Regularity of Dirac-harmonic maps

Details Regularity of Dirac-harmonic maps Regularity of Dirac-harmonic maps

2008-10-10

arxiv.org/abs/0810.1958v1

Mathematics

Analysis of PDEs

30 pages

Scientific paper

For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $\Sigma M$, and any compact Riemannian manifold $N$, we show an $\epsilon$-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 $n \ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $\psi\in W^{1,p}$ 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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