We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets.We also classify the Lorentz-invariant even generalized valuations.

In this paper we first establish an optimal Sobolev-type inequality for hypersurfaces in $\mathbb{H}^n$ (see Theorem 1.1). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. More precisely, we prove a geometric inequality in the hyperbolic space $\mathbb{H}^n$, which is a hyperbolic Alexandrov-Fenchel inequality,

$$ \begin{array}{rcl} \int_\Sigma \sigma_{2k}\ge C_{n-1}^{2k}\omega_{n-1}\left\{ \left( \frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 1k +\left( \frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 1k\frac {n-1-2k}{n-1}} \right\}^k, \end{array} $$

when $\Sigma$ is a horospherical convex and $2k\leq n-1$. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. Here $\sigma_{j}=\sigma_{j}(\kappa)$ is the $j$th mean curvature and $\kappa=(\kappa_1,\kappa_2,\cdots, \kappa_{n-1})$ is the set of the principal curvatures of $\Sigma$. Also, an optimal inequality for quermassintegral in $\mathbb{H}^n$ is

$$W_{2k+1}(\Omega)\geq\frac {\omega_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(\Omega)}{\omega_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}},$$

provided that $\Omega\subset\mathbb{H}^n$ is a domain with $\Sigma=\partial\Omega$ horospherical convex, where $2k\leq n-1$. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. Here $W_r(\Omega)$ is quermassintegrals in integral geometry.

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves $\mathcal{M}_{g,n}$. Their compactification $\overline{\mathcal{M}}_{g,n}$ is a projective algebraic variety, and as such it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmüller space by the action of the mapping class group gives a compactification of $\mathcal{M}_{g,n}$. We put an analytic structure on this quotient and prove that with respect to this structure, the compactification is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$.

We bring new insights into the longstanding Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons, and nilsolitons. We obtain that any homogeneous expanding Ricci soliton $M = G/K$ is diffeomorphic to a product $U / K \times N$, where $U$ is a maximal reductive Lie subgroup of $G$ and $N$ is the maximal nilpotent normal subgroup of $G$, such that the metric restricted to $N$ is a nilsoliton. Moreover, strong compatibility conditions between the metric and the action of $U$ on $N$ by conjugation must hold, including a nice formula for the Ricci operator of the metric restricted to $U / K$. Our main tools come from geometric invariant theory. As an application, we give many Lie theoretical characterizations of algebraic solitons, as well as a proof of the fact that the following a priori much stronger result is actually equivalent to Alekseevskii’s conjecture: Any expanding algebraic soliton is diffeomorphic to a Euclidean space.

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.