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Dupin cyclides are not of \(L_{1}\)-finite type. (English) Zbl 1410.53012

Summary: In this paper, we prove that the Dupin cyclides are not of \(L_1\)-finite type. An isometrically 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 \(L_1\psi_i=\lambda_i\psi_i\), \(\lambda_i \in \mathbb {R}\), \(0 \leq i \leq k\), \(L_1(f)=\operatorname{div}(P_1(\nabla f))\) for \(f \in \mathcal {C}^ \infty (M)\), \(L_1\psi =(L_1\psi_1, L_1\psi_2,L_1\psi_3)\), \(\psi =(\psi_1, \psi_2, \psi_3)\).

MSC:

53A05 Surfaces in Euclidean and related spaces
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