Related papers: No hyperbolic pants for the 4-body problem
We consider the Yang-Mills flow on hyperbolic 3-space. The gauge connection is constructed from the frame-field and (not necessarily compatible) spin connection components. The fixed points of this flow include zero Yang-Mills curvature…
In the $N$-body problem, a motion is called hyperbolic, when the mutual distances between the bodies go to infinity with non-zero limiting velocities as time goes to infinity. For Newtonian potential, in \cite{MV20} Maderna and Venturelli…
In this survey paper the ultrahyperbolic equation in dimension four is discussed from a geometric, analytic and topological point of view. The geometry centres on the canonical neutral metric on the space of oriented geodesics of…
We consider surfaces of constant Gaussian curvature immersed in 3-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is 3-dimensional hyperbolic space. This allows us to prove…
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$
We prove that a closed negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold. Equivalently, any closed analytic Riemannian manifold with…
A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…
The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of…
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays…
We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic…
In the $2$-dimensional $n$-body problem, $n\ge 3$, in spaces of constant curvature, $\kappa\ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then…
We prove that the existence of one flat horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension n with n greater than or equal to 3, implies that the manifold is homothetic…
The equations of motion for $N$ non-relativistic particles attracting according to Newton's law are shown to correspond to the equations for null geodesics in a $(3N+2)$-dimensional Lorentzian, Ricci-flat, spacetime with a covariantly…
The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a…
The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions…
The Busemann-Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller $n$-dimensional volume. It is known that the answer is affirmative if $n\le 4$ and…
We carry on a comprehensive study on static fluid distributions endowed with hyperbolical symmetry. Their physical properties are analyzed in detail. The energy density appears to be necessarily negative, which suggests that any possible…
In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed,…
We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict in the sense of Leray and Ohya and is equivalent to the Einstein equations. Its characteristics are the light cone and the…
We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form. In spatial dimension at least four we show the existence of continuous families of recollapsing models whenever at least one of…