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Register a new userA remark on the Laplacian flow and the modified Laplacian co-flow in G2-Geometry
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We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].
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We apply the general Ansatz in geometric flows on homogeneous spaces proposed by Jorge Lauret for the Laplacian co-flow of invariant $G_2$-structures on a Lie group, finding an explicit soliton on a particular almost Abelian $7$-manifold.
We prove the hypersymplectic flow of simple type on standard torus $mathbb{T}^4$ exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity-one $G_2$-Laplacian flow on a compact $7$-manifold which exists for all time and converges to a torsion-free $G_2$ structure modulo diffeomorphisms.
We study the existence of left invariant closed $G_2$-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $G_2$-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on $R^7$ in a similar way as in [23] we prove that the underlying metrics $g(t)$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as $t$ goes to infinity.
We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-infty,T)$, with $0<T<+infty$. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while in other cases the solution is ancient or eternal, that is it is defined on $(-infty, infty)$.
Matrices associated with graphs, such as the Laplacian, lead to numerous interesting graph problems expressed as linear systems. One field where Laplacian linear systems play a role is network analysis, e. g. for certain centrality measures that indicate if a node (or an edge) is important in the network. One such centrality measure is current-flow closeness. To allow network analysis workflows to profit from a fast Laplacian solver, we provide an implementation of the LAMG multigrid solver in the NetworKit package, facilitating the computation of current-flow closeness values or related quantities. Our main contribution consists of two algorithms that accelerate the current-flow computation for one node or a reasonably small node subset significantly. One sampling-based algorithm provides an unbiased estimation of the related electrical farness, the other one is based on the Johnson-Lindenstrauss transform. Our inexact algorithms lead to very accurate results in practice. Thanks to them one is now able to compute an estimation of current-flow closeness of one node on networks with tens of millions of nodes and edges within seconds or a few minutes. From a network analytical point of view, our experiments indicate that current-flow closeness can discriminate among different nodes significantly better than traditional shortest-path closeness and is also considerably more resistant to noise -- we thus show that two known drawbacks of shortest-path closeness are alleviated by the current-flow variant.
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