We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem; another interesting case corresponds to spheres inside duals of compact semisimple Lie algebras, our result can be used to fully …

2015-01-30math.DGmath.SG

Rigidity of the L^p norm of the Poisson bracket on surfaces

For a symplectic manifold $M$ let $\{\cdot,\cdot\}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G) \mapsto \|\{F,G\}\|_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on $C^\infty_c(M)$ when $\dim M = 2$ and $p < \infty$, extending previous rigidity results for $p = \infty$ in arbitrary dimension.

2016-09-28math.DGmath.SG

Rigidity of Hamiltonian actions on Poisson manifolds

This paper is about the rigidity of compact group actions in the Poisson context. The main resut is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results that we will explore in a future paper. We also review some classical rigidity results for differentiable actions of compact Lie groups and export it to the case of symplectic actions of compact Lie groups on symplectic manifolds.

2011-10-16math.DGmath.SG

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.

2015-09-11math-phmath.DGmath.MP

$C^0$-Poisson几何、共切子流形与清洁交点

$C^0$-Poisson geometry, coisotropic submanifolds, and clean intersection points

本文研究了Poisson同胚的刚性与非刚性，证明了共切子流形的$C^0$刚性，并引入了$C^0$-特征划分作为拓扑不变量。

2026-02-23math.DGmath.SG

Normal forms in Poisson geometry

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result generalizes Conn's theorem from fixed points to arbitrary symplectic leaves. We present two proofs of this result: a geometric one relying heavily on the theory of Lie algebroids and Lie groupoids (similar to the new p…

2013-01-23math.DGmath.SG

更多相关论文收起更多论文

Thurston's h-principle and Flexibility of Poisson Structures

We prove an analogue of Thurston's h-principle for $2$-dimensional foliations on manifolds of dimension bigger or equal to $4$, in the presence of a fiber-wise non-degenerate $2$-form. This helps us understand the flexibility of rank $2$ regular Poisson structures on open manifolds with dimension bigger or equal to $4$ and it also helps us understand the flexibility of Poisson structures (not regular) on closed $4$-manifolds.

2016-11-29math.DG

洛伦兹流形中共形嵌入的柔性与刚性

Flexibility and rigidity of conformal embeddings in Lorentzian manifolds

该研究探讨洛伦兹流形中共形嵌入的柔性与刚性，证明闭可定向曲面的类空嵌入可被光滑共形嵌入$C^0$逼近，且特定商空间中可等距嵌入的负曲率度量集在泰希米勒空间中投影为相对紧集。

2025-01-17math.DG

等距延拓的刚性与柔性

Rigidity and Flexibility of Isometric Extensions

本文研究$C^{1,θ}$等距延拓的刚性与柔性，证明赫尔德指数1/2为临界值，大于该值时切联络与Levi-Civita联络沿子流形一致，小于时可通过凸积分构造反例，并得到余维数更优的紧黎曼流形等距嵌入存在性定理。

2023-05-02math.APmath.DG

具有一个赤道的旋转球体的长度谱刚性与柔韧性

Length spectrum rigidity and flexibility of spheres of revolution with one equator

Ruhr Universität Bochum（波鸿鲁尔大学）；Sorbonne Université, Université Paris Cité, CNRS, IMJ-PRG（索邦大学、巴黎西岱大学、法国国家科学研究中心、IMJ-PRG）；Institute for Advanced Study, School of Mathematics（高等研究院数学学院）

本文研究了具有一个赤道的旋转球体的长度谱刚性与柔韧性，证明了在特定对称性下标记长度谱决定度量，并给出了等谱类的显式描述。

2026-09-29math.DGmath.DSmath.SG

The Poisson saturation of coregular submanifolds

This paper is devoted to coregular submanifolds in Poisson geometry. We show that their local Poisson saturation is an embedded Poisson submanifold, and we give a normal form for this Poisson submanifold around the coregular submanifold. This result recovers the normal form around Poisson transversals, and it yields Poisson versions of some normal form/rigidity results around constant rank submanifolds in symplectic geometry. As an application, we prove a uniqueness result concerning coisotropic embeddings of Dirac manifolds in Poisson manifolds. We also show how our results generalize to the …

2022-04-25math.DGmath.SG

正各向同性曲率下的刚性与柔性

Rigidity and flexibility under positive isotropic curvature

Hong Kong University of Science and Technology（香港科技大学）；Princeton University（普林斯顿大学）

本文构造了具有任意大Urysohn宽度和稳定极小圆盘半径的4-PIC度量，否证了正各向同性曲率下的宽度与圆盘半径界，并在闭偶维流形上证明了尖锐的谱估计及等号刚性。

2026-09-11math.DGmath.MGmath.SP

谱Ricci下界和均凸边界下的刚性与灵活性

Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary

University of Notre Dame（圣母大学）；Michigan State University（密歇根州立大学）

研究具有均凸边界的Riemann流形，其谱Ricci曲率非负，证明在特定γ范围内，流形必须分解为积结构，并给出关于相对基本群的拓扑刚性结果。

2026-05-12math.APmath.DG

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…

2022-11-03math.DG

Convex integration for the Monge-Ampère equation in two dimensions

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration à la Nash & Kuiper and establish the related h-principle for the Monge-Ampère equation in two dimensions

2016-05-02math-phmath.APmath.DG