Branched Singularities of Harmonic Maps

초록

In this paper we give an example of energy minimizing harmonic maps forwhich the set of singular points are two or more lines intersecting at a point.1. IntroductionIt is well known that a harmonic map between Riemannian manifolds are not neces-sarily continuous. In general, even energy minimizing harmonic maps can have pointsof discontinuity which we will call the singularity. The Hausdor dimension of thesingularity of a minimizing harmonic map is known to be less than or equal tom 3where m is the dimension of the domain [10].The shape of the singularity and the behavior of a minimizing harmonic map nearthe singularity have been studied by many mathematicians. For example, when thedimension of the domain m = 3 ; the singularity of minimizing harmonic map is consistof isolated points [10]. But whenm 4 the singular set can have more complicatedstructure. For the structure of singularity, it has been proved by Leon Simon that thesingular set of a minimizing harmonic map into a real analytic manifold is a union ofa pairwise disjoint locally (n 3) rectiable locally compact subsets [9]. But it isnot known whether there is a branch points in singular sets even for simplest case asmaps from B4 to S2: In fact, we do have only few explicit examples of minimizingharmonic maps with singularity.In this note, we introduce a minimizing harmonic map for which the singular setis made of nite lines crossing at a point. So far this is the only known example ofbranched singular set of minimizing harmonic maps.AMS subject classication : 53C43 58E20

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