Personne

Marc Troyanov

Publications associées (46)

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Les surfaces à courbure intégrale bornée au sens d'Alexandrov

Marc Troyanov

Dans les années 1940-1970, Alexandrov et l'``'Ecole de Leningrad" ont développé une théorie très riche des surfaces singulières. Il s'agit de surfaces topologiques, munie d'une métrique intrinsèque pour laquelle on peut définir une notion de courbure, qui ...

2009

On the Hodge decomposition in $\mathbb{R}^n$

Marc Troyanov

We prove a version of the

L^p

hodge decomposition for differential forms in Euclidean space and a generalization to the class of Lizorkin currents. We also compute the

L_{qp}-

cohomology of

\mathbb{R}^n

. ...

2009

Steiner's Invariants and Minimal Connections

Marc Troyanov, Stephen Ducret

The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure. ...

2008

On the naturality of the exterior differential

Marc Troyanov

We give sufficient conditions for the naturallity of the exterior differential under Sobolev mappings. In other words we study the validity of the equation

d\, f^* \alpha = f^*\; d\alpha

for a smooth form

\alpha

and a Sobolev map

f

. ...

2008

On the moduli space of singular Euclidean surfaces

Marc Troyanov

The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces. ...

Eur. Math. Soc., Zürich2007

Weak metrics on Euclidean domains

Marc Troyanov

A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. The aim of this paper is to present some interesting weak metrics and to study some of their properties. In particular, we introduce a ...

2007

Sobolev inequalities for differential forms and $L_{q,p}$-cohomology

Marc Troyanov

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold

(M,g)

and the

L_{q,p}

-cohomology of that manifold. The

L_{q,p}

-cohomology of

(M,g)

is defined to be the quotient of the space of closed differential ...

2006

Piecewise flat Metrics on Surfaces and the Moduli Space

Marc Troyanov

We show how the theory of deformations of geometric structures (development and holonomy) applied to the case of piecewise flat surfaces leads to some interesting geometric structures on the moduli space of a punctured surface. ...

2006

Thurston's weak metric on the Teichmüller space of the torus

Marc Troyanov

We define and study a natural weak metric on the Teichmüller space of the torus. A similar metric has been defined by W. Thurston on the Teichmüller space of higher genus surfaces and our definition is motivated by Thurston's definition. However, we shall ...

2005

Tangent Spaces to Metric Spaces: Overview and Motivations

Marc Troyanov

This is the text of an introductory lecture I made at the Séminaire Borel 2003 (IIIème Cycle Romand de Mathématiques). The general theme of that seminaire was Tangent Spaces of Metric Spaces. ...

2003

On the Hodge decomposition in $\mathbb{R}^n$

Marc Troyanov

We prove a version of the

L^p

hodge decomposition for differential forms in Euclidean space and a generalization to the class of Lizorkin currents. We also compute the

L_{qp}-

cohomology of

\mathbb{R}^n

. ...

2009

Sobolev inequalities for differential forms and $L_{q,p}$-cohomology

Marc Troyanov

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold

(M,g)

and the

L_{q,p}

-cohomology of that manifold. The

L_{q,p}

-cohomology of

(M,g)

is defined to be the quotient of the space of closed differential ...

2006

On the naturality of the exterior differential

Marc Troyanov

We give sufficient conditions for the naturallity of the exterior differential under Sobolev mappings. In other words we study the validity of the equation

d\, f^* \alpha = f^*\; d\alpha

for a smooth form

\alpha

and a Sobolev map

f

. ...

2008