New expanding Ricci solitons starting in dimension four (Q6611139)

An expanding gradient Ricci soliton is a complete Riemannian manifold \((M,g)\) and a function \(f\in C^{\infty}(M)\) such that\N\[\N\mathrm{Ric}(g)+\nabla^{2}f+\frac{1}{2\tau}g=0,\N\]\Nfor some \(\tau>0\). Such metrics generalise Einstein manifolds with negative scalar curvature (which are recovered by taking the function \(f\) to be constant); they also appear as singularity models for the Ricci flow. In particular, in order to be able to continue the flow past a time where a singularity has developed, it is expected that one might have to glue in an expanding gradient Ricci soliton (see e.g. [\textit{P. Gianniotis} and \textit{F. Schulze}, Geom. Topol. 22, No. 7, 3925--3977 (2018; Zbl 1405.53087)]). In the paper under review the authors prove an existence theorem for expanding Ricci solitons; more precisely, given \(d_{1}\in \mathbb{N}\) and Einstein manifolds \((M_{i},g_{i})\) for \({i=2,\ldots r}\), they prove the existence of an \(r\)-parameter family of expanding solitons (and an \((r-1)\)-parameter family of negative Einstein metrics) on the product\N\[\N\mathbb{R}^{d_{1}+1}\times M_{2}\times \cdots \times M_{r}.\N\]\NFurthermore, in the case that the Einstein constants of the manifolds \((M_{i},g_{i})\) are nonnegative, they prove that the solitons are asymptotically conical. If \(d_{1}\geq 2\) and the \((M_{i},g_{i})\) are all Ricci-flat, then they show that all cones with links of the form\N\[\N\left(S^{d_{1}}\times M_{2}\times \cdots \times M_{r}, \sigma_{1}^{-2}g_{1}+\cdots +\sigma_{r}^{-2}g_{r}\right)\N\]\Noccur as an asymptotic cone (\(g_{1}\) is the round metric on the sphere \(S^{d_{1}}\) and \(\sigma_{i}>0\) are constants).\N\NThe existence results generalise those of \textit{A. S. Dancer} and \textit{M. Y. Wang} [Int. Math. Res. Not. 2009, No. 6, 1107--1133 (2009; Zbl 1183.53060)] and \textit{M. Buzano} et al. [Pac. J. Math. 273, No. 2, 369--394 (2015; Zbl 1316.53051)] where the Einstein metrics had positive scalar curvature. The proof goes by considering a warped product construction on\N\[\N(0,T)\times S^{d_{1}}\times M_{2}\times \cdots \times M_{r}\N\]\Nand a potential function \(f:(0,T) \rightarrow \mathbb{R}\). This turns the soliton equation into a system of ODEs which are relatively easy to solve. It is slightly more work to show that the resulting metrics are complete. This is done in section 2 of the paper. The remaining sections, which make up the majority of the paper, use various ODE comparison techniques to establish the asymptotic geometry of the solitons.