Graphical translating solitons for the mean curvature flow and isoparametric functions (Q6619629)

In this interesting paper, the author considers the case of a translating soliton for the mean curvature flow starting from a graph of a function on a domain in a unit sphere which is constant along each leaf of isoparametric foliation.\N\NFirst, it is proved that such a function is given as a composition of an isoparametric function on the sphere and a function which is given as a solution of a certain ordinary differential equation.\N\NSubsequently, analysing the shape of the graphs of the solutions of the ordinary differential equation, a full classification of the shape of such translating solitons is given.\N\NLastly, the author investigates the case where the number of distinct principal curvatures of the isoparametric hypersurface defined by the regular level set for the isoparametric function is \(1\), \(2\), or \(3\), and he studies a domain of the function which is given as a composition of the isoparametric function and the solution of the ordinary differential equation.