The mixed volume preserving mean curvature flow
The author studies a class of mean curvature flows he calls ``mixed volume'' flows, which encompasses both the surface area preserving flow that he previously considered in [Asian J. Math 7, No. 1, 7--30 (2003)], and the standard volume preserving flow [\textit{G. Huisken}, J. Reine Angew. Math. 382, 35--48 (1987; Zbl 0621.53007)]. He proves that each flow has a smooth solution for all times \(0\leq t<\infty\), and that the hypersurfaces converge in \(C^{\infty}\) to a sphere with the same ``mixed volume'' that \(M_{0}\) has. More precisely, let \(M_{0}\) be a compact, strictly convex hypersurface, without boundary, of dimension \(n\geq2\) that is smoothly embedded in \(\mathbb{R}^{n+1}\), and is represented locally by a diffeomorphism \(F_{0}.\) The flows are given by \[ {\frac{\partial}{\partial t}}F(x,t)=({h_{k}(t)-H(x,t)})\nu(x,t), \] \[ F(\;,0)=F_{0}, \] \(k=-1,0,\dots,n-1,\) where \(H\) is mean curvature of \(M_{t}=F_{t}(M),\) \(\nu\) is the outer unit normal to \(M_{t}\), and \[ h_{k}(t)={\frac{\int_{M_{t}}HE_{k+1}\, d\mu}{\int_{M_{t}}E_{k+1}\,d\mu}.} \] Here \(E_{l}\) is the \(l\)-th symmetric function of the principal curvatures of \(M_{t}\). This is the usual volume preserving mean curvature flow when \(k=-1\), and the surface area preserving mean curvature flow when \(k=0\). The ``mixed volume'' is the quantity \[ V_{n-k}=\begin{cases} \text{Vol}(\phi_{t}) & k=-1\\ \left\{(n+1)\begin{pmatrix} n\\ k \end{pmatrix} \right\} ^{-1}\int_{M_{t}}E_{k}\,d\mu & k=0,1,\dots,n-1 \end{cases}, \] where \(\phi_{t}\) is the region contained inside of \(M_{t}.\) The associated flow preserves the ``mixed volume'' for each \(k\). In the proof the author uses the Aleksandrov-Fenchel inequality to bound the ``mixed volumes'', which together with a maximum principle argument on the support function, yields a bound on \(H\).
scientific article; zbMATH DE number 2078058