Immersions with fractal set of points of zero Gauss-Kronecker curvature (Q2576706)

Let \(M^n\) be an orientable hypersurface of \(\mathbb R^{n+1}\) and let \(N: M^n\to S^n\) be the associated Gauss map in the given orientation. Then, \(A= dN\) is selfadjoint and its eigenvalues are the principal curvatures, \(H_n= \text{det}(dN)\) is the Gauss-Kronecker curvature and \(\text{rank}(N)= \min_{p\in M}(\text{rank\,}d_pN)\). A compact set \(F\subset S^n\) is called a good Cantor set if \(S^n- F= \bigcup_{i\in M}U_i\) is the disjoint union of open balls \(U_i\) in \(S^n\) of radius bounded by a small constant. The authors main theorem is: For any \(F\subset S^{n-1}\) a good Cantor set, there are immersions \(x: S^n\to \mathbb{R}^{n+1}\) such that \(\text{rank}(x)= n-1\). The Gauss-Kronecker curvature is nonnegative and \(\{p\in S^n: H_n(p)= 0\}= F\times D^1\), where \(D^1\) is the one-dimensional disk.