Complete noncompact \(\mathrm{G}_2\)-manifolds from asymptotically conical Calabi-Yau 3-folds (Q2059018)

A \(G_2\)-manifold is a Riemannian manifold whose holonomy group is the compact exceptional Lie group \(G_2\). In the present paper, the authors give an analytic method for constructing complete non-compact \(G_2\)-manifolds. This gives the first general analytic construction of complete Ricci-flat metrics on non-compact manifolds of any odd-dimension, establishing a link with the Cheeger-Fukaya-Gromov theory of collapsing with bounded curvature. In the non-compact setting, the only tools available for producing Ricci-flat metrics are holonomy reduction methods and symmetry reduction. In even dimensions, holonomy reduction techniques related to Kähler geometry have been useful for constructing such metrics in both the compact and complete non-compact settings. For odd-dimensional manifolds, the only possible irreducible holonomy reduction is that of \(G_2\)-holonomy for 7-folds. In recent years, the number of known families of compact \(G_2\)-manifolds has grown considerably, but there remain only a handful of known families of complete non-compact irreducible \(G_2\)-manifolds. In the Kähler setting, there has been great success in finding circle-invariant Einstein metrics on the total space of circles bundles (or more generally, torus-invariant metrics on torus bundles) over compact Kähler-Einstein Fano manifolds. The absence of Killing fields rules out such bundle constructions in the compact irreducible Ricci-flat case. In the non-compact case, however, this provides a natural ground for searching for such metrics, particularly in light of the great utility of the Gibbons-Hawking ansatz as a method of producing interesting complete non-compact circle-invariant hyper-Kähler metrics in dimension four. The idea is to look for families of complete circle-invariant Ricci-flat metrics \(g_{\varepsilon}\) with submaximal volume growth on the total space \(M\) of a circle bundle over an asymptotically conical Ricci-flat manifold \((B,g_B)\) that collapse in the Gromov-Hausdorff sense as \(\varepsilon \to 0\) to \((B,g_B)\). To make this idea into a powerful generally applicable tool for constructing large families of interesting new complete Ricci-flat metrics, several ingredients are necessary: \begin{itemize} \item[(i)] tools to construct enough interesting asymptotically conical Ricci-flat metrics for use as the base metric; \item[(ii)] a good understanding of the deformation theory of such asymptotically conical Ricci-flat metrics; \item[(iii)] a well-behaved perturbation theory to allow for the correction by PDE methods of almost Ricci-flat metrics on \(M\) to genuine Ricci-flat metrics. \end{itemize} In the general Ricci-flat setting, all three issues cause serious difficulty. In the special case where we try to construct \(G_2\)-holonomy metrics on circle bundles over asymptotically conical Calabi-Yau threefolds, we have additional tools available. Surprisingly, the tools now available are sufficiently powerful that it works in complete generality, independent of the particular asymptotically conical Calabi-Yau threefold or circle bundle in question. The main theorem is the following: Theorem. Let \((B,g_0,\omega_0, \Omega_0)\) be an asymptotically Calabi-Yau 3-fold, asymptotic with rate \(\mu <0\), to the Calabi-Yau cone \((C(\Sigma), g_C, \omega_C, \Omega_C)\) over a smooth Sasaki-Einstein 5-fold \(\Sigma\). Let \(M \to B\) be a principal \(\text{U}(1)\)-bundle such that \(c_1(M) \neq 0\), but \(c_1(M) \cup [\omega_0]=0\in H^4(B)\). Then there exists \(\varepsilon_0 >0\) such that for every \(\varepsilon \in (0, \varepsilon_0)\), the 7-fold \(M\) carries an \(\mathbb{S}^1\)-invariant torsion-free \(G_2\)-structure \(\varphi_{\varepsilon}\) with the following properties. Let \(g_{\varepsilon}\) denote the Riemannian metric on \(M\) induced by \(\varphi_{\varepsilon}\). \begin{itemize} \item[(i)] \(g_{\varepsilon}\) has restricted holonomy \(\text{Hol}^0(g_{\varepsilon}) = G_2\). \item[(ii)] \((M,g_{\varepsilon})\) is an asymptotically locally conical manifold: outside of a compact set \(K\), \(M\) is identified with the total space of a principal \(\text{U}(1)\)-bundle over an exterior region \(\{ r > R \}\) in the cone \(C(\Sigma)\). Under this identification \[g_{\varepsilon} = g_C + \varepsilon^2 \vartheta_{\infty} + O(r^{-\min \{ 1, -\mu \}})\] with analogous decay for all covariant derivatives. Here, \(\vartheta_{\infty}\) is a connection on the principal circle bundle \(M \backslash K \to \{ r \geq R\} \subset C(\Sigma)\). \item[(iii)] There exists a connection \(\vartheta\) on \(M \to B\) such that the difference between \(g_{\varepsilon}\) and the Riemannian submersion with fibers of constant \(2\pi \varepsilon\) \[g_0 + \varepsilon^2 \vartheta^2\] converges to zero as \(\varepsilon \to 0\) in \(C^{k,\alpha}\) for every \(k \geq 0\) and \(\alpha \in (0,1)\). In particular, \((M,g_{\varepsilon})\) collapses with bounded curvature to \((B,g_0)\) as \(\varepsilon \to 0\). \end{itemize} Two illustrations of the strength of the construction are: Infinitely many diffeomorphism-types of complete non-compact simply connected \(G_2\)-manifolds are constructed. Previously, only a handful of such diffeomorphism-types were known. They are used it to prove the existence of continuous families of complete non-compact \(G_2\)-metrics of arbitrarily high dimension. Previously, only rigid or 1-parameter families of complete non-compact \(G_2\)-metrics were known.