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A monotonicity theorem for subharmonic functions on manifolds

Aleksei Kulikov, Fabio Nicola, Joaquim Ortega-Cerdà, Paolo Tilli

TL;DR

The paper proves a sharp monotonicity theorem for subharmonic-type distributions on manifolds with an isoperimetric inequality, giving a differential bound on the distribution function $\mu(t)$ under $\Delta_M u \ge -c$. This framework yields unified, sharp Wehrl-entropy-type inequalities across spherical, Euclidean, and hyperbolic geometries, with extremals identified as reproducing kernels corresponding to coherent states; it also establishes local Faber–Krahn-type inequalities and analyzes equality cases. The method integrates coarea formulas, isoperimetric inequalities, and ODE comparison, and extends to non-Morse functions via Morse-function approximation. Together, these results unify several Wehrl conjectures (and prove uniqueness in the $SU(2)$ case) and provide a general, versatile approach to concentration phenomena in reproducing-kernel Hilbert spaces 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.

A monotonicity theorem for subharmonic functions on manifolds

TL;DR

The paper proves a sharp monotonicity theorem for subharmonic-type distributions on manifolds with an isoperimetric inequality, giving a differential bound on the distribution function under . This framework yields unified, sharp Wehrl-entropy-type inequalities across spherical, Euclidean, and hyperbolic geometries, with extremals identified as reproducing kernels corresponding to coherent states; it also establishes local Faber–Krahn-type inequalities and analyzes equality cases. The method integrates coarea formulas, isoperimetric inequalities, and ODE comparison, and extends to non-Morse functions via Morse-function approximation. Together, these results unify several Wehrl conjectures (and prove uniqueness in the case) and provide a general, versatile approach to concentration phenomena in reproducing-kernel Hilbert spaces 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.
Paper Structure (9 sections, 11 theorems, 56 equations)

This paper contains 9 sections, 11 theorems, 56 equations.

Table of Contents

  1. Main results and applications
  2. Proof of Theorem \ref{['main-diff']}
  3. Proof of the Theorem \ref{['main-contr']}
  4. Applications, the generalized Wehrl conjecture
  5. Bloch coherent states
  6. Glauber coherent states
  7. SU(1,1) coherent states
  8. Local estimates: Faber-Krahn inequalities
  9. Appendix, Proof of Theorem \ref{['main-mon']}

Key Result

Theorem 1.1

Let $M$ be an $n$-dimensional Riemannian manifold satisfying isop and let $u:M\to \mathbb{R}$ be a Morse function on $M$, $u\in C^2(M)$, such that for all $t\in \mathbb{R}$ the superlevel sets $u^{-1}([t, +\infty))$ are compact and $\Delta_M u \ge -c$, for some constant $c > 0$ where $\Delta_M$ is t for almost all $t\in (-\infty, t_0)$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['main-contr']}
  • Definition 1
  • Theorem 4.1
  • ...and 11 more