Volume-preserving flow by powers of the \(m\text{-th}\) mean curvature in the hyperbolic space (Q1688578)

The authors consider a general class of volume-preserving flows and give a condition under which hypersurfaces converge to round spheres. An early result of this type is Huisken's result that all convex hypersurfaces converge to round spheres under volume preserving mean curvature flow [\textit{G. Huisken}, J. Reine Angew. Math. 382, 35--48 (1987; Zbl 0621.53007)]. However, the flows considered by the authors are more general. Mean curvature is the sum of the principal curvatures; the authors consider more generally the \(m\)th mean curvature \(H_m\), defined as the \(m\)th elementary symmetric polynomial of the principal curvatures, which the authors then raise to the power \(\beta\), where \(\beta\) is a fixed constant at least \(\frac1m\). Such flows were previously considered in [\textit{E. Cabezas-Rivas} and \textit{C. Sinestrari}, Calc. Var. Partial Differ. Equ. 38, No. 3--4, 441--469 (2010; Zbl 1197.53082)] for hypersurfaces in Euclidean space. The main innovation of the authors is to replace Euclidean space by hyperbolic space with constant sectional curvature \(-a^2\), where the flow behaves quite differently in some ways. In [loc. cit.], Cabezas-Rivas and Sinestrari showed that a hypersurface converges to a round sphere under the condition that the initial surface is convex and close to being round in the sense that the ratios between the arithmetic and geometric means of the principal curvatures are bounded by a fixed constant. The authors prove the corresponding result in a hyperbolic ambient space, with the principal curvatures \(\lambda_i\) replaced by \textit{shifted} principal curvatures \(\tilde\lambda_i:=\lambda_i-a\).