arXiv research

Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for the existence of a complete uniform differential structure on a given differenti…

Transforms Aronszajn to Sikorski subcartesian spaces.

Constructs differential forms on $C^\infty$-ringed spaces.

Right inverse found for Cartan differential in rank-1 symmetric spaces.

Study of differential forms and vector fields on orbit spaces.

The paper defines and analyzes $abla$-Sobolev spaces and operators on manifolds.

Lectures explore how differential methods improve understanding of algebraic group orbit spaces.

We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying b…

Survey on moduli spaces of differentials from algebraic geometry perspective.

Formula for Euler characteristic of moduli spaces of Abelian differentials.

New formalization of curved spaces using pointwise affine spaces.

The distance function $\varrho(p,q)$ (or $d(p,q)$) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $\mathbb R^n$, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjectu…

The abstract discusses the linear and smooth structures of mapping spaces.

Analyzes geometric structures on profinite diffeological spaces.

Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize distribution spaces in our study of wave and heat equations.

Tensoring $p$-weak differentiable structures preserves their properties.

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the norm…

Develops second order infinitesimal structures on Teichmüller space.

Defines vector fields and differential forms on local C-infinity-ringed spaces.

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the actio…

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $δ$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilat…

The paper studies the Poisson transform of differential forms on hyperbolic spaces.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

The uniform structure on a differential space defined by a family of generators is considered.

In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just as grad (ient), div (ergence) and rot (ation) are uniquely determined geometric…

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfie…

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given.

Diffeology extends differential geometry to complex spaces.

Study shows solutions of differential inclusions are homotopy equivalent in $W^{1,p}$-topology.

Two types of differentials are shown equivalent for compactifying moduli spaces.

New geometric Joyce structures on moduli spaces of quadratic differentials.

In this paper the notion of an M-th order invariant bilinear differential pairing is introduced and a formal definition is given. If the manifold has an AHS structure, then various first order pairings are constructed. This yields a classification of all first order invariant bilinear differential pairings on homogeneo…

The paper constructs cohomology classes on curve strata.

Proves even degrees can be realized in Abelian differential strata.

Study cohomology spaces of sl(2) acting on n-ary differential operators.

New stability conditions identified from quadratic differentials on surfaces.

Paper solves a class of differential equations with specific solutions.

On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as …

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential…

We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iter…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

Unified approach to stability conditions on surfaces with quadratic differentials.

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left…

Critical graphs of quadratic differentials equidistribute in moduli space.

Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.

SageMath package diffstrata calculates intersection theory on abelian differentials.