Dirac-harmonic maps from index theory
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Cites background or result from "Dirac-harmonic maps from index theo..."
...armonic maps have already been obtained. This includes the regularity of solutions [CJLW05], [Zhu09], [WX09] and the energy identity [CJLW05]. In addition, an existence result for uncoupled solutions [AG12], for the boundary value problem [CJW], [CJWZ13] and for a nonlinear version of Dirac-geodesics [Iso12] have been established. A heat flow approach for Dirac-harmonic maps has been studied in [Bra13a],...
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...lution of the Euler-Lagrange equations (3.4) and (3.5) uncoupled, if φis a harmonic map. Using tools from index theory, a general existence result for uncoupled Diracharmonic maps could be derived in [AG12]. Since the index of the twisted Dirac-operator does not change when considering a connection with torsion on φ−1TNthe arguments from [AG12] can also be applied in our case. Thus, let us briefly recall...
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..., where D/Tor 0 ψ∈ Γ(ΣM⊗ φ−1 0 TN), we have to construct for any smooth variation φtof φ0 a smooth variation of ψtsatisfying d dt R Mhψt,D/ Tor t ψti t=0 = 0. This is the same argument as Cor. 5.2 in [AG12]. Note that D/and D/Tor have the same principal symbol and the same index. Hence, this smooth variation can be constructed by assuming that the index α(M,σ,E) is non-trivial, see [AG12], Prop. 8.2 and...
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30 citations
Cites methods from "Dirac-harmonic maps from index theo..."
...There have been other approaches, such as [18, 7, 2, 4]....
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Cites background from "Dirac-harmonic maps from index theo..."
...This includes several analytical results [11], [23], [14], [26] and an existence result for uncoupled solutions [2]....
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...In particular, if f = 0, then φ ∈ W 2,p loc for all p ∈ [1, 2) and φ ∈ W 1,q loc for all q ∈ [1,∞)....
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17 citations
Cites background from "Dirac-harmonic maps from index theo..."
...These solutions constitute new examples of nontrivially coupled Dirac-harmonic maps; see [14] for an explicit example of coupled Dirac-harmonic map from surfaces and [1;5] for constructions and existence of uncoupled Dirac-harmonic maps (in the sense that the map part is an ordinary harmonic map) from surfaces and high dimensional manifolds; in section 4, we prove the global existence of the Dirac-geodesic flow (Theorem 1....
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15 citations
Cites background from "Dirac-harmonic maps from index theo..."
...Other examples can be found in [18], [3]....
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References
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"Dirac-harmonic maps from index theo..." refers background or result in this paper
...e the closed target manifold N has non-positive sectional curvature. Then there exists an energy-minimizing (hence harmonic) map in every homotopy class of smooth maps from any closed manifold Minto N[14, 30]. As an application, pick any closed connected Riemannian spin manifold N with non-positive sectional curvature and dimension n≡ 1 (8). Let N′ be any n-dimensional closed Riemannian spin manifold with...
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...s harmonic but this time fb∗TCP2 becomes trivial, hence no non-trivial Dirac-harmonic map can be found using our methods. In an analogous way, existence results for harmonic maps by e.g. EellsSampson [14] (see also [30]) or Y.L. Xin [35] give Dirac-harmonic maps provided the corresponding α-genus does not vanish. We summarize the results in the following two theorems, where we assume all surfaces to b...
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"Dirac-harmonic maps from index theo..." refers background or result in this paper
...One easily verifies that the only parts in [29] using orientation are parts of the introduction, the definition of φ, Lemma 1....
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...In order to facilitate the comparison to [29] we adapt to their notation to a large extend....
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...As already said above, in [29] it is claimed that M should be orientable....
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...of the results of [29] also hold in this non-orientable case....
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...The following theorem can be deduced from the proofs in [29]:...
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