In the mathematical field ofanalysis,quasiregular maps are a class of continuous maps between Euclidean spacesRⁿ of the same dimension or, more generally, betweenRiemannian manifolds of the same dimension, which share some of the basic properties withholomorphic functions of one complex variable.

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

One drawback of this theory is that it deals only with maps between two-dimensional spaces (Riemann surfaces). The theory of functionsof several complex variables has a different character, mainly because analytic functions of several variables are notconformal. 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 ofMöbius transformations only.This is a theorem ofJoseph Liouville; relaxing the smoothness assumptions does not help, as proved byYurii Reshetnyak.^[1]

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.

Adifferentiable mapf of a regionD inRⁿ toRⁿ is calledK-quasiregular if the following inequality holds at all points inD:

${\displaystyle \Vert Df(x){\Vert }^n\le K|J_f(x)|}$.

HereK ≥ 1 is a constant,J_f is theJacobian determinant,Df is the derivative, that is the linear map defined by theJacobi matrix, and ||·|| is the usual (Euclidean)norm of the matrix.

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 theSobolev spaceW^1,n
_loc whose partial derivatives in the sense ofdistributions have locally summablen-th power, and such that the above inequality is satisfiedalmost everywhere. This is a formal definition of aK-quasiregular map. A map is calledquasiregular if it isK-quasiregular with someK. Constant maps are excluded from the class of quasiregular maps.

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

Quasiregular maps are open and discrete.

This means that the images ofopen 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 ahomeomorphism, so that the result is an analytic function. This is a theorem ofSimion Stoilov.

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

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

${\displaystyle (r,\theta ,z)\mapsto (r,2\theta ,z).}$

This map is 2-quasiregular. It is smooth everywhere except thez-axis. A remarkable fact is that all smooth quasiregular maps are local homeomorphisms. Even more remarkable is that every quasiregular local homeomorphismRⁿ → Rⁿ, wheren ≥ 3, is a homeomorphism (this is atheorem 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 ofRⁿ to itself are quasiconformal.

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 ofPicard's theorem which is due to Seppo Rickman:^[3]

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

Whenn = 2, this omitted set can contain at most one point (this is a simple extension of Picard's theorem). But whenn > 2, the omitted set can contain more than one point, and its cardinality can be estimated from above in terms ofn and K. In fact, any finite setcan be omitted, as shown byDavid Drasin and Pekka Pankka.^[4]

Iff is an analytic function, then log |f| issubharmonic, andharmonic away from the zeros off. The corresponding fact for quasiregular maps is that log |f| satisfies a certain non-linearpartial differential equation ofelliptic type.This discovery of Reshetnyak stimulated the development ofnon-linear potential theory, which treats this kind of equationsas the usualpotential theory treats harmonic and subharmonic functions.

• Yurii Reshetnyak
• Vladimir Zorich

• Mathematical analysis

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