We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on Λ(x,t)=(|∇T(x,t)|^2g(t)+|Rm(x,t)|^2g(t))^{1/2} will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ(x,t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x,t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structure φ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to φ will converge to a torsion-free G2 structure which is in the orbit of φ under diffeomorphisms isotopic to the identity. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.

This work is devoted to the numerical computation of complex band structure $\k=\k(\omega)\in\mathbb C^3$ for positive $\omega$ of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is to fix two degrees of freedom in $\k\in\mathbb C^3$ and to view the remaining one as the eigenvalue of a quadratic operator pencil derived from Maxwell's equations. Then Yee's scheme is employed to discretize $\nabla\times$ and $\k\times$ operators in this quadratic operator pencil. Distinct from the others' works which either ignore or directly exploit the Hamiltonian structure of the spectrum of the resulting QEP, we reformulate this QEP into an equivalent $\top$-palindromic QEP to facilitate the use of superior structure-preserving algorithms. Ultimately we rely on the structured Arnoldi algorithm, namely the G$\top$SHIRA algorithm, to compute eigenvalues of a $\top$-skew-Hamiltonian pair which are near or in $[-2,2]$, a much narrower region than the whole positive real axis in the origin problem. Moreover, to accelerate the inner iterations of the G$\top$SHIRA algorithm, we propose the preconditioning technique, making most of the eigenmatrix, which can essentially be seen as the Kronecker product of three discrete Fourier transformation matrices, of the commutative discretized $\partial_x,\partial_y,\partial_z$ operators. The advantage of our method is discussed in detail and corroborated by several numerical results.

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [36]. The proof extends the argument of Huang-Wang [26].

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature 􏰢 1 in the sense of Alexandrov.