Quantitative stability of the sharp estimates

Establish quantitative stability versions of the optimal extinction-time estimate, the optimal arrival-time-function estimates, and the optimal parabolic isoperimetric estimate, quantifying how closeness to equality forces closeness to the shrinking round sphere or round-ball configuration.

Background

The paper proves sharp inequalities and characterizes their equality cases by the round ball and the multiplicity-one smoothly shrinking round sphere. It does not provide quantitative rigidity or stability estimates measuring the deficit in these inequalities against a geometric distance from the equality configuration. The authors explicitly identify quantitative stability for all three principal theorems as an unresolved problem.