A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle
A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle
复制标题
克莱因瓶上拉普拉斯算子最小特征值的唯一极值度量
DOI：
--
复制
发表时间：
2006
期刊：
影响因子：
--
通讯作者：
M. Jazar
中科院分区：
文献类型：
--
作者：
A. Soufi;H. Giacomini;M. Jazar
本文证明了Jakobson，Nadirashvili和Polterovich最近提出的猜想：在Klein瓶$mathbb{K}$上，旋转度规$g_0 = {9+（1+ 8cos^2v）^2over 1+ 8cos^2v}左（du^2 + {dv^2over 1+ 8cos^2v}
8），$0le u <fracpi 2$，$0le v <pi$，是拉普拉斯算子的第一特征值的唯一极值度量，它被看作是给定面积的所有黎曼度量空间上的泛函。证明导致我们研究的哈密顿动力系统，原来是完全可积的求积。
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich cite{JNP}: on the Klein bottle $mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8cos ^2v)^2over 1+8cos ^2v} left(du^2 + {dv^2over 1+8cos ^2v}
ight),$$ $0le u <fracpi 2$, $0le v <pi$, is the emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.