The sigma model (dual) representation for a two-parameter family of integrable quantum field theories

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

“…We shall give a detailed presentation of Fateev's examples of ancient solutions on S 3 [18] and discuss their properties. First we start with a parametrization of S 3 .…”

“…The example of Fateev [18], which will be presented here in detail together with a thorough analysis of its properties, provides a negative answer to the last two questions raised above in dimension three. In [18], the solution was described by a set of functions given with explicit formulae.…”

“…In this paper we present examples of ancient solutions on spheres (as well as some other compact manifolds). The three dimensional example on S 3 is originally due to Fateev [18]. Because of many interests in this example, we add here details (in the appendix of this paper) to make it more accessible to mathematicians.…”

“…In [18], the solution was described by a set of functions given with explicit formulae. It is by no means easy to check that these functions indeed give an ancient solution to Ricci flow.…”

“…Extra analytic part is needed here since we have to solve the Ricci flow equation, which is reduced to a nonlinear ODE system, instead of Einstein equation, which is equivalent to a quadratic algebraic equation in this formulation. When solving the nonlinear system equivalent to the Ricci flow equation, instead of finding the solutions explicitly as in [18], we prove the global existence by finding a first integral, since the ODE system involved in the high dimensional case of this paper seems a bit more complicated than Fateev's three dimensional case. Here, we merely focus on solutions on spheres and the total space of the generalized Hopf fibrations, even though some of the techniques can be adapted to other manifolds.…”