Relationship between the volume-ratio ε-regularity theorem and backward pseudolocality

Determine the precise relationship between the volume-ratio ε-regularity theorem for Ricci flows with a locally type-I scalar-curvature bound and Bamler’s backward pseudolocality theorem.

Background

The paper establishes an ε-regularity theorem asserting that a Ricci flow satisfying a locally type-I scalar-curvature bound and sufficiently strong local volume-ratio lower bounds has a quantitative lower bound for its curvature radius. The authors note that this result resembles Bamler’s backward pseudolocality theorem, which provides a different form of curvature control under hypotheses involving backward-time geometric and entropy conditions.

The precise logical or quantitative relationship between the two results is not established in the paper. Clarifying whether one theorem implies, generalizes, or can otherwise be derived from the other would place the paper’s volume-ratio regularity result within the broader theory of Ricci-flow pseudolocality.