论文标题
维度的椭圆形的自焦点$ \ geq 3 $
Self-focal points of ellipsoids of dimension $\geq 3$
论文作者
论文摘要
Riemannian歧管$（M，G）$的自我点是一个点$ P $，因此从$ p $开始的每个大地测量都在某个积极的时间返回到$ p $。如果所有通过$ p $关闭的所有大地测量学均已关闭，则称为杆，如果所有地球循环都返回但并非全部平稳地关闭，则是非极性自焦点。二维三轴椭圆形的脐带点是非极性自焦点点。关于Riemannian dimensian $ \ geq 3 $的Riemannian流形的自我点数知之甚少。我们证明，尺寸的椭圆形$ \ geq 3 $，至少有4个不同的轴没有自焦点。尺寸的某些椭圆形$ \ geq 3 $带有三个不同轴的椭圆形确实具有非极性自焦点点。带有$ \ leq 2 $不同轴的椭圆形总是具有自焦点点。 自我焦点在研究$ l^{\ infty} $ laplace eigenfunctions的规范中起着重要作用。我们的结果表明，Laplace eigenfunctions on Dimension $ \ geq 3 $，至少4美元不同的轴从未实现最大SUP-NORM的增长。
A self-focal point of a Riemannian manifold $(M,g)$ is a point $p$ so that every geodesic starting from $p$ returns to $p$ at some positive time. It is called a pole if all geodesics through $p$ are closed, and a non-polar self-focal point if all geodesics loop back but not all are smoothly closed. Umbilic points of two dimensional tri-axial ellipsoids are non-polar self-focal points. Little is known about existence of self-focal points for Riemannian manifolds of dimension $\geq 3$. We prove that ellipsoids of dimension $\geq 3$ with at least 4 distinct axes have no self-focal points. Certain ellipsoids of dimension $\geq 3$ with three distinct axes do have non-polar self-focal points. Ellipsoids with $\leq 2$ distinct axes always have self-focal points. Self-focal points play an important role in the study of $L^{\infty}$ norms of Laplace eigenfunctions. Our results imply that Laplace eigenfunctions on ellipsoids of dimension $\geq 3$ with at least $4$ distinct axes never achieve maximal sup-norm growth.
