The authors study the renormalized volume for 4-dimensional Ricci-flat ALE manifolds and orbifolds \((M,g)\) with at most finitely many isolated orbifold singularities. These spaces are defined as follows. Let \(\Gamma\subset \mathrm{SO}(4)\) be a finite, non-trivial group acting freely on \(\mathbb{S}^3\). A manifold or orbifold \((M,g)\) such that there exists a compact set \(K\subset M\) together with a diffeomorphism \(\Phi\colon (\mathbb{R}^4\setminus B_1(0))/\Gamma\to M\setminus K\) with \[ \vert \nabla_{g_0}^k(\Phi^*g-g_0)\vert_{g_0}=\mathcal{O}(r^{-4-k}) \] for all \(k\geq 0\) as \(r\to \infty\), is called an ALE space. In this setting three theorems are proved.
{Theorems A & C}. Let \((M,g)\) be a 4-dimensional Ricci-flat ALE space. Then:
(1) One can define a renormalized volume \(\mathcal{V}(M,g)\) by subtracting the volume of orbiballs from the volume of a particular exhaustion \(\Omega_{\rho}\) of \(M\): \[ \mathcal{V}(M,g)= \lim_{\rho\to \infty} \left(\mathrm{Vol}_g(\Omega_{\rho})-\mathrm{Vol}_{g_0}(B_{\rho}(0)/\Gamma)\right), \] and this limit exists;
(2) The renormalized volume satisfies a negative mass theorem: \[ \mathcal{V}(M,g)\leq 0, \] with equality if and only if \((M,g)\) is isometric to a flat orbifold \((\mathbb{R}^4/\Gamma,g_0)\);
(3) When \((M,g)\) is a Kronheimer’s hyper-Kähler ALE space, there is an explicit formula for \(\mathcal{V}(M,g)\) in terms of the value of the Kronheimer period point \(\zeta\).
Their third theorem, Theorem B, is more technical and is used in proving the above results. Roughly stated:
{Theorem B}. Let \((M,g)\) be a 4-dimensional Ricci-flat ALE space. Then one can construct a \(C^{k_0}\)-diffeomorphism \(\Phi \colon (\mathbb{R}^4\setminus B_1(0))/\Gamma)\to M\setminus K \) of arbitrary high regularity \(k_0\) such that:
(1) \(\Phi(\partial B_{\rho}(0)/\Gamma)=\Sigma_{\rho}\) (see [O. Chodosh et al., J. Differ. Geom. 105, No. 1, 1–19 (2017; Zbl 1364.53035)]);
(2) One has a very good control of the leading term in \(\Phi^*g-g_0\);
(3) The remainder term in \(\Phi^*g-g_0\) and its \(k_0\) first derivatives are an order of magnitude smaller than the leading term.
Theorem B can be used to study other problems about 4-dimensional Ricci-flat ALE spaces, and will be of independent interest.