On open flat sets in spaces with bipolar comparison
Nina Lebedeva
TL;DR
This paper proves that in certain Riemannian manifolds, the presence of an open flat subset implies the entire manifold is flat, establishing the strength of (3,3)-bipolar comparison over Alexandrov comparison.
Contribution
It demonstrates that (3,3)-bipolar comparison implies flatness if an open flat subset exists, and compares its strength to Alexandrov comparison.
Findings
(3,3)-bipolar comparison implies flatness with open flat subset
The same result holds for a modified MTW condition without perpendicularity
(3,3)-bipolar comparison is strictly stronger than Alexandrov comparison
Abstract
We show that if a Riemannian manifold satisfies (3,3)-bipolar comparisons and has an open flat subset then it is flat. The same holds for a version of MTW where the perpendicularity is dropped. In particular we get that the (3,3)-bipolar comparison is strictly stronger than the Alexandrov comparison.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Point processes and geometric inequalities · Numerical methods in inverse problems