Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

Abstract

For a sequence of coupled fields {(φ n , ψ n )} from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.

Asymptotic analysis for Dirac-harmonic maps from degenerating spin surfaces and with bounded index

“…When the domain is fixed, the blow-up theory for a sequence of Dirac-harmonic maps with uniformly bounded energy has been systematically studied in [2,19,26] for Diracharmonic maps and in [11] for the case of a more general model. To study the existence problem for Dirac-harmonic maps, a heat flow approach was investigated in [4,5,12,15]; see [13][14][15] for the blow-up analysis of the corresponding approximate Dirac-harmonic maps. Roughly speaking, the results of these works assert that the failure of strong convergence occurs at finitely many energy concentration points.…”

From Harmonic Maps to the Nonlinear Supersymmetric Sigma Model of Quantum Field Theory: at the Interface of Theoretical Physics, Riemannian Geometry, and Nonlinear Analysis

“…These equations were derived in [9], and the investigation of the regularity of their solutions has been started in [8]. Further aspects and extensions of the model have been studied in [5,6,10,11,12,14,15,16,39,44,42,70,76,85,84] and many other papers. For the regularity theory, the method of Rivière [64] turned out to be very useful.…”

“…A general superdiffeomorphism then combines those two types of translations. Under the coordinate transformation (42), D is transformed to…”

Harmonic maps and their generalizations

“…这些方程在文献 [60] 中被推导出来, 关于这些方程解的正则性理论在文 献 [61] 中开始建立起来. 这一模型进一步的侧面和延伸 2) 可参见文献 [63][64][65][66][67][68][69] 和 [70][71][72][73][74][75] 等. 在正则性 理论中, Rivière [76] 的方法被证明是非常有效的.…”