Circumcenter extension maps for non-positively curved spaces

Area of Science:

• Geometric analysis
• Topology
• Differential geometry

Background:

• Hadamard manifolds are complete non-positively curved Riemannian manifolds.
• Cross ratio preserving homeomorphisms relate to conformal geometry.
• Understanding maps between manifold boundaries is crucial.

Purpose of the Study:

• To extend cross ratio preserving homeomorphisms on Hadamard manifold boundaries.
• To analyze the properties of this extension, termed circumcenter extension.
• To investigate conditions for Hölder continuity, rough isometries, and isometries.

Main Methods:

• Utilizing visibility conditions on Hadamard manifolds.
• Describing regions for Hölder continuity.
• Analyzing cocompact group actions for rough isometry properties.
• Improving quasi-isometry constants for 2D manifolds.

Main Results:

• Every cross ratio preserving homeomorphism extends to a circumcenter extension under visibility conditions.
• The map is Hölder-continuous on specific regions.
• The map is a rough isometry for manifolds with cocompact group actions.
• Improved quasi-isometry constants for 2D Hadamard manifolds.
• A sufficient condition for the map to be an isometry on Hadamard surfaces.

Conclusions:

• The circumcenter extension is a robust tool for studying Hadamard manifold boundaries.
• The study provides new insights into the geometric properties of these extensions.
• Results contribute to the understanding of geometric structures and mappings in non-positively curved spaces.