Related papers: Regular Maps on Euclidean Spaces and Tori
Quasiconformal maps in the complex plane are homeomorphisms that satisfy certain geometric distortion inequalities; infinitesimally, they map circles to ellipses with bounded eccentricity. The local distortion properties of these maps give…
We prove a regularity theorem for harmonic maps into Teichm\"uller space. More specifically, if $u$ is a harmonic map from a Riemannian domain to the metric completion of Teichm\"uller space with respect to the Weil-Petersson metric, and…
We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,\beta}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $\beta \in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the…
Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of…
A continuous map C^d -> C^N is a complex k-regular embedding if any k pairwise distinct points in C^d are mapped by f into k complex linearly independent vectors in C^N. Our central result on complex k-regular embeddings extends results of…
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by…
We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional…
In this paper we investigate the space of harmonic maps from a 2-torus to $\mathbb{S}^3$ using the spectral curve correspondence and Whitham deformations. In an open and dense subset of a parameter space we find that the space of harmonic…
Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold…
We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further…
We show that a $K$-quasiregular $\omega$-curve from a Euclidean domain to a Euclidean space with respect to a covector $\omega$ is locally $(1/K)(\lVert \omega\rVert/|\omega|_{\ell_1})$-H\"older continuous. We also show that quasiregular…
The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, projective plane and the…
Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric…
This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds is a classical subject in…
A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear…
A map which is non-orientable or has non-empty boundary has a canonical double cover which is orientable and has empty boundary. The map is called stable if every automorphism of this cover is a lift of an automorphism of the map. This note…
We record two facts on spaces of derived maps between the operads $E_d$ of little $d$-cubes. Firstly, these mapping spaces are equivalent to the mapping spaces between the non-unitary versions of $E_d$. Secondly, all endomorphisms of $E_d$…
Given a positive integer $p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $\mathcal{N}$. The target manifold is required to satisfy suitable topological conditions; in particular,…
This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of…
We generalize the Ruh-Vilms problem by characterizing the submanifolds in Euclidean spaces with proper biharmonic Gauss map and we construct examples of such hypersurfaces.