Multiple integrals in the calculus of variations and nonlinear elliptic systems

We introduce a functional that couples the nonlinear sigma model with a spinor field: 
$L=\int_M[|d\phi|^2+(\psi,\D\psi)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.

Dirac-Harmonic Maps

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.

Conformal Riemannian Maps between Riemannian Manifolds, Their Harmonicity and Decomposition Theorems

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle Σ M, and any compact Riemannian manifold N, we show an ϵ-regularity theorem for weakly Dirac-harmonic maps (ϕ, ψ):M ⊗ Σ M → N ⊗ ϕ * TN. As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when n = 2. For n ≥ 3, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in . If, in addition, ψ ∈ W 1,p for some , then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.

https://www.math.purdue.edu/~wang2482/dirac_harmonic.pdf

Regularity of Dirac-Harmonic Maps

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space Rn, the hyperbolic space Hn, and a Riemannian manifold Sn (n⩾3) with the Schwarzschild metric to any Riemannian manifold N.

/pdf/liouville-theorems-for-dirac-harmonic-maps-4q0fg596aa.pdf

Liouville Theorems for Dirac-Harmonic Maps

In this note, we prove conformal lower bounds for Dirac operators of submanifolds in terms of conformal and extrinsic quantities

Extrinsic conformal lower bounds of eigenvalue for Dirac operator

We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from a Riemann surface to a sphere $\S^n$. We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.

pdf/regularity-theorems-and-energy-identities-for-dirac-harmonic-295m5ux46j.pdf

Regularity Theorems and Energy Identities for Dirac-Harmonic Maps

VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jager–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

On VT -harmonic maps

/pdf/dirac-harmonic-maps-between-riemann-surfaces-4rsdc24cw3.pdf

Dirac-harmonic maps between Riemann surfaces

Dirac-geodesics are Dirac-harmonic maps from one dimensional domains. In this paper, we introduce the heat flow for Dirac-geodesics and establish its long-time existence and an asymptotic property of the global solution. We classify Dirac-geodesics on the standard 2-sphere $S^2(1)$ and the hyperbolic plane $\mathbb {H}^2$, and derive existence results on topological spheres and hyperbolic surfaces. These solutions constitute new examples of coupled Dirac-harmonic maps (in the sense that the map part is not simply a harmonic map).

/pdf/dirac-geodesics-and-their-heat-flows-2fn9wrtybz.pdf

Dirac-geodesics and their heat flows

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