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We consider the quantum correction to the Lagrangean by the massless freeboson in the curved background in three dimensions where one of the coordinatesis periodic. The correction term is given by an expansion of the metric withrespect to the derivative and the first term expresses to the usual Casimirenergy. As an application, we investigate the change of the geometry in threedimensional black hole due to the quantum effect and we show that the geometrybecomes like that of the Reissner-Nordstr\o m solution.Comment: Latex file, 10 page

Casimir Effect in the Curved Background and the Black Hole in Three Dimensions