Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces

Abstract

We give a finite negative-shift criterion for preserving a prescribed exterior noncollapsing bound under fully nonlinear curvature flows whose hypersurfaces need not be convex. The criterion uses the optimized two-point expression of Andrews and Langford. We verify it by a uniform global two-matrix estimate for the explicit speed \[ F_{\varepsilon}(A)= tr A- \varepsilon\frac{\norm{A^{\circ}}^{2}}{ tr A} \] on the open cone \[ Γ=\{A\in\Sym(n): tr A>0, \norm{A^{\circ}}< tr A\}.\] Here $A^\circ$ denotes the trace-free part of $A$. This cone contains the positive cone and also matrices with a negative eigenvalue, while the displayed speed is concave and non-convex. For an explicit range of parameters depending on the initial exterior bound, the cone and that bound are preserved throughout the smooth embedded existence interval. Together with the known interior estimate for concave speeds, this gives two-sided tangent-ball bounds for this nonlinear family, including closed nonconvex initial hypersurfaces.