Abstract
We obtain supremum of the \(k\)-th normalized Steklov eigenvalues of all rotationally symmetric conformal metrics on \([0,T]\times \mathbb {S}^1\), \(k>1\). This generalizes the corresponding result of Fraser and Schoen for the case \(k=1\). We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We obtain some results on the comparison of the normalized Steklov eigenvalues of rotationally symmetric metrics and general conformal metrics on \([0,T]\times \mathbb {S}^1\). We also construct an example of a conformal metric on \([0,T]\times \mathbb {S}^1\) whose first normalized Steklov eigenvalue is larger than that of the corresponding rotationally symmetric conformal metric.
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The authors would like thank the referee for providing many valuable and helpful suggestions.
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Communicated by R. Schoen.
Xu-Qian Fan: Research partially supported by the National Natural Science Foundation of China (10901072, 11101106).
Luen-Fai Tam: Research partially supported by Hong Kong RGC General Research Fund #CUHK 403011.
Chengjie Yu: Research partially supported by GDNSF S2012010010038, the National Natural Science Foundation of China (11001161) and a supporting project from the Department of Education of Guangdong Province with contract no. Yq2013073.
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Fan, XQ., Tam, LF. & Yu, C. Extremal problems for Steklov eigenvalues on annuli. Calc. Var. 54, 1043–1059 (2015). https://doi.org/10.1007/s00526-014-0816-8
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DOI: https://doi.org/10.1007/s00526-014-0816-8