Related papers: Alexandrov's theorem revisited

Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their…

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to…

We consider the variational problem of minimizing an anisotropic perimeter functional under a volume constraint in a Euclidean convex domain. We extend to this setting analytical properties of the isoperimetric profile, topological features…

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting…

Let $M_j$ be a sequence of Riemannian manifolds with sectional curvature bound below collapsing to a compact Alexandrov space $X$ of dimension $k$. Suppose that all but finitely many points of $X$ are $(k,\delta)$-strained and that the…

Bertrand's theorem proves that inverse square and Hooke's law-type central forces are the only ones for which all bounded orbits are closed. Similar analysis was used to show that for other central force laws there exist closed orbits for a…

We consider variational problems with regular H{\"o}lderian weight or boundary singularity, and Dirichlet condition. We prove the boundedness of the volume of the solutions to these equations on analytic domains.

We prove some qualitative properties for singular solutions to a class of strongly coupled system involving a Gross--Pitaevskii-type nonlinearity. Our main theorems are vectorial fourth order counterparts of the classical results of [J.…

Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional…

In this paper, we give a simple proof that the density at infinity of fibers of a definable function is locally Lipschitz outside the set of asymptotic critical values.

Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space.…

We study finite orbits for non-elementary groups of automorphisms of compact projective surfaces. In particular we prove that if the surface and the group are defined over a number field k and the group contains parabolic elements, then the…

Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$…

Let $K$ be a number field and $\phi\in K(z)$ a rational function. Let $S$ be the set of all archimedean places of $K$ and all non-archimedean places associated to the prime ideals of bad reduction for $\phi$. We prove an upper bound for…

We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to…

In this paper, we prove that manifolds of finite volume with Anosov geodesic flow have dense periodic orbits. The same result works for conservative Anosov flows in non-compact cases.

In this paper, we study analytic self-maps of the unit disk for which the hyperbolic diameters of the images of hyperbolic balls of radius 1 are uniformly bounded below. We give several characterizations of such maps involving the behaviour…

Let $\Omega \subset \mathbb{R}^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $\Omega \cap B_r (x)$ is equal to a positive constant when $x$ varies in the essential boundary of…

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic…

The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.