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Let Q 4 (c) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss-Kronecker curvature of a complete minimal hypersurface in Q 4 (c), c ≤ 0, whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form Q 4 (c) with constant Gauss-Kronecker curvature K . For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of Q 4 (c) with K constant.

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Let H, K and R be, respectively, the mean curvature, the GaussKronecker curvature and the scalar curvature of a closed hypersurface immersed in the unit sphere S 4 . In this paper we completely classify the closed hypersurfaces of S 4 where any two of H, K and R are constant.

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