Geometric analysis of a mixed elliptic-parabolic conformally invariant boundary value problem
Jürgen Jost, Lei Liu, and Miaomiao Zhu
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Submission date: 19. Jun. 2018
MSC-Numbers: 53C43, 58E20
Keywords and phrases: Supersymmetric nonlinear sigma model, Dirac-harmonic maps, $\alpha$-Dirac-harmonic maps, $\alpha$-Dirac-harmonic map flow, blow-up, energy identity, neck analysis
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In this paper, we develop a general spectrum of methods (Pohozaev identity, three circle method, blow-up analysis, energy identities, energy decay estimates etc.) for conformally invariant variational problems at a particularly challenging example. We study the reﬁned blow-up behaviour and asymptotic analysis for a sequence of α-Dirac harmonic maps from a closed Riemann surface into a general Riemannian manifold with uniformly bounded energy. We prove an energy identity for the spinor part and a generalized energy identity for the map part. We also show that the map parts of the Dirac-harmonic necks converge to some geodesics on the target manifold. Moreover, we give a length formula for the limiting geodesic near a blow-up point if there is only one bubble. In particular, if the Ricci curvature of the target manifold has a positive lower bound and the sequence of α-Dirac harmonic maps has bounded index, then the limit of the map part of the necks consist of geodesics of ﬁnite length and the energy identity for the map part hold. In technical terms, these results are achieved by establishing a new decay estimate of the tangential energies of both the map part and the spinor part as well as a new decay estimate for the energy of the spinor as α ↘ 1.