#### Bulg. J. Phys. vol.33 no.s3 (2006), pp. 091-108

#### Geometric Flows and (Some of) Their Physical Applications

I. Bakas

go back^{1,2}^{1}*Theory Division, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland*^{2}*Department of Physics, University of Patras, GR-26500 Patras, Greece*Abstract.The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis of non-linear sigma models and in general relativity. They are divided into classes of intrinsic and extrinsic curvature flows. Here, we review the main aspects of intrinsic geometric flows driven by the Ricci curvature, in various forms, and explain the intimate relation between Ricci and Calabi flows on Kähler manifolds using the notion of super-evolution. The integration of these flows on two-dimensional surfaces relies on the introduction of a novel class of infinite dimensional algebras with infinite growth. It is also explained in this context how Kac'sKsimple Lie algebra can be used to construct metrics on_{2}Swith prescribed scalar curvature equal to the sum of any holomorphic function and its complex conjugate; applications of this special problem to general relativity and to a model of interfaces in statistical mechanics are also briefly discussed.^{2}