射影平面上拉普拉斯算子第二非零特征值的等周不等式
微分几何
2018-05-15 v5 谱理论
摘要
我们证明了实射影平面上 Laplace-Beltrami 算子的第二非零特征值的等周不等式。对于单位面积的度量，该特征值不大于 。此值由射影平面上面积序列为1的度量在极限下达到。极限度量是奇异的，可实现为具有标准度量且面积比为 3:2 的射影平面与球在点处相切的并。我们还证明了射影平面上第二非零特征值的重数至多为 6。
引用
@article{arxiv.1608.07334,
title = {An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane},
author = {Nikolai S. Nadirashvili and Alexei V. Penskoi},
journal= {arXiv preprint arXiv:1608.07334},
year = {2018}
}
备注
28 pages, v2: minor errors corrected, v3: major revision, the proof is simplified, v4: minor errors corrected, some proofs and the expository part are extended, v5: the statement of the main theorem is made more precise