Dupin Cyclides are Not of $$L_{1}$$ L 1 -Finite Type

Abstract

In this paper, we prove that the Dupin cyclides are not of \(L_1\)-finite type. An isometric immersed surface \(\psi : M\rightarrow \mathbb {E}^3 \) is said to be of \(L_1 \)-finite type if \(\psi =\sum _{i=0} ^k\psi _i\) for some positive integer k, \(\psi _i:M \rightarrow \mathbb {E}^3 \) is smooth and \(L_1\psi _i=\lambda _i\psi _i\), \(\lambda _i \in \mathbb {R}\), \(0 \le i \le k\), \(L_1(f )=\text {div}(P_1(\nabla f))\) for \(f \in \mathcal {C}^ \infty (M)\), \(L_1\psi =(L_1\psi _1, L_1\psi _2,L_1\psi _3), ~\psi =(\psi _1, \psi _2, \psi _3)\).