Measuremetric boundary and Liouville theorem in Alexandrov geometry
Introductory Workshop: Modern Riemannian Geometry January 18, 2016  January 22, 2016
Location: SLMath:
Tags/Keywords
differential geometry
Riemannian geometry
modern geometry
curvature
curvature estimates
Ricci curvature
Ricci curvature lower bounds
measuremetric spaces
Liouville measure
Alexandrov spaces
geodesic flow
14438
I will introduce the notion of measuremetric boundary on measuremetric 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 measuremetric 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 measuremetric boundary.
Kapovitch

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