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A Variational Problem and Classification Theorem for Extremal Hypersurfaces in the Space Form

The classification of the isoparametric extremal hypersurfaces in the space form is obtained in this paper. We also derived the Euler-Lagrange equation for extremal hypersurface and obtained Simons’ type integral inequality. When the integral equality holds, we can obtain the characteristics of isoparametric extremal hypersurfaces by using this classification theorem. The classification of isoparametric extremal hypersurfaces in the space form is the first study that has not been seen in the previous literature.