[Paper Review] On Ricci solitons of cohomogeneity one

Andrew Dancer, McKenzie Y. Wang|ArXiv.org|Feb 6, 2008

Geometric Analysis and Curvature Flows26 references4 citations

TL;DR

This paper develops a general framework for constructing explicit cohomogeneity one Kähler-Ricci solitons using ODEs derived from symmetry reduction. It produces new complete examples of expanding, steady, and shrinking Kähler-Ricci solitons on manifolds foliated by circle bundles over products of Fano Kähler-Einstein manifolds or coadjoint orbits, with smooth compactifications via prescribed collapsing conditions at special orbits.

ABSTRACT

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansatze of cohomogeneity one type to produce new explicit examples of complete Kahler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kahler-Einstein manifolds or over coadjoint orbits of a compact connected semisimple Lie group.

Motivation & Objective

• To develop a general formalism for cohomogeneity one Ricci solitons using symmetry reduction.
• To unify and generalize existing cohomogeneity one Kähler-Ricci soliton constructions, particularly those involving circle bundles over Fano Kähler-Einstein manifolds.
• To produce new explicit examples of complete Kähler-Ricci solitons of expanding, steady, and shrinking types with non-trivial isometry groups.
• To analyze smooth compactification conditions at special orbits via prescribed collapsing of spheres in the normal bundle.
• To establish consistency conditions for soliton equations under various asymptotic and boundary behaviors, including completeness and conicality.

Proposed method

• Formulate the cohomogeneity one Ricci soliton equations as a system of ordinary differential equations (ODEs) in a transverse variable.
• Use an ansatz where the metric is invariant under a group action with generic orbits of codimension one, reducing curvature equations to ODEs.
• Model the metric as a warped product with circle fibers over a base manifold, with fiber radii and base metrics depending on a single variable.
• Apply the gradient Ricci soliton equation in the form $\mathrm{Ric}(g) + \mathrm{Hess}(u) + \frac{\epsilon}{2}g = 0$ with $u$ a radial function.
• Impose smoothness conditions at special orbits by requiring the normal bundle to collapse to a sphere with round metric, leading to ODE boundary conditions at endpoints.
• Derive consistency conditions such as $E^* = k+1$ and $c_i^2 = 1/(k+1)$ for smooth extension, linking soliton type to geometric parameters.

Experimental results

Research questions

• RQ1 Can a general framework be developed for constructing cohomogeneity one Kähler-Ricci solitons using symmetry reduction?
• RQ2 What are the necessary and sufficient conditions for completeness and smooth compactification of such solitons at special orbits?
• RQ3 How do the asymptotic behaviors (e.g., conical or constant radius) of the solitons depend on the soliton type (expanding, steady, shrinking)?
• RQ4 Can new explicit examples be constructed on circle bundles over arbitrary products of Fano Kähler-Einstein manifolds, even when the base lacks isometries?
• RQ5 What are the precise relations between the soliton constant $\epsilon$, the geometry of the base, and the collapsing parameters at special orbits?

Key findings

• The paper constructs new explicit examples of complete Kähler-Ricci solitons of expanding, steady, and shrinking types on manifolds foliated by circle bundles over products of Fano Kähler-Einstein manifolds.
• For steady solitons with $\kappa_1 < 0$, the metric is complete at infinity and the circle fibers have asymptotically constant radius.
• Expanding solitons with $\kappa_1 < 0$ are asymptotically conical, and shrinking solitons with $\kappa_1 > 0$ are also asymptotically conical.
• Smooth compactification at special orbits is achieved when the collapsing sphere $H/K$ has dimension $k$, with $c_i^2 = 1/(k+1)$ and $E^* = k+1$.
• The consistency condition $E^* = \frac{\epsilon a_i + 2}{b_i}$ ensures the soliton equation is satisfied, and for $\epsilon = 0$, the form $\sum a_i \Theta_{KE}^*|_{\mathfrak{p}_i}$ must be closed.
• Compact shrinking solitons exist when the integral (3.24) vanishes for some choice of $\kappa_1$, and these include cases previously found by other methods.