Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Liouville-Type Theorems on the Hyperbolic Space

Sanghoon Lee

Abstract

In this paper, we establish Liouville-type theorems for a one-parameter family of elliptic PDEs on the standard upper half-plane model of the hyperbolic space, under specific geometric assumptions. Our results indicate that the Euclidean half-plane is the only compactification of the hyperbolic space when the scalar curvature of the compactified metric has a designated sign.

Liouville-Type Theorems on the Hyperbolic Space

Abstract

In this paper, we establish Liouville-type theorems for a one-parameter family of elliptic PDEs on the standard upper half-plane model of the hyperbolic space, under specific geometric assumptions. Our results indicate that the Euclidean half-plane is the only compactification of the hyperbolic space when the scalar curvature of the compactified metric has a designated sign.
Paper Structure (17 sections, 28 theorems, 61 equations)

This paper contains 17 sections, 28 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Background and setup
  3. Main results
  4. Organization of the paper
  5. Preliminaries
  6. Scattering theory on CCE manifolds and adapted metrics
  7. $y^a$-weighted Sobolev spaces
  8. Some Liouville-type theorems
  9. Liouville-type theorems for degenerate elliptic PDEs
  10. Liouville-type theorem for subharmonic functions
  11. Conformal infinity with positive scalar curvature
  12. Previously known properties
  13. Proof for Theorem \ref{['mainthm2.1']}
  14. Liouville-type theorems on the hyperbolic space
  15. Liouvile-type theorems for adapted metrics with $0<\gamma<\frac{n}{2}$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(X^{n+1}, \partial X = M , g_+)$ be a CCE manifold with conformal infinity $(M, [h])$. Assume that the scalar curvature $R_h$ of $(M, h)$ is positive. For $0<\gamma<1$, it follows that $|{\nabla_s^*} \rho_s|^2 < 1$ on $\mathring{X}$, or equivalently, $\mathrm{sgn}(R_s^*) = \mathrm{sgn}(\gamma -

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 46 more