Measure-metric boundary and Liouville theorem in Alexandrov geometry

Tags/Keywords

• differential geometry

• Riemannian geometry

• modern geometry

• curvature

• curvature estimates

• Ricci curvature

• Ricci curvature lower bounds

• measure-metric spaces

• Liouville measure

• Alexandrov spaces

• geodesic flow

Primary Mathematics Subject Classification

• 01-01 - Introductory exposition (textbooks, tutorial papers, etc.) pertaining to history and biography

• 51Lxx - Geometric order structures [See also 53C75]

• 51A30 - Desarguesian and Pappian geometries

• 51D10 - Abstract geometries with exchange axiom

• 51D15 - Abstract geometries with parallelism

• 51A40 - Translation planes and spreads in linear incidence geometry

Secondary Mathematics Subject Classification No Secondary AMS MSC

I will introduce the notion of measure-metric boundary on measure-metric spaces and discuss various motivational examples. I will then describe some recent joint work on measuremetric boundary with Alexander Lytchak and Anton Petrunin.  We show that if an Alexandrov space has a zero measure metric boundary then for almost every point in almost every direction there exists an infinite geodesic and the geodesic flow preserves the Liouville measure. We conjecture that any Alexandrov space without boundary has zero measure-metric boundary and hence the above result should hold for all such spaces. While we can not prove it in general we show that this is true for convex hypersurfaces in smooth manifolds. We also show that finite dimensional Alexandrov spaces have finite measure-metric boundary.