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 (revised version: October 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 compact Riemann surface with smooth boundary into a general compact Riemannian manifold with uniformly bounded energy. We prove generalized energy identities for both the map part and the spinor part. We also show that the map parts of the α-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 target manifold has a positive lower bounded Ricci curvature or has a ﬁnite fundamental group and the sequence of α-Dirac harmonic maps has bounded Morse index, then the limit of the map part of the necks consist of geodesics of ﬁnite length which ensures the energy identities 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.