The Linearized System for Isometric Embeddings and Its Characteristic Variety

The Isometric Embedding Problem in a Null Cone

In the first part of this paper, we extend the result of Li-Wang on the linearized embedding problem to a compact manifold of arbitrary dimension. Using this, we then show that any metric perturbation of a embedded $n$-sphere is also isometrically embedded up to a solution of the homogenous Codazzi equation, irrespective of the ambient geometry. In the second part we specialize to dimension two, and study these results within an ambient Null Cone. Specifically, given a path of metrics on the 2-sphere and an initial isometric embedding, we develop a small parameter existence and uniqueness theo…

球簇上线性化线丛的赋值不变量

Valuative invariants for linearized line bundles on a spherical variety

该研究给出射影球簇上线性化充足线丛的赋值不变量公式，将其用于研究法诺球簇的g-孤立子，并证明法诺球极流形对应锥总存在Calabi-Yau锥度量。

Almost isometric actions, property $(T)$, and local rigidity

Let $Γ$ be a discrete group with property $(T)$ of Kazhdan. We prove that any Riemannian isometric action of $Γ$ on a compact manifold $X$ is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in \cite{FM2}. One definition of property $(T)$ is that a group $Γ$ has property $(T)$ if every isometric $Γ$ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with proper…

通过复伪双曲空间中的交替曲面构造非极大约束拟等距 $\mathrm{PU}_{2,n+1}$-表示

Non-maximal quasi-isometric $\mathrm{PU}_{2,n+1}$-representations via alternating surfaces in complex pseudo-hyperbolic spaces

本文为每个可容许非极大Toledo不变量构造了曲面群到$\mathrm{PU}_{2,n+1}$的不可约表示，其轨道映射为拟等距嵌入，并构造了Shilov边界上的不连续域，商为纤维丛。