Quasiregular map

Last updated May 03, 2019

In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces Rⁿ of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable.

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space(M, g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g_p(X|_p, Y|_p) is a smooth function. The family g_p of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighbourhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

Motivation

The theory of holomorphic (=analytic) functions of one complex variable is one of the most beautiful and most useful parts of the whole mathematics.

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x₀ converges to the function in some neighborhood for every x₀ in its domain.

One drawback of this theory is that it deals only with maps between two-dimensional spaces (Riemann surfaces). The theory of functions of several complex variables has a different character, mainly because analytic functions of several variables are not conformal. Conformal maps can be defined between Euclidean spaces of arbitrary dimension, but when the dimension is greater than 2, this class of maps is very small: it consists of Möbius transformations only. This is a theorem of Joseph Liouville; relaxing the smoothness assumptions does not help, as proved by Yurii Reshetnyak.^[1]

In mathematics, a conformal map is a function that preserves orientation and angles locally. In the most common case, the function has a domain and an image in the complex plane.

Joseph Liouville FRS FRSE FAS · was a French mathematician.

Yurii Grigorievich Reshetnyak is a Soviet Russian mathematician and academician.

This suggests the search of a generalization of the property of conformality which would give a rich and interesting class of maps in higher dimension.

Definition

A differentiable map f of a region D in Rⁿ to Rⁿ is called K-quasiregular if the following inequality holds at all points in D:

\|Df(x)\|^{n}\leq K|J_{f}(x)|

.

Here K ≥ 1 is a constant, J_f is the Jacobian determinant, Df is the derivative, that is the linear map defined by the Jacobi matrix, and ||·|| is the usual (Euclidean) norm of the matrix.

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices.

The development of the theory of such maps showed that it is unreasonable to restrict oneself to differentiable maps in the classical sense, and that the "correct" class of maps consists of continuous maps in the Sobolev space W^1,n
_loc whose partial derivatives in the sense of distributions have locally summable n-th power, and such that the above inequality is satisfied almost everywhere. This is a formal definition of a K-quasiregular map. A map is called quasiregular if it is K-quasiregular with some K. Constant maps are excluded from the class of quasiregular maps.

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L^p-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In measure theory, a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of almost everywhere is a companion notion to the concept of measure zero. In the subject of probability, which is largely based in measure theory, the notion is referred to as almost surely.

Properties

The fundamental theorem about quasiregular maps was proved by Reshetnyak:^[2]

Quasiregular maps are open and discrete.

This means that the images of open sets are open and that preimages of points consist of isolated points. In dimension 2, these two properties give a topological characterization of the class of non-constant analytic functions: every continuous open and discrete map of a plane domain to the plane can be pre-composed with a homeomorphism, so that the result is an analytic function. This is a theorem of Simion Stoilov.

Reshetnyak's theorem implies that all pure topological results about analytic functions (such that the Maximum Modulus Principle, Rouché's theorem etc.) extend to quasiregular maps.

Injective quasiregular maps are called quasiconformal. A simple example of non-injective quasiregular map is given in cylindrical coordinates in 3-space by the formula

(r,\theta ,z)\mapsto (r,2\theta ,z).

This map is 2-quasiregular. It is smooth everywhere except the z-axis. A remarkable fact is that all smooth quasiregular maps are local homeomorphisms. Even more remarkable is that every quasiregular local homeomorphism Rⁿ → Rⁿ, where n ≥ 3, is a homeomorphism (this is a theorem of Vladimir Zorich ^[2]).

This explains why in the definition of quasiregular maps it is not reasonable to restrict oneself to smooth maps: all smooth quasiregular maps of Rⁿ to itself are quasiconformal.

Rickman's theorem

Many theorems about geometric properties of holomorphic functions of one complex variable have been extended to quasiregular maps. These extensions are usually highly non-trivial.

Perhaps the most famous result of this sort is the extension of Picard's theorem which is due to Seppo Rickman:^[3]

A K-quasiregular mapRⁿ → Rⁿcan omit at most a finite set.

When n = 2, this omitted set can contain at most two points (this is a simple extension of Picard's theorem). But when n > 2, the omitted set can contain more than two points, and its cardinality can be estimated from above in terms of n and K. In fact, any finite set can be omitted, as shown by David Drasin and Pekka Pankka.^[4]

Connection with potential theory

If f is an analytic function, then log |f| is subharmonic, and harmonic away from the zeros of f. The corresponding fact for quasiregular maps is that log |f| satisfies a certain non-linear partial differential equation of elliptic type. This discovery of Reshetnyak stimulated the development of non-linear potential theory, which treats this kind of equations as the usual potential theory treats harmonic and subharmonic functions.

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References

↑ Yu. G. Reshetnyak (1994). Stability theorems in geometry and analysis. Kluwer.
1 2 Yu. G. Reshetnyak (1989). Space mappings with bounded distortion. American Mathematical Society.
↑ S. Rickman (1993). Quasiregular mappings. Springer Verlag.
↑ D. Drasin; Pekka Pankka (2015). "Sharpness of Rickman's Picard theorem in all dimensions". Acta Math. 214. pp. 209–306.

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[resh-1] Yu. G. Reshetnyak (1994). Stability theorems in geometry and analysis. Kluwer.

[resh2-2] 1 2 Yu. G. Reshetnyak (1989). Space mappings with bounded distortion. American Mathematical Society.

[3] S. Rickman (1993). Quasiregular mappings. Springer Verlag.

[4] D. Drasin; Pekka Pankka (2015). "Sharpness of Rickman's Picard theorem in all dimensions". Acta Math. 214. pp. 209–306.