The Ricci flows perturbation equations and the Weyl anomaly coefficients are derived in the two-dimensional bosonic string sigma model. These equations correspond to the two-loop flow equations for the graviton field g(mu nu) and the dilaton field Phi. The one-loop perturbation solution of the cigar soliton can be expressed in terms of the hypergeometric functions. The two-loop asymptotic perturbation solution of the cigar soliton is reduced by using a small parameter expansion method. Moreover, analytic solutions of the second basic form l and n are obtained in accordance with the perturbation Gauss-Codazzi equations. The modified expression of the deformed principal curvatures of a two-dimensional surface can then be given in terms of l and n. The influence of quantum Ricci flows on the space-time geometry is analyzed and discussed, and the physical meaning of the Weyl anomaly coefficients varying with the momentum scale lambda is also explained.
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