TANGENT UNIT-VECTOR FIELDS:NONABELIAN HOMOTOPY INVARIANTS AND THE DIRICHLET ENERGY  

作　　者：A.Majumdar J.M.Robbins M.Zyskin 

机构地区：[1]Mathematical Institute,University of Oxford,24-29 St.Giles,Oxford OX1 3LB,UK, [2]School of Mathematics,University of Bristol,University Walk,Bristol BS8 1TW,UK [3]Department of Mathematics,SETB 2.454-80 Fort Brown,Brownsville,TX 78520,USA

出　　处：《Acta Mathematica Scientia》2010年第5期1357-1399,共43页数学物理学报（B辑英文版）

基　　金：supported by a Royal Commission for the Exhibition of 1851 Research Fellowship between 2006-2008;supported by Award No.KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) to the Oxford Centre for Collaborative Applied Mathematics

摘　　要：Let O be a closed geodesic polygon in S2. Maps from O into S2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S2, we compute the infimum Dirichlet energy 6（H） for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for C（H） involves a topological invariant - the spelling length - associated with the （non-abelian） fundamental group of the n-times punctured two-sphere, π1（S2 - {s1,..., sn}, ＊）. The lower bound for C（H） is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes （classes containing wholly conformal and anticonformal representatives respectively）, the expression for C（H） reduces to a previous result involving the degrees of a set of regular values sl,…… sn in the target 82 space. These degrees may be viewed as invariants associated with the abelianization of vr1（S2 - {s1,..., sn}, ＊）. For nonconformal classes, however, ε（H） may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy （one-constant Oseen-Frank energy） of reflection-symmetric tangent unitvector fields in a rectangular prism.Let O be a closed geodesic polygon in S2. Maps from O into S2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S2, we compute the infimum Dirichlet energy 6（H） for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for C（H） involves a topological invariant - the spelling length - associated with the （non-abelian） fundamental group of the n-times punctured two-sphere, π1（S2 - {s1,..., sn}, ＊）. The lower bound for C（H） is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes （classes containing wholly conformal and anticonformal representatives respectively）, the expression for C（H） reduces to a previous result involving the degrees of a set of regular values sl,…… sn in the target 82 space. These degrees may be viewed as invariants associated with the abelianization of vr1（S2 - {s1,..., sn}, ＊）. For nonconformal classes, however, ε（H） may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy （one-constant Oseen-Frank energy） of reflection-symmetric tangent unitvector fields in a rectangular prism.

关 键 词：harmonic maps conformal maps algebraic topology non-abelian homotopy invariants eombinatorics liquid crystals 

分 类 号：O157.5[理学—数学]