Some aspects of the global geometry of entire space-like submanifolds

Variational submanifolds of Euclidean spaces

Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) of the induced system with respect to a submanifold of a Euclidean space are studied, and the problem of existence of these "variational submanifolds" is formulated in general and solved for second-order systems. The variational sequence theory on sheaves of differential forms is …

整共形曲线要么是仿射的要么具有超欧几里得能量增长

Entire conformal curves are affine or have super-Euclidean energy growth

该研究证明高维整共形曲线仅分为仿射曲线和超欧几里得能量增长两类，推导了相关刚性结论与应用，并给出了子流形共形等价的相关结果。

Finiteness of the number of ends of minimal submanifolds in euclidean space

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimensionn meeting any n-plane passing through the origin in at most k points has no more c(n,N)k ends.

A neighbourhood theorem for submanifolds in generalized complex geometry

We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main result, which is a rigidity theorem for generalized complex deformations of holomorphic Poisson structures. Specifically, on a compact manifold with boundary we provide explicit conditions under which any generalized complex perturbation of a holomorphic Poisson structure is B-fiel…

单调拉格朗日子流形的解环性与无结性

Unlinking and unknottedness of monotone Lagrangian submanifolds

该研究在特定拓扑假设下证明了同单调性常数的单调拉格朗日子流形的无环绕性与纽结类型由同伦论数据决定，并推导出复维数≥5的奇数维辛向量空间中所有单调拉格朗日环面光滑同痕。

Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rig…

Variational aspects of the geodesic problem in sub-Riemannian geometry

We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds $\mathcal P$ and $\mathcal Q$ of a manifold $\mathcal M$ endowed with a distribution $\mathcal D\subset T\M$. We give a different proof, that holds in a more general context, of a result by Bismut (Large Deviations and the Malliavin Calculus, Birkhauser, 1984) stating that the normal extremizers that are not abnormal are critical points of the sub-Riemannian action functional. We use the Lagrangian multipliers method in a Hilbert manifold setting, which leads to a characterizat…

无限维流形中的喷雾不变集

Spray-Invariant Sets in Infinite-Dimensional Manifolds

研究无限维流形中喷雾不变集的几何性质，探讨其在不同正则性下的行为差异，扩展了对测地线保持性的理解。

极小子流形的一个二分性

A dichotomy for minimal submanifolds

本文综述最新成果，提出极小子流形的二分性，通过体积加倍定理等机制推导其受限性带来的几何与函数论限制，将$\bf{R}^3$嵌入极小圆盘理论弱推广至所有维数。

The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show that Moser's theorem nevertheless extends to codimension 2, i.e., a minimal $p$-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a $p$-plane. Our method depends on convexity properties of Grassmannians which come into play as the targets of the (harmonic) Gauss maps of our minimal subma…

正分次群上的斯托克斯定理

Stokes' theorem on positively graded groups

本文研究正分次李群（特别是Carnot群）上适应非交换几何的微分子复形上斯托克斯定理的有效性，通过引入几何条件，证明Rumin复形和与de Rham复形齐次权重过滤相关的新谱复形上的斯托克斯型公式，并揭示这些复形可通过莱布尼茨法则和特定子流形（R-流形和谱流形）上的积分从经典de Rham复形直接解释，最后提出适应这些子复形的流概念。

Curved Flats, Pluriharmonic Maps and Constant Curvature Immersions into Pseudo-Riemannian Space Forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions o…

具有积分条件的时空中的广义 Alexandrov 定理

Generalized Alexandrov theorems in spacetimes with integral conditions

本文利用任意权重的 Minkowski 公式，将时空中的 Alexandrov 型定理推广到更弱的平均曲率积分条件，并给出余维二子流形位于无剪切零超曲面上的充要条件及刚性结果。

On the Geometry of Conformally Stationary Lorentz Spaces

In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply it to the construction of complete, noncompact maximal submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hy…

Lightlike sets with applications to the rigidity of null geodesic incompleteness

An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified concusions may arise, showing that those conclusions will fail only in special cases, at least some of which may be described. These are the so-called rigidity theorems, and have many important examples in the especialized literature. In this paper, we prove rigidity results for generalized plane waves and certain globally hyperbolic spacetimes in the prese…

Nonpositively curved metric in the positive cone of a finite von Neumann algebra

In this paper we study the metric geometry of the space $Σ$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $τ$. The trace induces an incomplete Riemannian metric $<x,y>_a=τ(ya^{-1}xa^{-1})$, and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an…

Blaschke猜想与复几何

Blaschke Conjecture and Complex Geometry

本文通过引入大二次曲面纤维化，结合bend-and-break论证与Hwang-Mok VMRT识别，证明了在切丛具有适配复结构条件下Blaschke猜想成立。