Personne

Marc Troyanov

Isoperimetry in Finitely Generated Groups

Marc Troyanov, Bruno Emanuel Ferreira De Sousa Correia

We revisit the isoperimetric inequalities for finitely generated groups introduced and studied by N. Varopoulos, T. Coulhon and L. Saloff-Coste. Namely we show that a lower bound on the isoperimetric quotient of finite subsets in a finitely generated group ...

Springer2024

Double Forms, Curvature Integrals and the Gauss-Bonnet Formula

Marc Troyanov

The Gauss-Bonnet Formula is a significant achievement in nineteenth century differential geometry for the case of surfaces and the twentieth century cumulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for higher-dimensional R ...

Springer2024

Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear form

Bernard Dacorogna, Marc Troyanov

We generalize the celebrated results of Bernhard Riemann and Gaston Darboux: we give necessary and sufficient conditions for a bilinear form to be flat. More precisely, we give explicit necessary and sufficient conditions for a tensor field of type (0, 2) ...

Springer Heidelberg2023

Riemannian Surfaces with Simple Singularities

Marc Troyanov

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 simple sin-gularities.Wefirstd ...

Springer International Publishing2023

On Alexandrov’s Surfaces with Bounded Integral Curvature

Marc Troyanov

During the years 1940–1970, Alexandrov and the “Leningrad School” have investigated the geometry of singular surfaces in depth. The theory developed by this school is about topological surfaces with an intrinsic metric for which we can define a notion of c ...

Springer International Publishing2023

On the equation (Du)(t) H Du = G

Bernard Dacorogna, Marc Troyanov

In this article, we want to find a map u : (Omega) over bar -> R-n solving, in Omega, the equation ...

PERGAMON-ELSEVIER SCIENCE LTD2022

The Myers-Steenrod Theorem For Finsler Manifolds Of Low Regularity

Marc Troyanov

We prove a version of Myers-Steenrod's theorem for Finsler manifolds under the minimal regularity hypothesis. In particular we show that an isometry between C-k,C-alpha-smooth (or partially smooth) Finsler metrics, with k + alpha > 0, k is an element of N ...

Amer Mathematical Soc2017

The connectivity at infinity of a manifold and Sobolev inequalities

Marc Troyanov

The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an L-q,L-p-Sobolev inequality (2

Elsevier2014

The Holder-Poincaré duality for Lqp-cohomology

Marc Troyanov

We prove the following version of Poincaré, duality for reduced L (q,p) -cohomology: For any 1 < q, p < a, the Lqp -cohomology of a Riemannian manifold is in duality with the interior Lp'q'-cohomology for 1/p + 1/p' = 1/q + 1/q' = 1. ...

Springer Netherlands2012

A Short Proof of the Holder-Poincare Duality for Lp-Cohomology

Marc Troyanov

We give a short proof of the duality theorem for the reduced L-P-cohomology of a complete oriented Riemannian manifold. ...

2010