The starting point of Perelman's proof is the observation that (a modified version of) the Ricci flow is a gradient flow. The connection to 2d non-linear sigma

The starting point of Perelman's proof is the observation that (a modified version of) the Ricci flow is a gradient flow. The connection to 2d non-linear sigma models is briefly discussed in section 2 of the introduction in https://arxiv.org/pdf/math/0211159.pdf https://arxiv.org/pdf/math/0211159.pdf and section 1.4*. The overall intuition is that string theory captures the geometry of the back ground Riemannian manifold.