arXiv research

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the constructio…

A new method for non-rigid point set registration reduces computational complexity.

The paper explores $ρ$-regularity for real analytic maps and its relation to Milnor fibrations.

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a ration…

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of…

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex manifold) is uniquely determined. If M is regular and the…

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over Riemann surfac…

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V…

Harmonic maps depend analytically on representations.

New framework uses elliptic operators to study projective maps.

Let $M$ and $N$ be Nash manifolds, and $f$ and $g$ Nash maps from $M$ to $N$. If $M$ and $N$ are compact and if $f$ and $g$ are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, $C^infty$ R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash…

New proof of harmonic map uniqueness with analytic targets.

Analytic completeness criterion applied to constant mean curvature surfaces.

New method normalizes Milnor fibrations for real analytic maps.

We prove short time existence and uniqueness of the Laplacian flow starting at an arbitrary closed $G_2$-structure. We establish long time existence and convergence of the Laplacian flow starting near a torsion-free $G_2$-structure. We analyze the limit map of the Laplacian flow in relation to the moduli space of torsi…

This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that appear in the realm of special holonomy as well as some of the topological and …

In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real ana…

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…

We study mappings on sub-Riemannian manifolds which are quasi-regular with respect to the Carnot-Caratheodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit …

Real analytic maps can be unstable even if infinitesimal changes are stable.

Analytic metrics are uniquely determined by their scattering map.

Researchers compute zeta-determinants and analytic torsion for metric mapping tori.

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…

Paper excludes the lowest energy level as an accumulation point for harmonic maps into analytic manifolds.

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the per…

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Let $X$ be a (real or complex) Banach space, and $\mathcal{I}(X)$ be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on $X$ whose squares equal themselves. We show that the Banach submanifold $\mathcal{I}(X)$ of $\mathcal{L}(X)$ is a locally trivial analytic affine-Banach bundle o…

The paper proves properties of Finsler submanifolds and analytic maps.

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f:Y -> X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) in X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are p…

We show that every complete metric space is homeomorphic to the precise locus of zeros of an entire analytic map from a Hilbert space to a Banach space. As a corollary, every complete separable metric space is homeomorphic to the precise locus of zeros of an entire analytic map between two separable complex Hilbert spa…

We give necessary and sufficient local conditions for the simultaneous unitarizability of a set of analytic matrix maps from an analytic 1-manifold into SL_n(C) under conjugation by a single analytic matrix map. We apply this result to the monodromy arising from an integrable partial differential equation to construct …

This note provides a new proof of the real analyticity of the Liouville map.

The paper defines wave-front singularities using explicit analytic functions.

Analyzes complex structure deformations using cohomology contraction methods.

The purpose of this paper is to define semi- and subanalytic subsets and maps in the context of real analytic orbifolds and to study their basic properties. We prove results analogous to some well-known results in the manifold case. For example, we prove that if $A$ is a subanalytic subset of a real analytic quotient o…

Develops second order infinitesimal structures on Teichmüller space.

Part 2 discusses compatibility between analytic and topological Gysin maps.

We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphas…

We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group…

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral form…

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which ar…

In \cite{Boed}, C.-F. Bödigheimer constructed a finite cell-complex $\mf{Par}_{g,n,m}$ and a bijective map $\cH: \mf{Dip}_{g,n,m} \to \mf{Par}_{g,n,m}$ (the Hilbert-uniformization) from the moduli space of dipole functions on Riemann surfaces with $n$ directions and $m$ punctures to $\mf{Par}_{g,n,m}$. In \cite{Boed} a…

Geometric analysis on real analytic manifolds using seminorms.

We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith maps are solutions of a conformally invariant nonlinear first order P…

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated mini…

An analytico-geometric reflection principle is established by means of normal deformations of analytic discs.