Ishida, Masashi Small four-manifolds without non-singular solutions of normalized Ricci flows. (English) Zbl 1308.53100 Asian J. Math. 18, No. 4, 609-622 (2014). The paper has as starting point the following conjecture due to F. Fang et al. [Math. Ann. 340, No. 3, 647–674 (2008; Zbl 1131.53034)]. Conjecture 1. Let \(X\) be a closed oriented smooth Riemannian 4-manifold with \(\| X\|\neq 0\) and \(\overline{\lambda}(X)<0\), where \(\| X\|\) is the Gromov’s simplicial volume of \(X\) and \(\overline{\lambda}(X)\) is Perelman’s \(\overline{\lambda}\) invariant. If there is a non-singular solution to the normalized Ricci flow on \(X\), then the Gromov-Hitchin-Thorpe type inequality holds: \[ 2\chi(X)-3|\tau(X)|\geq \frac{1}{1296\pi^2}\| X\|.\tag{1} \]Perelman’s \(\overline{\lambda}\) invariant [G. Perelman, arXiv e-print service, Paper No. 0211159, 39 p. (2002; Zbl 1130.53001); ibid., Paper No. 0303109, 22 p. (2003; Zbl 1130.53002)] is defined by \(\overline{\lambda}(X)=\sup_{g\in \mathcal{R}_X}\lambda_{g(\mathrm{vol}_g)}^{2/n}\), where \(\mathcal{R}_X\) is the space of all Riemannian metrics on \(X\), \(\lambda_g\) is the lowest eigenvalue of the elliptic operator \(4\triangle_g+s_g\), and \(\triangle_g=d^*d=-\nabla\cdot\nabla\) is the positive-spectrum Laplace-Beltrami operator associated with \(g\). The authors assert that to their knowledge, Conjecture 1 still remains open. But in a joint work with R. I. Baykur and M. Ishida [J. Geom. Anal. 24, No. 4, 1716–1736 (2014; Zbl 1322.57023)] the author of the present paper has shown that the converse of Conjecture 1 does not hold in general. Namely, these authors have proved that for sufficiently large positive integers \(\ell_1\), \(\ell_2>0\) and for sufficiently large positive odd integers \(g\), \(h>3\), a connected sum of the type \[ M:=X\#K3\#(\Sigma_g\times \Sigma_h)\#\ell_1(S^1\times S^3)\#\ell_2 \overline{\mathbb{C}P^2} \] has \(\|M\|\neq 0\) and satisfies a strict case of the inequality (2). In addition, \(M\) admits infinitely many distinct smooth structures for which Perelman’s \(\overline{\lambda}\) invariant is negative and there is no non-singular solution to the normalized Ricci flow for any initial metric. These properties of a space of the type of \(M\) are called \(\infty\)-property \(\mathcal{R}\). Now it follows that the existence of 4-manifolds with \(\infty\)-property \(\mathcal{R}\) implies that the converse of Conjecture 1 does not hold in general. By this, the author concludes that the existence and non-existence of non-singular solutions are not controled by a topological information like (1). In addition, in this way we obtain new examples of 4-manifolds without Einstein metrics because an Einstein metric is an example of an non-singular solution. But because the above constructions of spaces with the \(\infty\)-property \(\mathcal{R}\) were achieved by taking sufficiently large integers \(\ell_1\), \(\ell_2\), \(g\), \(h\), in this paper the author studies the existence of 4-manifolds with \(\infty\)-property \(\mathcal{R}\) for small \(\ell_1\), \(\ell_2\), \(g\), \(h\). More precisely, the author proves that there still exist 4-manifolds with \(\infty\)-property \(\mathcal{R}\) for \(\ell_1=\ell_2=0\) and \(g=h=3\). Thus, if \(N_p\) denotes a 4-manifold with fundamental group \(\mathbb{Z}_p\), \(p\) odd, which is obtained from the product \(L(p,1)\times S^1\) by performing a surgery along \(\{pt\}\times S^1\), then the main result of the paper is the following theorem. Theorem A. For any positive integer \(n\) with \(0\leq n\leq 7\), there exists an irreducible symplectic 4-manifold \(X_n\) which is homeomorphic to 1. \(4\mathbb{C}P^2\#(13+n)\overline{\mathbb{C}P^2 }\#(S^1\times S^3)\) or 2. \(3\mathbb{C}P^2\#(12+n)\overline{\mathbb{C}P^2}\#N_p\) or 3. \(3\mathbb{C}P^2\#\overline{\mathbb{C}P^2} \) and a connected sum \(X_n\#K3\#(\Sigma_3\times \Sigma_3)\) has the \(\infty\)-property \(\mathcal{R}\). Reviewer: Ioan Pop (Iaşi) Cited in 1 Document MSC: 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 57R57 Applications of global analysis to structures on manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) Keywords:four-manifold; Ricci flow; non-singular solution Citations:Zbl 1131.53034; Zbl 1130.53001; Zbl 1130.53002; Zbl 1322.57023 × Cite Format Result Cite Review PDF Full Text: DOI Euclid