Kemper, Matthias: Gromov hyperbolic manifolds, weighted isoperimetry and bubbles. 2020
Inhalt
- Contents
- Introduction
- I Potential Theory on Hyperbolic Spaces
- 1 Basic Concepts
- 1.1 Geometric Structures
- 1.1.1 Bounded Geometry
- 1.1.2 Gromov Hyperbolic Spaces
- 1.1.3 Gromov Boundary and Visuality
- 1.1.4 Connecting Two Points: Harnack and Phi-Chains
- 1.2 Analytic Structures
- 1.3 Hyperbolic Unfoldings
- 2 Potential Theoretic Results
- 2.1 Local Maximum Principles and Harnack Inequalities
- 2.2 Global Results from Resolvent Equations and Bounded Geometry
- 2.3 Hyperbolicity and Boundary Harnack Inequalities
- 2.3.1 Global Behaviour: Phi-Chains
- 2.3.2 Boundary Harnack Inequality
- 2.3.3 The Hyperbolic Martin Boundary
- 2.3.4 Sharpness
- 2.4 Ray Expansion of Harmonic Functions
- 2.5 Application to Uniform Spaces
- II Isoperimetry and Bubbles
- 3 Weighted Linear Isoperimetric Inequalities
- 3.1 Laplacian and Harmonic Measure
- 3.2 Onion Cover
- 3.3 A Weighted Mesoscale Friedrichs Inequality
- 3.4 Infinitesimal Friedrichs and Isoperimetric Inequalities
- 3.5 Examples for Weight Functions and Generalisations
- 3.6 Local Sobolev Inequalities from Friedrichs Inequalities
- 4 Bubbles and Mean Convex Exhaustion
- 4.1 Regularity of Bubbles
- 4.2 Mean Convexity at Infinity
- 4.3 Mean Convexity from Isoperimetric Inequalities
- 5 Application to the Conformal Laplacian
- A Generalised Hypersurfaces
- Bibliography