Minimal hypersurfaces with zero Gauss-Kronecker curvature

Abstract

We investigate complete minimal hypersurfaces in the Euclidean space R 4 , with Gauss-Kronecker curvature identically zero. We prove that, if f : M 3 → R 4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then f (M 3 ) splits as a Euclidean product L 2 × R, where L 2 is a complete minimal surface in R 3 with Gaussian curvature bounded from below.2000 Mathematics Subject Classification. 53C42.

The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

“…Combining Theorems 1.2 and 1.4 with their results we obtain the following statements. We point out that Theorem 1.5(i) and (ii) are proved in [6,7], respectively, and (ii) of Theorem 1.6 is proved in [8].…”

“…In Lemma 2.4 we obtain an expression for the Laplacian of the function F = log |K | and as consequences we show that function F = log |K | is superharmonic if c = 0 and strongly superharmonic when c < 0. We also state the local version of the results of Cheng [2] and Hasanis et al [6]. Summarizing we have Theorem 1.1 Let M 3 be a connected three-dimensional Riemannian manifold minimally immersed in Q 4 (c), c ≤ 0, with nowhere vanishing Gauss-Kronecker curvature K .…”

“…In particular, Cheng proved, in [2], that a complete minimal hypersurface in Q 4 (c), c ≤ 0, with scalar curvature R bounded from below and constant Gauss-Kronecker curvature K , must have K identically zero. For complete minimal hypersurfaces in R 4 (c = 0) with constant K , Hasanis et al [6] , by using the Principal Curvature Theorem of Smyth and Xavier [13], were able to remove the hypothesis on the scalar curvature and arrived to the same conclusion.…”

“…Hasanis et al [6][7][8], gave examples and a classification of complete minimal hypersurfaces in Q 4 (c) with K identically zero. Combining Theorems 1.2 and 1.4 with their results we obtain the following statements.…”

“…Recently there has been a great interest in studying the complete minimal hypersurfaces in a space form Q 4 (c) with constant Gauss-Kronecker curvature, see the very interesting papers of Cheng [2] and Hasanis et al [6][7][8]. In particular, Cheng proved, in [2], that a complete minimal hypersurface in Q 4 (c), c ≤ 0, with scalar curvature R bounded from below and constant Gauss-Kronecker curvature K , must have K identically zero.…”