Abstract
In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere \(\mathbb {S}^{n}\) and on the real hyperbolic space \(\mathbb {H}^{n}\). It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on \(\mathbb {H}^{n}\) and prove the Hong–Krahn–Szego type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.
Highlights
Let M be a complete, connected, connected Riemannian manifolds of constant sectional curvature
The Rayleigh–Faber–Krahn inequality has been extended to many other operators, see e.g. [7] for further references
We refer to Henrot [7] and Brasco and Franzina [5] for more historic remarks on isoperimetric inequalities, namely the Rayleigh–Faber–Krahn inequality and the Hong–Krahn–Szegö inequality
Summary
Let M be a complete, connected, connected Riemannian manifolds of constant sectional curvature. In this paper we are interested in isoperimetric inequalities of the convolution type operator K for the first and the second eigenvalues. In this note we prove the Rayleigh–Faber–Krahn inequality for the integral operator K , i.e. it is proved (in Theorem 2.1) that the geodesic ball is a maximiser of the first eigenvalue of the integral operator K among all domains of a given measure in M.
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