Related papers: A systolic inequality for geodesic flows on the tw…

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental…

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple…

The systolic ratio of a contact form $\alpha$ on the three-sphere is the quantity \[ \rho_{\mathrm{sys}}(\alpha) = \frac{T_{\min}(\alpha)^2}{\mathrm{vol}(S^3,\alpha\wedge d\alpha)}, \] where $T_{\min}(\alpha)$ is the minimal period of…

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any…

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined…

A classic result by Gromov and Lawson states that a Riemannian metric of non--negative scalar curvature on the Torus must be flat. The analogous rigidity result for the standard sphere was shown by Llarull. Later Goette and Semmelmann…

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the…

The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at…

We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is…

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and…

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a…

By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an…

We prove that if $M$ is a three-manifold with scalar curvature greater than or equal to -2 and $\Sigma\subset M$ is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of…

We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by a smooth metric whose geodesic flow has an elliptic closed geodesic.

We investigate the rudiments of Riemannian geometry on orbit spaces $M/G$ for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space $M/G$ and they can hit…

Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures between $0$ and $1$. A minimal $2$-sphere immersed in $M$ has area at least $4\pi$. If an embedded minimal sphere has area $4\pi$, then $M$ is isometric to the unit…

We study integral curvature conditions for a Riemannian metric $g$ on $S^4$ that quantify the best bilipschitz constant between $(S^4,g)$ and the standard metric on $S^4$. Our results show that the best bilipschitz constant is controlled by…

We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define…

This paper proves that in any closed Riemannian surface $M$ with diameter $d$, the length of the $k^\text{th}$-shortest geodesic between two given points $p$ and $q$ is at most $8kd$. This bound can be tightened further to $6kd$ if $p = q$.…

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an…