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: May 2019)
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 show the existence of Dirac-harmonic maps from a compact spin Riemann surface with smooth boundary to a general compact Riemannian manifold via a heat ﬂow method when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor. Technically, we solve a new elliptic-parabolic system arising in geometric analysis that is motivated by the nonlinear supersymmetric sigma model of quantum ﬁeld theory. The corresponding action functional involves two ﬁelds, a map from a Riemann surface into a Riemannian manifold and a spinor coupled to the map. The ﬁrst ﬁeld has to satisfy a second order elliptic system, which we turn into a parabolic system so as to apply heat ﬂow techniques. The spinor, however, satisﬁes a ﬁrst order Dirac type equation. We carry that equation as a nonlinear constraint along the ﬂow. In order to solve this system, we adapt the idea of Sacks-Uhlenbeck to raise the integrand of the harmonic map action to a power α > 1; the solutions of the resulting Euler-Lagrange equations are called α-Dirac harmonic maps. Because of the (unchanged) spinor action, the analysis is more diﬃcult than that of Sacks-Uhlenbeck. Nevertheless, we can carry out the limit α ↘ 1 to solve our original problem.
Then we develop a general spectrum of methods (Pohozaev identity, three-circle method, blow-up analysis, energy identities, energy decay estimates etc.) for the compactness problem of the space of α-Dirac harmonic maps and for a further analysis of the limiting problem. 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. In particular, if the target manifold has a positive lower bound on the 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 consists 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.