Discretization of vector bundles and rough Laplacian

Discretization of Riemannian manifolds applied to the Hodge Laplacian

An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of balls of sufficiently small radius). We exhibit then a lower bound for the first positive eigenvalue of the combinatorial Laplacian and deduce a lower bound for the first positive eigenvalue of the Hodge Laplacian.

黎曼流形上拉普拉斯算子的图离散化： Ricci曲率有界条件下

Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature

本文在Ricci曲率有界条件下，通过图近似方法证明了流形拉普拉斯算子特征值的均匀收敛性。

Small eigenvalues of the rough and Hodge Laplacians under fixed volume

For each degree p, we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that any positive eigenvalues of the rough and Hodge Laplacians acting on differential p-forms converge to zero. In particular, on the sphere, we can choose these Riemannian metrics as those of non-negative sectional curvature. This is a generalization of the results by Colbois and Maerten in 2010 to the case of higher degree forms.

具下Ricci界的黎曼流形的加权离散化

A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds

针对具下Ricci界的紧黎曼流形，引入其ε-离散化上的加权组合拉普拉斯算子，证明其与拉普拉斯-贝尔特拉米算子的特征值一致可比，应用该结果证明测度Gromov-Hausdorff收敛下的谱稳定性，并重新得到Schoen-Wolpert-Yau不等式。

1-形式上粗糙拉普拉斯算子的特征值估计及其应用

Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications

本文研究了闭合2n维黎曼流形上粗糙拉普拉斯算子作用于1-形式的第一个正特征值的估计，并推导了在特定曲率条件下欧拉特征数消失的结果。

有限差分法在平坦表面上与平坦单位向量丛的结合

Finite difference method on flat surfaces with a flat unitary vector bundle

本文研究了半翻译表面与平坦单位向量丛的离散拉普拉斯算子谱与弗里德里克扩展谱的渐进行为，同时推导了近谐函数的哈纳克型估计，并探讨了其与生成树、生成森林及加权循环根生成森林数量的渐近展开关系。