Two-sided non-collapsing curvature flows (Q2807109)

The authors investigate the flows of embedded \(n\)-dimensional hypersurfaces in a complete, simply connected space form with the velocity \(\partial_tX(x,t)\) being equal to \(-F(x,t)\nu(x,t)\), where \(\nu\) is a choice of unit normal to the evolving hypersurface \(X\), and the speed is \(F=f(\kappa_1,\dots,\kappa_n)\), where \(f\) is a smooth, symmetric, degree-one homogeneous function defined on the positive cone in \(\mathbb{R}^n\) and is monotone increasing in each argument, and \(\kappa_i\) are the principal curvatures of \(X\). They show that if in addition \(f\) is inverse-concave, then \(X\) is exterior non-collapsing.NEWLINENEWLINEThis extends some previous results of the first author and his collaborators in [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 1, 23--32 (2013; Zbl 1263.53059); Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 1, 331--338 (2015; Zbl 1322.53068)]. The proof is based on techniques used also in [Zbl 1263.53059; Zbl 1322.53068] and a new observation. (Some relevant techniques were surveyed by the first author in [in: Regularity and evolution of nonlinear equations. Somerville, MA: International Press. 1--47 (2015; Zbl 1329.53053)], and by \textit{S. Brendle} in [Bull. Am. Math. Soc., New Ser. 51, No. 4, 581--596 (2014; Zbl 1432.53131)].)NEWLINENEWLINECombined with some previous results of the first author and his collaborators in [Zbl 1263.53059; Zbl 1322.53068], the main result of this paper implies that the curvature flows of hypersurfaces in space forms by concave, inverse-concave speed functions are two-sided non-collapsing. The authors also give some applications.