Regularity of Dirac-harmonic maps

2008/10/10 by Wang, Changyou, Xu, Deliang
#35J20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0810.1958

Abstract

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle ΣM, and any compact Riemannian manifold N, we show an ε-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when n = 2. For n ≥ 3, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in Rⁿ. If, additions, ψ∈ W^1,p for some p>2n/3, then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.

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