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We study fully nonlinear geometric flows that deform strictly $k$-convexhypersurfaces in Euclidean space with pointwise normal speed given by a concavefunction of the principal curvatures. Specifically, the speeds we consider areobtained by performing a nonlinear interpolation between the mean and the$k$-harmonic mean of the principal curvatures. Our main result is a convexityestimate showing that, on compact solutions, regions of high curvature areapproximately convex. In contrast to the mean curvature flow, the fullynonlinear flows considered here preserve $k$-convexity in a Riemannianbackground, and we show that the convexity estimate carries over to thissetting as long as the ambient curvature satisfies a natural pinchingcondition.

Convexity estimates for hypersurfaces moving by concave curvature functions