Minimal hypersurfaces ofS 4 with constant Gauss-Kronecker curvature

Closed Hypersurfaces of $$\mathbb{S}^4$$ with Two Constant Curvature Functions

“…Moreover, the geodesic radius of such a tube is π/2 and the second fundamental form of Σ in each normal direction is non-zero. This was proved by S. C. Almeida and F. G. B. Brito in[1]. In particular, of S 4 is isometric to the boundary of a tube of geodesic radius π/2 of the Veronese surface of S 4 .…”

“…Saying that H = K ≡ 0 is the same as assuming M 3 to be a minimal hypersurface with Gauss-Kronecker curvature vanishing everywhere. In that case, there are infinitely many non-isoparametric examples of those hypersurfaces (see [1] and [14]). Notice also that there are many non-isoparametric examples of closed hypersurfaces of S 4 where just one of the curvature functions is constant.…”

Complete minimal hypersurfaces in with zero Gauss–Kronecker curvature

“…As a by-product of this approach they were able to describe locally the minimal hypersurfaces in the (n + 1)-dimensional space form whenever the rank of the nullity distribution is constant. Almeida and Brito [2] initiated the study of compact minimal hypersurfaces in the unit sphere S 4 with vanishing Gauss-Kronecker curvature. In fact they proved that such compact hypersurfaces are boundaries of tubes of minimal 2-spheres in S 4 , provided that the second fundamental form never vanishes.…”

Complete minimal hypersurfaces in the hyperbolic space ℍ⁴ with vanishing Gauss-Kronecker curvature

“…Let A be the shape operator associated with a unit normal ξ. Then the principal curvatures of f are 2 on U x ∩ U x and so the local function u introduced in Section 3 can be extended to a smooth global one.…”

“…As a by-product of this approach they were able to locally describe the minimal hypersurfaces in the (n + 1)-dimensional space form whenever the rank of the nullity distribution is constant. Almeida and Brito [2] initiated the study of compact minimal hypersurfaces in the unit sphere S 4 with vanishing Gauss-Kronecker curvature. In fact they proved that such compact hypersurfaces are boundaries of tubes of minimal 2-spheres in S 4 , provided that the second fundamental form never vanishes.…”