Cohomogeneity two Ricci solitons with sub-Euclidean volume

TL;DR

This work constructs and classifies new four-dimensional Ricci solitons of cohomogeneity two with volume-collapsing ends by reducing the soliton equation to a degenerate Monge–Ampère equation for the conformal factor coupled to ODEs. The authors develop a robust reduction via auxiliary functions, obtain explicit complete expanding solitons, and establish existence results for shrinking and steady solitons with boundary, including several new singular examples and connections to Schwarzschild and Plebański–Demiański metrics. A central achievement is the rigidity/classification result: local Monge–Ampère solutions determine the soliton geometry, yielding a clear dichotomy between conformally cylindrical bases and conformally flat/scalar-flat cases, and a comprehensive treatment of degenerate/resolvent phenomena. The findings illuminate how volume-collapsing ends and boundary phenomena arise in cohomogeneity-two Ricci solitons and provide a framework potentially usable for gluing-desingularization constructions of new solitons with controlled ends. Overall, the paper expands the landscape of explicit cohomogeneity-two Ricci solitons, linking classical Einstein metrics with novel expanding, shrinking, and singular solitons through a unified Monge–Ampère–ODE approach.

Abstract

We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Ampère equation for the conformal factor coupled with ODEs. We obtain explicit complete expanding solitons as well as abstract existence results for shrinking and steady solitons with boundary. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics. We also classify local solutions of this Monge-Ampère equation to prove rigidity for these solitons.

Figures (4)

• Figure 1: A numerically approximated solution $f$ to ODE (\ref{['eqn:resulting-ODE']}) with initial data $f'(0) = 0$ and $f"(0) = -1$ as given in Proposition \ref{['prop:SexBuffet']}, showing the behavior of such solutions. At the boundary points where $f = 0$, the derivatives of $f$ blow up.
• Figure 2: The shaded regions above represent $\Omega$, the possible domains of definition for the \ref{['metric:SStar']} metrics with $b_0 > 0$. The conformally flat domains ($c = 0$) would be the same, without the constraint $|x| < \sqrt{a_0 / c}$ coming from $A > 0$. The origin bullet represents an asymptotically cylindrical end of the base metric. The solid lines denote $\{q = 0\}$, which are either ends or boundary points if the derivatives of $q$ are finite or infinite, respectively. If $b_0 < 0$, then the regions would be intersected with the half plane where $y > \sqrt{b_0}$.
• Figure 3: Figure (a) shows a conformally scalar flat soliton, (e.g. ${\rm (}$\ref{['cor:SexBuffet']}${\rm )}$ with $c \neq 0$) where the path $y \to \infty$ has finite length. In the conformally flat case (b), (e.g. ${\rm (}$\ref{['cor:SexBuffet']}${\rm )}$ with $c = 0$), this length is infinite. Figure (c) shows paths of either finite or infinite length depending on the value of $f'$ near the boundary. For metrics where $f$ has bounded derivative where it vanishes, these lengths are infinite, so the boundary line is an end. For metrics ${\rm (}$\ref{['cor:SexBuffet']}${\rm )}$ when $c = 0$ or $c \neq 0$, these paths have finite length approaching the boundary where the curvature blows up. Figure (d) shows a path of finite length where the manifold closes up spherically, smoothly if $a_0 = 1$ or with a conical singularity otherwise. We can also approach a point where $A,q$ both vanish as shown in Figure (e), which can either be finite or infinite length. Figure (f) shows a path along with the metric closes up as in \ref{['metric:q=sqrtx2+y2']} solitons with $b_0 < 0$. The $(y,t)$ surface is hyperbolic as shown above in equation \ref{['eqn:metric:x2-b0']}, smoothly if and only if $b_0 = -1$, otherwise this is a conical singularity.
• Figure 4: Two geometric realizations of the boundary for the ${\rm (}$\ref{['cor:SexBuffet']}${\rm )}$ metrics, viewed (a) from the boundary and (b) from the interior. These visualizations exhibit the boundary of the cylinder as the limit of neighboring circles growing to infinity. In Figure (a), the neighboring circles expand by stereographic projection; the blue line represents a curve as in \ref{['fig:offToRadialy=infty']} of finite length. In Figure (b), the concentric circles expand by increasingly condensed corrugation near the boundary to preserve the metric in the radial direction. This behavior is similar to the singularity in the Ooguri-Vafa metric where the circular fibers blow up in finite radius. Our metric singularities can therefore be seen as a cohomogeneity two analogue.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Example 2.1
• Claim 2.2
• proof
• Claim 2.3
• proof
• Corollary 2.4
• Lemma 2.5
• proof