On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

An approach to spectral problems on Riemannian manifolds

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polyharmonic functions with appropriate fixed set of singularities.

几何奇点与Hodge理论

Geometric singularities and Hodge theory

本文研究非光滑几何数据下Hodge-Dirac算子核的稳定性，证明其在均匀扰动下保持同构，并应用于粗糙黎曼度量，得出紧致情形下核与上同调同构的结论。

流体力学中黎曼指数映射的奇异集

Singular Sets of Riemannian Exponential Maps in Hydrodynamics

该研究通过谱分析证明二维无粘Euler方程等的拉格朗日解映射奇异集为高斯零集，明确相关算子的Schatten-von Neumann类，并指出三维对应方程中解映射不具备Fredholm性质是本质性的。

The Riemannian manifold of all Riemannian metrics

The space of all Riemannian metrics on a smooth second countable finite dimensional manifold is itself a smooth manifold modeled on the space of symmetric (0,2)-tensor fields with compact support. It carries a canonical Riemannian metric which is invariant under the action of the diffeomorphism group. We determine its geodesics, exponential mapping, curvature, and Jacobi fields in a very explicit manner.

Banach流形上的受控曲线与端点映射的奇点。I. Banach流形结构

Regulated curves on a Banach manifold and singularities of endpoint map. I. Banach manifold structure

本文推广了Banach流形上受控曲线的Banach流形结构构建方法，取消了强Riemannian流形和绝对连续性的限制，使其适用于任意Banach流形，并将在后续工作中探讨可控性问题。

Grand Lebesgue Spaces norm estimates for eigen functions for Laplace-Beltrami operator defined on the closed compact smooth Riemannian manifolds

We derive a sharp Grand Lebesgue Space norm estimations for normalized eigen functions for the Laplace-Beltrami operator defined on the compact smooth Riemann manifold. These estimates allow us to deduce in particular the exponential decreasing tail of distribution for these eigen functions.