Tohoku Mathematical Journal
2024
September
SECOND SERIES VOL. 76, NO. 3

Tohoku Math. J.
76 (2024), 391-410

Title GRAPHICAL TRANSLATING SOLITONS FOR THE MEAN CURVATURE FLOW AND ISOPARAMETRIC FUNCTIONS

Author Tomoki Fujii

(Received October 14, 2022, revised January 10, 2023)
Abstract. In this paper, we consider 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. First, we show 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. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons. Finally, we investigate a domain of the function which is given as a composition of the isoparametric function and the solution of the ordinary differential equation in 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.

Mathematics Subject Classification. Primary 53E10.
Key words and phrases. Mean curvature flow, translating soliton, isoparametric function.

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