Inverse mean curvature flow with forced term
This paper concerns an inverse mean curvature flow \textit{with forced term} for convex hypersurfaces in \(\mathbb R^{n+1}\), when the evolution of hypersurfaces, \(F(\cdot, t): S^n\to \mathbb R^{n+1}\), is governed by the equation \[ \frac{dF}{dt} = \left( \frac{1}{H} - c\right)\nu, \] where \(H\) stands for the mean curvature of the evolving hypersurface with respect to the outer unit normal vector \(\nu\), and \(c\) is a constant. The main results are the following. {Theorem 1.} Suppose the initial hypersurface \(M_0 = F(S^n, 0)\) in \(\mathbb R^{n+1}\) is strictly convex. Then the strict convexity of the hypersurface is preserved under the evolution. {Theorem 2.} Suppose the initial hypersurface \(M_0\) in \(\mathbb R^{n+1}\) is strictly convex. If \(c<\frac{1}{H}\) on \(M_0\), then the flow in question will expand the hypersurface for all time and \(\tilde M_t = \;e^{\frac{1}{n}t}\, F(S^n,t)\) will converge in \(C^\infty\)-topology to a sphere. If \(c>\frac{1}{H}\) on \(M_0\), then the flow will contract the hypersurface to a point in finite time. To prove the convexity of the evolving hypersurface, the author applies methods used in [\textit{F. Schulze}, Math. Z. 251, No. 4, 721--733 (2005; Zbl 1087.53062)]. Moreover, to prove that the evolving hypesurface expands under the flow and converges to a sphere after rescaling, in the case of \(c<\frac{1}{H}\) the author applies the method of support function used in [\textit{J. I. E. Urbas}, J. Differ. Geom. 33, No. 1, 91--125 (1991; Zbl 0746.53006); corrigendum ibid. 35, No. 3, 763--765 (1992)].
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