Regularity for Dirac-harmonic map with Ricci type spinor potential
Dirac-harmonic maps were introduced in [\textit{Q. Chen} et al., Math. Z. 251, No. 1, 61--84 (2005; Zbl 1091.53042); ibid. 254, No. 2, 409--432 (2006; Zbl 1103.53033)] as pairs \((\phi, \psi)\) of maps that are critical points of the functional \[ L(\phi, \psi)=\frac{1}{2}\int_M\big(|d \phi|^2+\langle \mathcal{D}\psi, \psi\rangle\big)dv_g, \] where \(\phi: (M^2, g)\longrightarrow (N^n,h)\) is a map from a \(2\)-dimensional manifold into an \(n\)-dimensional manifold, \(\mathcal{D}\) is the Dirac operator along \(\phi\), and \(\psi\) is a section of the bundle \(\Sigma M\otimes \phi^{*}TN\longrightarrow M\) with \(\Sigma M\) denoting the usual spinor bundle on \(M\). The paper under review studies Dirac-harmonic maps with Ricci type spinor potential that are defined to be the critical points of the functional \[ L_{\lambda}(\phi, \psi)=\int_M\big\{\frac{1}{2}\big(|d \phi|^2+\langle \mathcal{D}\psi, \psi\rangle \big)+\lambda \text{R}_{ij}(\phi) \langle \psi^i, \psi^j\rangle\big\}dv_g. \] The authors derive the Euler-Lagrange equation of the functional and call a solution of it a weak solution of the Dirac-harmonic map with Ricci type potential. They prove that any weak solution of the Dirac-harmonic map with Ricci type potential from a compact Riemann surface into a compact manifold is smooth. They also obtain an approximate compactness result for such maps.
Dirac-harmonic maps with Ricci type spinor potential
scientific article; zbMATH DE number 6145943
scientific article