Higher regularity of the inverse mean curvature flow
The authors study the inverse mean curvature flow (IMCF) dealing with a closed hypersurface \(F: M^n\times [0,T]\to \mathbb{R}^{n+1}\), whose evolution in time is governed by the equation \({\frac{dF}{dt} = \frac{1}{H} \nu}\), where \(H\) and \(\nu\) are the mean curvature and exterior unit normal of \(F\). It is known that if \(F_0=F(M^n,0)\) is a smooth star-shaped hypersurface of strictly positive mean curvature then IMCF has a smooth solution for all times which approaches a homothetically expanding spherical solution as \(t\to \infty\), see [\textit{C. Gerhardt}, J. Differ. Geom. 32, No. 1, 299--314 (1990; Zbl 0708.53045); \textit{J. I. E. Urbas}, Math. Z. 205, No. 3, 355--372 (1990; Zbl 0691.35048)]. On the other hand, if the initial hypersurface \(F_0\) is not star-shaped, then the singularities may develop; moreover it is proved by \textit{K.~Smoczyk} that in the case \(n=2\) such singularities can occur if the mean curvature \(H\) tends to zero somewhere during the evolution [Asian J. Math. 4, No. 2, 331--335 (2000; Zbl 0989.53040)]. The main result of the paper is the following lower bound for the mean curvature of star-shaped solutions to IMCF. Theorem. Let \(F: M^n\times [0,T]\to \mathbb{R}^{n+1}\) be a smooth star-shaped solution to IMCF. Suppose that the following estimates hold at \(t=0\): \[ 0<R_1\leq \langle F,\nu\rangle \leq R_2. \] Then there is a constant \(0<C_n<\infty\) depending only on \(n\) such that the estimates \[ \frac{1}{H\langle F,\nu\rangle}\leq C_n\max\left(\frac{1}{\sqrt{t}},1\right)\frac{1}{R_1}| F_0|^{1/n} \] and \[ H\geq \frac{1}{C_n}\min\left(\sqrt{t},1\right) e^{-t/n}\frac{R_1}{R_2}| F_0|^{-1/n} \] hold everywhere on \(M^n\times [0,T]\). Remark that if \(F\) is star-shaped and satisfies \(0<R_1\leq \langle F,\nu\rangle \leq R_2\) at \(t=0\), then during the evolution under IMCF it remains star-shaped and satisfies \(e^{t/n}R_1\leq \langle F,\nu\rangle \leq R_2e^{t/n}\) [cf Zbl 0708.53045]. Thus in the star-shaped setting \textsl{the mean curvature has to grow at least of order \((t-t_0)^{1/2}\) when starting from a hypersurface of nonnegative mean curvature at time \(t_0\).} Besides the authors prove an original estimate for the full second fundamental form of hypersurfaces under IMCF: Theorem. Let \(F: M^n\times [0,T)\to \mathbb{R}^{n+1}\) be a smooth solution of IMCF satisfying uniform bounds \( 0<H_1\leq H \rangle \leq H_2\). Then the full second fundamental form satisfies an estimate \[ | A| \leq c_n \frac{H_2^2}{H_1}\frac{1}{t} \] everywhere on \(M^n\times [0,T)\). This extends results of K.~Smoczyk to all dimensions and leads to the following blowup criterion for IMCF. Theorem. Let \(F: M^n\times [0,T)\to \mathbb{R}^{n+1}\) be a smooth solution to IMCF with \(H>0\), \(0<T<\infty\). If the mean curvature \(H\) remains bounded from below by a constant \(H_1>0\) for all \(t\in (0,T)\), the solution can be extended beyond \(T\). In particular, if \([0,T)\), \(T<\infty\), is the maximal time interval of existence for a smooth solution to IMCF, then the speed \(1/H\) is unbounded for \(t\to T\). Finally, the results obtained are applied to weak (level set) solutions to IMCF [cf. J. Differ. Geom. 59, No. 3, 353--437 (2001; Zbl 1055.53052)]. It is shown that one can construct smooth solutions to IMCF for star-shaped \(C^1\)-smooth initial hypersurfaces with positive weak mean curvature. Moreover, every weak solution to IMCF is regular after the first instant \(t_0\) where a level set \(F_{t_0}\) is star-shaped; this implies that weak solutions to IMCF are regular outside compact sets.
scientific article; zbMATH DE number 5504164