We obtain an estimate for the volumes of neighborhoods of sets of large curvature in three-dimensional Kähler-Einstein manifolds. The key technical step is to prove that a version of monotonicity for $L^2$ energy holds as long as the underlying region does not “carry homology” (in the sense that the normalized energy in a ball controls the normalized energy in an interior ball).

This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains. We prove a “low energy” result in all dimensions, in the sense that if normalized energy in a large ball is small enough, then the normalized energy in any interior ball must also be small.

We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the $L^2$ metric. The primary motivation for studying this problem comes from Teichmüller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmüller space with respect to a class of metrics that generalize the Weil-Petersson metric.

For a simply connected solvable Lie group $G$ with a cocompact discrete subgroup $\Gamma$, we consider the space of differential forms on the solvmanifold $G/\Gamma$ with values in a certain flat bundle so that this space has a structure of a differential graded algebra (DGA). We construct Sullivan’s minimal model of this DGA. This result is an extension of Nomizu’s theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa’s result of formality of nilmanifolds and Benson-Gordon’s result of hard Lefschetz properties of nilmanifolds.

We propose a finite-dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston’s gluing equation and Haken’s normal surface equation. The action functional is the volume.

Let $M$ be a closed orientable 3-manifold admitting an $\mathbb{H}^2 \times \mathbb{R}$ or $\widetilde{\mathrm{SL}_2}(\mathbb{R})$ geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group $\mathrm{Diff}(M)$ is either contractible or homotopy equivalent to $S^1$, according as the center of $\pi_1(M)$ is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of $\mathrm{Diff}(M)$ and of the space of Seifert fiberings $\mathrm{SF}(M)$ for compact orientable aspherical 3-manifolds. We also prove that when $M$ has an $\mathbb{H}^2 \times \mathbb{R}$ or $\widetilde{\mathrm{SL}_2}(\mathbb{R})$ geometry and the base orbifold has underlying manifold the 2-sphere with three cone points, the inclusion $\mathrm{Isom}(M) \to \mathrm{Diff}(M)$ is a homotopy equivalence.