A monotonicity theorem for subharmonic functions on manifolds

TL;DR

This paper establishes a sharp monotonicity theorem for subharmonic functions on manifolds, leading to new insights and proofs for entropy conjectures and extremal functions in complex analysis and geometric contexts.

Contribution

It introduces a measure-theoretic monotonicity theorem for subharmonic functions, unifying and extending results on entropy and extremal functions on various manifolds.

Findings

Proves the Wehrl entropy conjecture for SU(2) with extremals as coherent states.

Provides contractivity estimates for analytic functions on key complex domains.

Characterizes extremal subsets of fixed measure in the context of the theorem.

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 Poincar\'e 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.