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MiS Preprint

Geometric analysis of a mixed elliptic-parabolic conformally invariant boundary value problem

Jürgen Jost, Lei Liu and Miaomiao Zhu


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 $\alpha >1$; the solutions of the resulting Euler-Lagrange equations are called $\alpha$-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 $\alpha \searrow 1$ 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 $\alpha$-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 $\alpha$-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 $\alpha$-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 $\alpha$-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 $\alpha \searrow 1$.

MSC Codes:
53C43, 58E20
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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2018 Repository Open Access
Jürgen Jost, Lei Liu and Miaomiao Zhu

Geometric analysis of a mixed elliptic-parabolic conformally invariant boundary value problem