On the manifold $\mathcal{M}(M)$ of all Riemannian metrics on a compact manifold $M$, one can consider the natural $L^2$-metric as described first by D.G. Ebin, The manifold of Riemannian metrics. In this paper we consider variants of this metric, which in general are of higher order. We derive the geodesic equations; we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a “universal lifting theorem” for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.

We study the mean curvature evolution of smooth, closed, two-convex hypersurfaces in $\mathbb{R}^{n+1}$ for $n \ge 3$. Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in G. Huisken & C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces,—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain $L^p$-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.

In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume $V^{g,n}$ of the moduli spaces of hyperbolic surfaces of genus $g$ with $n$ punctures as $g \to \infty$. Then we discuss basic geometric properties of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure as $g \to \infty$.

We give a complete classification of irreducible symmetric spaces for which there exist proper $SL(2,\mathbb{R})$-actions as isometries, using the criterion for proper actions by T. Kobayashi and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem.

This work uncovers the tropical analogue, for measured laminations, of the convex hull construction in decorated Teichmüller theory; namely, it is a study in coordinates of geometric degeneration to a point of Thurston’s boundary for Teichmüller space. This may offer a paradigm for the extension of the basic cell decomposition of Riemann’s moduli space to other contexts for general moduli spaces of flat connections on a surface. In any case, this discussion drastically simplifies aspects of previous related studies as is explained. Furthermore, a new class of measured laminations relative to an ideal cell decomposition of a surface is discovered in the limit. Finally, the tropical analogue of the convex hull construction in Minkowski space is formulated as an explicit algorithm that serially simplifies a triangulation with respect to a fixed lamination and has its own independent interest.