A monotonicity theorem for subharmonic functions on manifolds

Abstract

We provide a sharp monotonicity theorem about the distribution of subharmonic functions on manifolds, which can be regarded as a new, measure theoretic form of the uncertainty principle. As an illustration of the scope of this result, we deduce contractivity estimates for analytic functions on the Riemann sphere, the complex plane and the Poincaré disc, with a complete description of the extremal functions, hence providing a unified and illuminating perspective of a number of results and conjectures on this subject, in particular on the Wehrl entropy conjecture by Lieb and Solovej. In this connection, we completely prove that conjecture for SU(2), by showing that the corresponding extremals are only the coherent states. Also, we show that the above (global) estimates admit a local counterpart and in all cases we characterize also the extremal subsets, among those of fixed assigned measure.

arXiv commentFixed a few typos

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• Proof of an entropy conjecture for Bloch coherent spin states and its generalizations
• The Faber-Krahn inequality for the Short-time Fourier transform
• Proof of the Wehrl-type Entropy Conjecture for Symmmetric SU(N) Coherent States
• Contractive inequalities for Bergman spaces and multiplicative Hankel forms
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Cited by in corpus (1)

• Multiple sampling and interpolation in a space of polynomials