Related papers: Boundary rigidity and stability for generic simple…

We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of…

Fix a smooth closed manifold $M$. Let $R_M$ denote the space of all pairs $(g,L)$ such that $g$ is a $C^3$ Riemannian metric on $M$ and the real number $L$ is not the length of any closed $g$-geodesics. A locally constant geodesic count…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a…

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…

We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become…

In this paper, we investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation.…

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. {\color{blue}For a metric space, we define its boundary to be the completion of the space minus…

Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize $(M,g)$ as a domain…

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…

We study the Lorentzian Calder\'on problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This…

We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation $\mathcal{S}^\sharp$ known on a lateral boundary. We show that, under a non-conjugacy assumption, every defining function…

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as…

In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator…

Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization…

Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero…

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly…