Numerical Kähler-Ricci soliton on the second del Pezzo

The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kähler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density). In a recent paper [1], Doran, Herzog, Kantor, and the present authors presented several numerical methods for solving the Einstein equation on toric manifolds. We applied those methods to the third del Pezzo surface (CP#3CP, or dP3), which is known by work of Tian-Yau [2, 3] and Siu [4] to admit a unique Kähler-Einstein metric. Two of the numerical methods involved simulation of (normalized) Ricci flow, ∂gμν ∂t = −2Rμν + 2gμν , (1) which is guaranteed by a theorem of Tian-Zhu [5] to converge starting from any Kähler metric in the same class as the Ricci form. In this note we show that simulation of Ricci flow can also be applied effectively to toric manifolds that do not admit a Kähler-Einstein metric but rather a Kähler-Ricci soliton.1 A (shrinking) Kähler-Ricci soliton consists of a Kähler metric gμν and a holomorphic vector field ξ satisfying Rμν = gμν −∇(μξν). (2) Such a metric is a fixed point up to diffeomorphisms of the flow (1), and the Tian-Zhu theorem again guarantees convergence (up to diffeomorphisms) to it. In this paper we focus specifically on the second del Pezzo surface (dP2). The existence and uniqueness of a Kähler-Ricci soliton on dP2 follow from theorems of Wang-Zhu [7] and Tian-Zhu [8, 9] respectively. However, this metric is not known explicitly. We will describe a numerical approximation to it obtained using one of the methods of [1], and explore some of its geometrical properties. (The first del Pezzo surface is also toric, and, like dP2, admits a Kähler-Ricci soliton [10]; however, that soliton is co-homogeneity 1 and is already known in a fairly explicit form [11].) We begin by reviewing some basic facts about toric metrics and dP2. A toric manifold admits two natural coordinate systems: complex coordinates u + iθ, and symplectic coordinates (xi, θ ). The θ are periodic (θ ∼ θ + 2π), and the U(1) isometries (where n is the complex dimension) of any toric metric act by translating them, leaving u and xi fixed. The real parts u of the complex coordinates range over R, and the metric can be expressed in terms of the Kähler potential f(u) as ds = Fij(du du + dθdθ), (3) where Fij = ∂2f ∂ui∂uj . (4) The symplectic coordinates xi range inside a Delzant polytope, which in the case of dP2 is a pentagon, and the metric can be expressed in terms of the symplectic potential g(x) as ds = Gdxidxj +Gijdθ dθ, (5) See [6] for a review of Ricci solitons.