International Congress of Mathematicians

692 Juha HeinonenThese are nonconstant mappings / : 0 —t R n in the Sobolev space Wj o '"(0;R"),where O c R n is a domain and n > 2, satisfying the following requirement: thereexists a constant K > 1 such that\f'(x)\ n < KJ f (x) (2.1)for almost every x £ ii, where |/'(x)| denotes the operator norm of the (formal)differential matrix f'(x) with J/(x)=det/'(x) its Jacobian determinant. One alsospeaks about K-quasiregular mappings if the constant in (2.1) is to be emphasized. 1Requirement (2.1) had been used as the analytic definition for quasiconformalmappings since the 1930s, with varying degrees of smoothness conditions on/. Quasiconformal mappings are by definition quasiregular homeomorphisms, andReshetnyak was the first to ask what information inequality (2.1) harbours per se.In a series of papers in 1966-69, Reshetnyak laid the analytic foundations for thetheory of quasiregular mappings. The single deepest fact he discovered was thatquasiregular mappings are branched coverings (as defined above). It is instructiveto outline the main steps in the proof for this remarkable assertion, which akinto Riemann's result exerts significant topological information from purely analyticdata. For the details, see, e.g., [28], [29], [18].To wit, let / : 0 —t R n be K^quasiregular. Fix y £ R n and consider thepreimage Z = f^1(y).One first shows that the function u(x) = log \f(x) —y\ solvesa quasilinear elliptic partial differential equation-divA(x,Vu(xj)=0, A(x,Ç)-Ç~\Ç\ n , (2.2)in the open set 0 \ Z in the weak (distributional) sense. In general, A in (2.2)depends on /, but its ellipticity only on K and n. For holomorphic functions, i.e.,for n = 2 and K =1, equation (2.2) reduces to the Laplace equation —divVu = 0.Now u(x) tends to ^oo continuously as x tends to Z. Reshetnyak developssufficient nonlinear potential theory to conclude that such polar sets, associatedwith equation (2.2), have Hausdorff dimension zero. It follows that Z is totallydisconnected,i.e., the mapping / is light. This is the purely analytic part of theproof. The next step is to show that nonconstant quasiregular mappings are sensepreserving.This part of the proof mixes analysis and topology. What remains is apurely topological fact that sense-preserving and light mappings between connectedoriented manifolds are branched coverings.Initially, Reshetnyak's theorem served as the basis for a higher dimensionalfunction theory. In the 1980's, it was discovered by researchers in nonlinear elasticity.In the following, we shall discuss more recent, different types of applications.3. The branch setBranched coverings between surfaces behave locally like analytic functions accordingto a classical theorem of Stoïlow. By a theorem of Chernavskiï, for every1 The definition readily extends for mappings between connected oriented Riemannian n-manifolds.