A two-point function, non-collapsing property and pinching estimates for inverse curvature flow in space forms

In this paper, we consider the problem of exterior non-collapsing estimates for a fully nonlinear inverse curvature flow for inverse-concave speed functions in hyperbolic space, Euclidean space and the sphere. We obtain a large class of two-sided non-collapsing inverse curvature flows. Using our non-collapsing estimate, we can derive pinching estimate for inverse curvature flow in space forms, which was proved by Wei (J Geom Anal 29:1555-1570, 2019). Since the convex speed function cannot be used to obtain the exterior non-collapsing estimate for inverse curvature flow, we make use of the method in Andrews and Langford (Ann Scuola Norm Pisa Cl Sci 5:543-560, 2016) and the inverse-concavity of speed function to generalize the exterior non-collapsing theorem to inverse curvature flows. Finally, we will give corresponding inscribed radius estimates for inverse curvature flow in space form, which is an improvement of Theorem 1.1 and Theorem 1.2 in Liu (Nonlinear Anal 155:198-206, 2017) and Theorem 1.1 in Liu and Ju (Commun Pure Appl Anal 16:945-952, 2017).