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Global well-posedness and scattering for the defocusing mass-critical Schrödinger equation in the three-dimensional hyperbolic space

Bobby Wilson, Xueying Yu

TL;DR

This work proves global well-posedness and scattering for the defocusing mass-critical NLS on the 3D hyperbolic space for radial data in $L^2(\mathbb{H}^3)$. The authors adapt the Kenig–Merle concentration-compactness framework to the hyperbolic setting by developing a robust profile-decomposition toolkit that mixes Euclidean and hyperbolic profiles, aided by a radial reduction that preserves frequency localization under a simple change of variables. Central to the argument are bilinear and improved Strichartz estimates on $\mathbb{H}^3$, Euclidean approximations of nonlinear profiles, and a Morawetz-based contradiction mechanism, culminating in two key propositions that rule out minimal-mass blow-up. The results extend critical NLS theory to a curved noncompact manifold, highlighting how negative curvature enhances dispersion while requiring delicate harmonic-analytic adaptations. The approach opens avenues for mass-critical scattering on other rotationally symmetric manifolds and underscores the utility of a Euclidean-blackbox strategy in non-Euclidean geometries.

Abstract

In this paper, we prove that the initial value problem for the mass-critical defocusing nonlinear Schrödinger equation on the three-dimensional hyperbolic space $\mathbb{H}^3$ is globally well-posed and scatters for data with radial symmetry in the critical space $L^2 (\mathbb{H}^3)$.

Global well-posedness and scattering for the defocusing mass-critical Schrödinger equation in the three-dimensional hyperbolic space

TL;DR

This work proves global well-posedness and scattering for the defocusing mass-critical NLS on the 3D hyperbolic space for radial data in . The authors adapt the Kenig–Merle concentration-compactness framework to the hyperbolic setting by developing a robust profile-decomposition toolkit that mixes Euclidean and hyperbolic profiles, aided by a radial reduction that preserves frequency localization under a simple change of variables. Central to the argument are bilinear and improved Strichartz estimates on , Euclidean approximations of nonlinear profiles, and a Morawetz-based contradiction mechanism, culminating in two key propositions that rule out minimal-mass blow-up. The results extend critical NLS theory to a curved noncompact manifold, highlighting how negative curvature enhances dispersion while requiring delicate harmonic-analytic adaptations. The approach opens avenues for mass-critical scattering on other rotationally symmetric manifolds and underscores the utility of a Euclidean-blackbox strategy in non-Euclidean geometries.

Abstract

In this paper, we prove that the initial value problem for the mass-critical defocusing nonlinear Schrödinger equation on the three-dimensional hyperbolic space is globally well-posed and scatters for data with radial symmetry in the critical space .
Paper Structure (34 sections, 35 theorems, 324 equations, 1 figure)

This paper contains 34 sections, 35 theorems, 324 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup
  3. History
  4. Motivation
  5. Main result and discussion
  6. Preliminaries
  7. Notations
  8. Hyperbolic geometry
  9. Fourier transforms
  10. Fourier transforms on $\mathbb{H}^d$
  11. Radial Fourier transform on $\mathbb{R}^d$
  12. A change of variables between $\mathbb{R}^d$ and rotationally symmetric manifolds
  13. Strichartz estimates
  14. In Euclidean spaces
  15. In hyperbolic spaces
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\phi \in L^2 (\mathbb{H}^3)$ and let $\phi$ be rotationally symmetric.

Figures (1)

  • Figure 2.1: Strichartz admissible pair regions for the hyperbolic space $\mathbb{H}^d$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Proposition 2.2: Euclidean Strichartz estimates in GV92KT98Yaji87
  • Definition 2.3: Strichartz spaces in $\mathbb{R}^d$
  • Proposition 2.4: Hyperbolic Strichartz estimates in AP09IS09
  • Remark 2.5
  • Definition 2.6: Strichartz spaces
  • Proposition 2.7: Theorem 1.2 in Kaiz14: Local Smoothing Estimates in $\mathbb{H}^d$
  • Remark 2.8
  • ...and 61 more