同胚于N维圆盘的凹边界黎曼流形上正交测地弦的存在性

A Multiplicity Result for Orthogonal Geodesic Chords in Finsler disks

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a N-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least N orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility assumption does not hold, then there are at least two orthogonal Finsler geodesic chords with different values of the energy functional.

利用障碍的非凸黎曼圆盘中的多重正交测地弦

Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles

利用非光滑临界点理论和带障碍测地线理论，给出了同胚于$N$ -圆盘的黎曼流形（带边界）中正交测地线弦的一个重数结果.这适用于自然哈密顿系统势阱中的制动轨道，为塞弗特著名猜想的证明提供了进一步的证据。

Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits

Using nonsmooth critical point theory, we prove the existence of at least N orthogonal geodesic chords in a class of Riemannian N-disk with strongly concave boundary. This yields a proof of a celebrated conjecture by H.Seifert on the number of brake orbits in a potential well of a natural Lagrangian/Hamiltonian system.

黎曼流形中的正交测地弦、Brake轨道和同宿轨道

Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds

通过经典的Maupertuis-Jacobi变分原理，将某些类Lagrange（或Hamilton）系统具有固定能量的解的研究归结为黎曼流形中测地线的研究.我们感兴趣的是研究刹车轨道和同宿轨道的存在性问题，在这种情况下，Maupertuis-Jacobi原理产生一个具有边界和在边界上以非平凡方式度量退化的黎曼流形.本文利用经典的Maupertuis-Jacobi原理，证明了如何消除边界上度量的退化，并证明了在具有{ \em正则}和{ \em强凹}边界的流形上，刹车轨道和同宿轨道的多重性问题可以归结为正交测地弦的多重性问题.