The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space R 4 . A Sobolev-type embedding theorem for the second fundamental forms of two dimensional surfaces is also proved, which might be of independent interest.General SMCF naturally arises in higher dimensional hydrodynamics. A singular vortex in a fluid is called a vortex membrane if it is supported on a codimension two subset. The law of locally induced motion of a vortex membrane can by deduced from the Euler equation by applying the Biot-Savart formula. In 2012, Shashikanth [33] first investigated the motion of a vortex membrane in R 4 and showed that it is governed by the two dimensional SMCF. Khesin [21] generalized this conclusion to any dimensional vortex membranes in Euclidean spaces and gave the formal definition of the SMCF which we apply here.The SMCF also emerges in the study of asymptotic dynamics of vortices in the context of superfluidity and superconductivity. Here the model is usually given by a PDE of a complex field and the vortices are just zero sets of the solutions. For example, for the Ginzburg-Landau heat flow, it was shown that asymptotically the energy concentrates on the codimension two vortices which moves along the mean curvature flow [26,19,3,20]. Similar phenomena are observed for wave and Schrödinger type PDEs. In particular, for the Gross-Pitaevskii equation which models the wave function associated with a Bose-Einstein condensate, physics evidences indicate that the vortices would evolve along the SMCF. This was first verified by Lin [29] for the vortex filaments in three space dimensions. For higher dimensions, Jerrard [17] proved this conjecture when the initial singular set is a codimension two sphere with multiplicity one in 2002. It is worth mentioning that he also proposed a notion of weak solution to the SMCF.Besides its physical significance, the SMCF is a rather canonical geometric flow for codimension two submanifolds which can be viewed as the Schrödinger-type counterpart of the well-known mean curvature flow(MCF). In fact, the SMCF has a notable Hamiltonian structure and is volume preserving. The infinite dimensional space of codimension two immersions of a Riemannian manifold admits a generalized Marsden-Weinstein symplectic structure [30], and the SMCF turns out to be the Hamiltonian flow of the volume functional on this space. This fact was first noted by Haller and Vizman [12] where they studied the non-linear Grassmannians. For completeness, we include a detailed explanation in Section 2.1 below.The SMCF is also related to another important Hamiltonian flow, namely, the Schrödinger flow [8,39,7]. The Schrödinger flow stems from the study of ferromagnetism and is the Hamiltonian flow of the energy functional defined on the space ...
We use Sacks–Uhlenbeck's perturbation method to find critical points of the Yang–Mills–Higgs functional on fiber bundles with 2-dimensional base manifolds. Such critical points can be regarded as a generalization of harmonic maps from surfaces, and also a generalization of the so-called twisted holomorphic maps [15]. We prove an existence result analogous to the one for harmonic maps. In particular, we show that the so-called energy identity holds for the Yang–Mills–Higgs functional.
In this paper, we show the uniqueness of Schrödinger flow from a general complete Riemannian manifold to a complete Kähler manifold with bounded geometry. While following the ideas of McGahagan[16], we present a more intrinsic proof by using the distance functions and gauge language.
An efficient scheme is proposed for generating n-qubit GreenbergerHorne-Zeilinger states of n superconducting qubits separated by (n − 1) coplanar waveguide resonators capacitively via adiabatic passage with the help of quantum Zeno dynamics in one step. In the scheme, it is not necessary to precisely control the time of the whole operation and the Rabi frequencies of classical fields because of the introduction of adiabatic passage. The numerical simulations for three-qubit Greenberger-Horne-Zeilinger state show that the scheme is insensitive to the dissipation of the resonators and the energy relaxation of the superconducting qubits. The three-qubit Greenberger-Horne-Zeilinger state can be deterministically generated with comparatively high fidelity in the current experimental conditions, though the scheme is somewhat sensitive to the dephasing of superconducting qubits.
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