On the Hodge decomposition in R^n

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 $\mathbb{R}^n$.Comment: 28 page

On the origin of Hilbert Geometry

In this brief essay we succinctly comment on the historical origin of Hilbert geometry. In particular, we give a summary of the letter in which David Hilbert informs his friend and colleague Felix Klein about his discovery of this geometry. The present paper is to appear in the Handbook of Hilbert geometry, (ed. A. Papadopoulos and M. Troyanov), European Mathematical Society, Z\"urich, 2014

On the Moduli Space of Singular Euclidean Surfaces

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.Comment: To appear in the Handbook of Teichmuller Theory, vol. 1, ed. A. Papadopoulos, European Math. Society Series, 200

The Myers-Steenrod theorem for Finsler manifolds of low regularity

We prove a version of Myers-Steenrod's theorem for Finsler manifolds under minimal regularity hypothesis. In particular we show that an isometry between $C^{k,\alpha}$-smooth (or partially smooth) Finsler metrics, with $k+\alpha>0$, $k\in \mathbb{N} \cup \{0\}$, and $0 \leq \alpha \leq 1$ is necessary a diffeomorphism of class $C^{k+1,\alpha}$. A generalisation of this result to the case of Finsler 1-quasiconformal mapping is given. The proofs are based on the reduction of the Finlserian problems to Riemannian ones with the help of the the Binet-Legendre metric.Comment: 14 page