Structures of manifolds and asymptoticproperties

本文研究了几何谱反问题的稳定性问题，即从具有边界的流形上的拉普拉斯特征函数的特征值和边值得到流形的几何信息的问题，以及双曲动力系统上素数闭轨道周期的渐近分布，它类似于数论及其他理论中的素数分布。针对前一类问题，从有边界流形的边界距离函数出发，得到了确定内度量的边界几何条件。对于后一种情况，我们应用表示理论、微扰理论、迭代积分和半经典分析得到了Dirichlet密度定理对于Heisenberg扩展的框架。

We consider stability of geometric spectral inverse problems, which is the problems to get geometric information of manifolds from the eigenvalues and the boundary values of eigenfunctions of the Laplacian on manifolds with boundary, and asymptotic distributions of periods of prime closed orbits on hyperbolic dynamical systems, which are analogy to the prime distribution of the number theory and beyond. Concerning the former problems, we obtain boarder geometric conditions for determination of interior metric from boundary distance function of manifolds with boundary. For the latter case, we get the frame applying the representation theory, perturbation theory, iterated integrals and semi-classical analysis for the Dirichlet density theorem for the Heisenberg extensions.