ of constant scalar curvatur…

Three circles theorems for Schrodinger operators on cylindrical ends and geometric applications

We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently for a surface, for a fixed potential and a dense set of metrics), the constant function zero is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. These results follow from a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution $u$ to a Schr…

Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ Σ\subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(Σ) = 4 π $, it was shown that $\ \parallel A_Σ- id \parallel_{L^2(Σ)} \leq C \parallel A^0_Σ\parallel_{L^2(Σ)}\ $, and building on this, closeness of $\ Σ $ to a round sphere in $\ W^{2,2}\ $ was established, when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\ $ is small. This was supplemented in [dLMu06] by giving a conformal parametrization $\ S^2 \stackrel{\approx}{\longrightarrow} Σ $ with small conformal factor…

Free-Boundary Problems for Holomorphic Curves in the 6-Sphere

We remark on two different free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball $B$ of the round 6-sphere that meets $\partial B$ orthogonally must be totally geodesic. Consequently, we obtain rigidity results for reflection-invariant holomorphic curves in $\mathbb{S}^6$ and associative cones in $\mathbb{R}^7$. Second, we consider holomorphic curves with boundary on a Lagrangian submanifold in a strict nearly-Kähler 6-manifold. By deriving a suitable second variation formula for area, we observe a topological …

Linearised perturbation of constant mass aspect function foliation in Schwarzschild black hole spacetime

We study the linearised perturbation of the constant mass aspect function foliation in a Schwarzschild black hole spacetime. In particular, we investigate the linearised perturbation of the asymptotic geometry of the foliation at null infinity. We show that there is a 4-dimensional linear space for the linearised perturbation of the initial leaf inside the event horizon, corresponding to which the linearised perturbation of the asymptotic geometry of the foliation at null infinity is preserved to be round. For such a linearised perturbation of the initial leaf in this 4-dimensional linear spac…

Nearly Kähler Geometry and $(2,3,5)$-Distributions via Projective Holonomy

We show that any dimension $6$ nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly Kähler manifold, if the cross product is definite, otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly Kähler and nearly para-Kähler parts separated by a hypersurface which canon…