Strengthening noncollapsing via MinL or systolic bounds

Investigate whether imposing a uniform positive lower bound on MinL(M) (the length of the shortest closed geodesic in a closed minimal surface within M) or on the systole Sys(M) can serve as effective noncollapsing hypotheses that yield stronger control of Sormani–Wenger intrinsic flat limits of 3-manifolds with nonnegative scalar curvature.

Background

The authors note Croke’s theorem implies MinL(M)\ge L_0>0 is stronger than MinA(M)\ge A_0>0, and that MinL bounds provide lower bounds on Gromov filling volumes, quantities related to SWIF distances. Such bounds could prevent convergence to the zero space and disappearing regions, as seen in prior work by Sormani–Wenger and Portegies–Sormani.

They suggest exploring MinL and systolic bounds as potential refinements of MinA to obtain better compactness and stability properties for limits under nonnegative scalar curvature.