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We consider the local uniqueness of a catenoid under the condition for the Gaussian curvature analogous to Bianchi surfaces. More precisely, if a nonplanar minimal surface in R3 has the 1 Gaussian curvature K = − (U (u)+V (v))2 for any functions U (u) and V (v) with respect to a line of curvature coordinate system (u, v), then it is part of a catenoid. To do this, we use the relation between a conformal line of curvature coordinate system and a Chebyshev coordinate system.
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