Moser Lemma in Generalized Complex Geometry

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…

The Poincaré Lemma in Subriemannian Geometry

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then the question arises: Can any two points be connected by a horizontal curve? Chow's Theorem gives an affirmative answer for bracket generating distributions. (A distribution is called bracket generating if horizontal vector fields and their iterated Lie brackets span the whole tang…

Generalized geometry, equivariant $\bar{\partial}\partial$-lemma, and torus actions

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold $M$ satisfies the $\bar{\partial}\partial$-lemma, we prove that they are both canonically isomorphic to $(S\g^*)^G\otimes H_H(M)$, where $(S\g^*)^G$ is the space of invariant polynomials over the Lie algebra $\g$ of $G$, and $H_H(M)$ is the $H$-twisted cohomology of $M$. Furthermore, we establish an equiva…

Bn-generalized geometry and G2(2)-structures

We introduce the concept of G2(2)-structure on an orientable 3-manifold M using the setting of generalized geometry of type Bn, study their local deformation by making use of a Moser-type argument, and give a description of the cone of G2(2)-structures.

辛动力学中的强闭合引理

Strong closing lemmas in Hamiltonian dynamics

本文探讨了利用谱不变量证明的辛动力学中强闭合引理，涵盖低维和高维情况，以及与极小超曲面的类比结果。

The geometry of recursion operators

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures t…

广义双曲圆堆积的边值问题与离散Schwarz-Pick引理

Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings

本文研究双曲几何中广义圆堆积的离散Schwarz-Pick引理，通过证明边值问题存在性与刚性，引入组合Calabi流方法，并提出极大值原理，最终证明了该引理。

在校准几何中关于双曲性和Schwarz引理

Hyperbolicity and Schwarz Lemmas in Calibrated Geometry

本文在校准几何中定义了双曲性概念并推导了Schwarz引理，扩展了复几何中的相关结果。

New aspects of the ddc-lemma

We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie groups and provide a thorough description of invariant structures on nilmanifolds, achieving a classification on 6-nilmanifolds. We study implications of the `dd^c-lemma' in the generalized complex setting. Similarly to the standard dd^c-lemma, its generalized version induces a de…

A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that…

On the $C^{\infty}$ closing lemma for Hamiltonian flows on symplectic $4$-manifolds

The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and the $C^r$ general density theorem for geodesic flows on closed Finsler surfaces by combining a result of Asaoka-Irie with the dual lens map technique. Then we extend our results to Hamitonian flows with certain restriction. We also list some applications of our results in differential geometry and contact topology.

Generalized geometry and the Hodge decomposition

In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the $dd^c$-lemma of Kähler geometry.

Spin(7)流形的$dd^Φ$-引理与Bott–Chern型上同调

A $dd^Φ$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds

该文研究8维无挠Spin(7)-流形上的$dd^Φ$-算子，证明其Hodge分解与类似Kähler几何的$dd^Φ$-引理，定义Bott–Chern型上同调并建立其与模空间、校准几何的关联，同时给出具有独立价值的Spin(7)结构相关恒等式。

On Lipschitz Geometry at infinity of complex analytic sets

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which i…

通过经典代数几何的凸积分中的一个分解引理

A Decomposition Lemma in Convex Integration via Classical Algebraic Geometry

本文证明了一个对称矩阵场的分解引理，用少于n(n+1)/2个秩一对称项实现分解，并优化了非线性PDE系统凸积分中的Hölder正则性。