Abstract
We study the Robin eigenvalue problem for the Laplace–Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded from above. Our second result asserts that geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue among bounded domains of the same volume.
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Notes
As defined in Theorem 1.1.
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Acknowledgements
We thank Professors Richard Schoen, Lei Ni and Zhou Zhang for their encouragement and support. We also thank the anonymous referee for valuable comments on the manuscript. X. Li is partially supported by Simons Collaboration Grant #962228 and a start-up grant at Wichita State University; K. Wang is partially supported by NSFC No.11601359; H. Wu is supported by ARC Grant DE180101348. Both K. Wang and H. Wu acknowledge the excellent work environment provided by the Sydney Mathematical Research Institute.
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Communicated by Sun-Yung Alice Chang.
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