Blaschke's problem for hypersurfaces (Q2466356)

Blaschke's problem for hypersurfaces is a very deep problem: which pairs of hypersurfaces \(f,\widetilde f: M^n\to\mathbb{R}^{n+1}\) envelop a common regular sphere congruence and induce conformal metrics on \(M^n\)? In the excellent work the authors give a nice proof of the following main theorem: Let \(f,\widetilde f: M^n\to\mathbb{R}^{n+1}\), \(n\geq 3\), be a nontrivial solution of Blaschke's problem. Then \(f(M)\) and \(\widetilde f(M)\) are, up to a Möbius transformation of \(\mathbb{R}^{n+1}\), open subsets of one of the following: (i) A cylinder over a plane curve. (ii) A cylinder \(C(\gamma)\times\mathbb{R}^{n-2}\), where \(C(\gamma)\) denotes the cone over a curve \(\gamma\) in \(\mathbb{S}\subset\mathbb{R}^3\). (iii) A rotation hypersurface over a plane curve. Conversely, for any hypersurface \(f: M^n\to\mathbb{R}^{n+1}\) that differs by a Möbius transformation of \(\mathbb{R}^{n+1}\) from a hypersurface as in each of the preceding cases there exists \(\widetilde f: M^n\to\mathbb{R}^{n+1}\) of the same type as \(f\) such that \((f,\widetilde f)\) is a nontrivial solution of Blaschke's problem. Moreover, \(\widetilde f\) is a Darboux transform of \(f\). The proof needs a lot of lemmas and has interesting relations with papers of X. Ma, C. L. Terg, T. Vlachos, H. Reckziegel and the authors.