Hyperbolic mean curvature flow (Q2378198)

A hypersurface \(M^n\) in \(\mathbb R^{n+1}\) is said to be evolved by the hyperbolic mean curvature flow if its position vector \(\vec r(\cdot, t)\) satisfies the equation \(\displaystyle{\frac{d\vec r^{\,\,2}}{dt^2} = H \vec n}\), where \(n\) and \(H\) stand for the unit inner normal and the mean curvature of \(M^n\), [see \textit{S.-T. Yau}, Asian J. Math. 4, No. 1, 235--278 (2000; Zbl 1031.53004)]. The authors prove that this hyperbolic version of the classical mean curvature flow admits a unique short-time smooth solution. Namely, if \(M^n\) is compact, then there exists a constant \(T>0\) such that the initial value problem \[ \displaystyle{\frac{d\vec r^{\,\,2}}{dt^2} = H \vec n}, \] \[ \vec r(\cdot, 0) =\vec r_0(\cdot),\quad \frac{d\vec r}{dt}(\cdot, 0 ) = \vec r_1(\cdot) \] has a unique smooth solution \(\vec r(\cdot, t)\) for \(t\in [0,T)\), where \(\vec r_1\) is a smooth vector-valued function. Some exact solutions, presented by evolved spheres and cylinders, show that an evolving hypersurface may expand first and then shrink to a point. Moreover, a nonlinear stability of hypersurfaces under the hyperbolic mean curvature flow is discussed, and it is demonstrated that the hyperplane in \(\mathbb R^{n+1}\), \(n>4\), is nonlinearly stable. Finally, the evolution of fundamental forms and curvatures of hypersurfaces under the hyperbolic mean curvature flow is analyzed.