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Abstract
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We estimate the upper bound of volume of a closed positively or nonnegatively curved
Alexandrov space
with strictly convex boundary. We also discuss the equality case. In particular, the
boundary conjecture holds when the volume upper bound is achieved. Our theorem
can also be applied to Riemannian manifolds with nonsmooth boundary, which
generalizes Heintze and Karcher’s classical volume comparison theorem. Our main tool
is the gradient flow of semiconcave functions.
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Keywords
Alexandrov space, volume comparison, convex boundary,
gradient flow
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Mathematical Subject Classification
Primary: 53C23
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Milestones
Received: 1 November 2020
Revised: 8 July 2021
Accepted: 14 August 2021
Published: 10 November 2021
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