Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow
Let \(x_{t}:M^{n}\rightarrow \mathbb{R}^{n+1}\) be a one-parameter family of smooth immersions and \(M_{t}=x_{t}(M^{n}).\) In the case when \(x_{t}\) satisfies the evolution equation \(\frac{d}{dt}x(p,t)=-H(p,t)\nu (p,t),\) \((t>0),\) where \( \nu (p,t)\) denotes a choice of unit normal to \(M_{t}\) at \(x(p,t)\), \(H(p,t)\) denotes the mean curvature with respect to this normal, it is said that the hypersurfaces \(M_{t}\) move by mean curvature. The mean curvature flow was studied by \textit{K. A. Brakke} [The motion of a surface by its mean curvature. (Princeton, New Jersey) (1978; Zbl 0386.53047)]. For compact, convex initial surfaces without boundary, \textit{G. Huisken} [J. Differ. Geom. 20, 237--266 (1984; Zbl 0556.53001)] proved that they converge asymptotically to round spheres. The question of the formation of singularities in the nonconvex case was analyzed by \textit{G. Huisken} [J. Differ. Geom. 31, No. 1, 285--299 (1990; Zbl 0694.53005)], \textit{M. A. Grayson} [Duke Math. J. 58, No. 3, 555--558 (1989; Zbl 0677.53059)], \textit{G. Dziuk} and \textit{B. Kawohl} [J. Differ. Equations 93, No. 1, 142--149 (1991; Zbl 0749.53001)]. The author is interested in the evolution of compact hypersurfaces \(M_{t}\) enclosing a prescribed constant volume \(V\). This means that \(x_{t}\) satisfies the following evolution equation: \(\frac{d}{dt}x(p,t)= -(H(p,t)-h(t))\nu (p,t),\) \((t>0)\), where \(h(t)=( \int_{M}\,dg_{t}) ^{-1}\cdot \int_{M}H \,dg_{t}\) is the average of the mean curvature, and \(g_{t}\) denotes the metric on \(M_{t}\). The initial hypersurface \(M_{0}\) is supposed to be smoothly embedded in the domain \(G=\{x\in R^{n+1}:0<x_{n+1}<d\},\) \((d>0)\) and \(\partial M_{0}\subset \partial G.\) The surface areas of \(M_{t}\) are decreasing under this flow, while the enclosed volume is constant. In this case, the hypersurfaces can be expected to converge to a surface of constant mean curvature which solves the isoperimetric problem. The author imposes volume constraints but takes a free boundary. Since in solving the isoperimetric problem using methods of calculus of variations, the minimizers prove to be surfaces of revolution intersecting the obstacle at a right angle [see \textit{M. Athanassenas}, J. Reine Angew. Math. 377, 97--107 (1987; Zbl 0604.53003), and \textit{M. Athanassenas}, Comment. Math. Helv. 72, No. 1, 52--66 (1997; Zbl 0873.35033)], the author assumes that the initial surface is rotationally symmetric and that the hypersurfaces \(M_{t}\) intersect \(\partial G\) orthogonally at the boundary. The author is concerned with the formation of singularities for the flow described above. This problem has been treated in the literature in the case of the mean curvature flow without a volume constraint [see \textit{S. Altschuler}, \textit{S. B. Angenent} and \textit{Y. Giga}, J. Geom. Anal. 5, No. 3, 293--358 (1995; Zbl 0847.58072)]. In section 1, a basic estimate for \(h\) is established. This is the key point in the paper, because it allows the author to use techniques similar to those in mean curvature flows. Also, the author controls the number of zeroes of the derivative of the radius function which describes the surface of rotation by means of a Sturmian theorem. In section 2 the fact is proved that ``too thin'' necks have to pinch-off, the flow thus develops singularities in finite time. Also, these necks will converge to single points along the axis of rotation, so that the singularities form a finite, discrete set on that axis. The asymptotic behaviour of singularities is analyzed in the last section. It is proved that in the case of the so-called type I singularities, after rescaling, the rescaled flow converges to a cylinder.
scientific article; zbMATH DE number 1969049