On the Rigidity of Horizontal Slices

Hawking mass and local rigidity of minimal two-spheres in three-manifolds

We study rigidity of minimal two-spheres $Σ$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $Σ$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\times Σ$. We also show that if $Σ$ is a critical point for the Hawking mass on the deSitter-Schwarzschild manifold $\mathbb{R}\times\Sph^2$ and can be written as a graph over a slice $Σ_r=\{r\}\times\mathbb{S}^2$, then $Σ$ itself must be a slice, and moreover that slices are inde…

On the Geometry of Classifying Spaces and Horizontal Slices

In this paper, we study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let $U$ be a neighborhood of the moduli space. Then we know the universal covering space $V$ of $U$ is a smooth manifold. Suppose $D$ is the classifying space of a polarized Calabi-Yau manifold with the automorphism group $G$. Let $D_1$ be the symmetric space associated with $G$. Then we proved that the map from $V$ to $D_1$ induced by the period map is a pluriharmonic map. We also give a Kahler metric on $V$, which is called the Hodge metric. We proved that the Ricci curvature of the Hodge met…

Schwarz引理与\\(\bar\partial_b\\)-调和映射的刚性

Schwarz lemmas and rigidity of \(\bar\partial_b\)-harmonic maps

本文证明从完备伪厄米流形到负曲率Kähler流形的\\(\bar\partial_b\\)-调和映射的Schwarz引理，并利用Bochner恒等式和Siu型散度恒等式，在Sasakian源和复双曲目标上分类达到水平界的映射，导出叶状结构、多重调和性和秩刚性。

海森堡群中黎曼逼近下的定量中心刚性

Quantitative Centre Rigidity under Riemannian Approximation in Heisenberg Groups

该论文证明了海森堡群中满足稳定性条件的有限黎曼周长集合的定量中心刚性，给出了水平中心导数等量的界，并指出周长超出量满足相同界。

渐近平坦切丛的正质量与刚性

Positive mass and rigidity for asymptotically flat tangent bundles

针对切丛上慢衰减渐近平坦度量，证明了其正质量定理与刚性定理，确定了广义渐近质量由底流形几何数据决定。

Local rigidity for Yamabe-type problems in warped products

Our aim in this paper is to study local rigidity for metrics defined on a compact manifold $M$ with boundary satisfying constant scalar curvature on $M$ and constant mean curvature on $\partial M$. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case. These conditions arise from the study of a spectral problem which is not included within the classical problems (Neumann, Steklov,...) that we call "mixed eigenvalue problem". Finally, we apply our previous results for the spatial slice of the de Sitter and Anti-de Sitter Schwa…