克莱因瓶上拉普拉斯算子最小值eigenvalue的唯一极值度量
A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle
摘要：
本文证明了Jakobson，Nadirashvili和Polterovich最近提出的一个猜想：在克莱因瓶\cite{JNP}上，旋转度量$$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ $0\le u <\fracπ2$，$0\le v <π$，是Laplacian的第一本征值的\emph{unique}极值度量.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。
英文摘要：
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ $0\le u <\fracπ2$, $0\le v <π$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
