For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian’s identity from Teichmüller theory. Any equality characterizes diagonal embeddings.

In this paper we solve two open problems of classical surface theory; we give an affirmative answer to a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres in $\mathbb{R}^3$ that satisfy a general elliptic prescribed curvature equation, and we prove as a consequence that round spheres are the only elliptic Weingarten spheres immersed in $\mathbb{R}^3$. For this, we first extend Hopf’s famous classification of constant mean curvature spheres in $\mathbb{R}^3$ to the general situation of surfaces modeled by elliptic PDEs in arbitrary three-manifolds that admit families of candidate examples.

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-closed portion, we extend previous results by the first two authors, deriving Regge–Wheeler type equations for two gauge-invariant master quantities without the earlier paper’s need of axisymmetry. For the closed portion, we build upon earlier work of Zerilli and Moncrief, wherein the authors derive an equation for a gauge-invariant master quantity in a spherical harmonic decomposition. We work with gauge-invariant quantities at the level of perturbed connection coefficients, with the initial value problem formulated on Cauchy data sets. With the choice of an appropriate gauge in each of the two portions, decay estimates on these decoupled quantities are used to establish decay of the metric coefficients of the solution, completing the proof of linear stability. Our result differs from that of Dafermos–Holzegel–Rodnianski, both in our choice of gauge and in our identification and utilization of lower-level gauge-invariant master quantities.

By employing the Green function, in this paper we provide a new and elementary proof for the interior regularity of solutions to the Monge–Ampère equation. This proof also applies to the complex Monge–Ampère equation and the $k$-Hessian equation.

Existence and uniqueness of the solution to the $L_p$ Minkowski problem for the electrostatic $\mathfrak{p}$-capacity are proved when $p \gt 1$ and $1 \lt \mathfrak{p} \lt n$. These results are nonlinear extensions of the very recent solution to the $L_p$ Minkowski problem for $\mathfrak{p}$-capacity when $p = 1$ and $1 \lt \mathfrak{p} \lt n$ by Colesanti et al. and Akman et al., and the classical solution to the Minkowski problem for electrostatic capacity when $p = 1$ and $\mathfrak{p} = 2$ by Jerison.