Let \(M\) be a \(C^ \infty\) compact, oriented manifold and \(M_ 1\) the space of \(C^ \infty\) Riemannian metrics of volume 1 on \(M\). The group \(D = \text{Diff}(M)\) acts on \(M_ 1\) by pullback, and the space \(M_ 1/D\) is the space of Riemannian structures on \(M\) – the moduli space of Riemannian metrics on \(M\). An Einstein metric \(g \in M_ 1\) is a metric such that \(\text{Ric}(g) = {\lambda\over n}g\), where \(n = \dim M\), \(\lambda\) denotes the scalar curvature of \(g\), and \(\text{Ric} =\) Ricci curvature. The subset \({\mathcal E} \subset M_ 1/D\) (endowed with the induced topology) is the moduli space of Einstein metrics of volume 1 on \(M\). Let \({\mathcal E}^ \mu\) (\(\mu = -,\circ,+)\) be the space of Einstein metrics with constant scalar curvature respectively negative zero, or positive.
Continuing his research published in [Res. Announ., Bull. Am. Math. Soc. 21, 163-167 (1983)] the author studies the global behaviour of the spaces \({\mathcal E}^ \mu \subset {\mathcal E}\) with respect to an extrinsic \(L^ 2\)- metric on \({\mathcal E}: \text{dis}_{L^ 2}(g_ 0,g_ 1) = \inf\{L(\gamma)\}\), \(g_ 0,g_ 1 \in {\mathcal E}^ \mu\), \(\gamma\) is an arbitrary curve in \(M_ 1/D\) with endpoints \(g_ 0\) and \(g_ 1\). Let \(\overline{D}\) denote the completion of \(D\) in the Lipschitz topology, acting on the space \(S^ 2(M)\) of symmetric bilinear forms on \(M\) with coefficients in \(L^ 2\); let \({\mathcal E}^ \mu_ s\) be the space of orbifold singular \(\lambda\)-Einstein metrics on \(M\), \(\mu = \text{sgn}(\lambda)\). The author proves several important results. Example: Let \(M\) be a compact, oriented 4-manifold. 1) The closure \(\overline{{\mathcal E}^ \mu}\) of \({\mathcal E}^ +\) in the extrinsic \(L^ 2\)-metric is contained in \({\mathcal E}^ 0 \cup {\mathcal E}^ 0_ s\), the space of Einstein metrics, regular and orbifold singular, of non-negative scalar curvature. For \(\lambda_ 0 > 0\), the space \(\overline{{\mathcal E}^{\lambda_ 0}}\) is a compact Hausdorff subspace of \(S^ 2(M)/\overline{D}\). 2) The closure \(\overline{{\mathcal E}^ 0}\) of \({\mathcal E}^ 0\) in the extrinsic \(L^ 2\) metric is a complete, generally non- compact Hausdorff subspace of \(S^ 2(M)/\overline{D}\), consisting of regular and orbifold singular Einstein metrics on \(M\). The space \(\overline{{\mathcal E}^ 0}\) is a locally compact subspace of \(S^ 2(M)/\overline{D}\). Finally, analogous problems are studied on the moduli of \(K3\)-surfaces. The methods are inspired by Differential geometry, Potential theory and Functional analysis.