2019
DOI: 10.1007/s12220-019-00266-4
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Spacelike Mean Curvature Flow

Abstract: We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space R n,m , which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the G 2 -Laplacian flow in cases related to coassociative fibrations. ContentsSpacelike mean curvature flow has been studied in codimension 1 by Ecker and Huisken [14], Ecker [11][12] and also Gerhardt [17]… Show more

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Cited by 22 publications
(15 citation statements)
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“…Fino-Raffero [17] studied so-called "extremely Ricci-pinched" G 2 -structures, a special class of G 2 -structure introduced by Bryant [4] (together with the predating examples of Bryant [4] and Lauret [25]), and showed the long time existence of the flow; in these cases the flow does not converge as t → ∞. In [29], Lambert-Lotay proved the long-time existence and convergence of the G 2 -Laplacian flow in the case of semi-flat coassociative T 4 -fibrations, via the reduction of the G 2 -Laplacian flow to a spacelike mean curvature flow in H 2 (T 4 ) = R 3,3 (the connection with spacelike mean curvature flow is another observation due to Donaldson [10]).…”
Section: The G 2 -Laplacian Flowmentioning
confidence: 99%
“…Fino-Raffero [17] studied so-called "extremely Ricci-pinched" G 2 -structures, a special class of G 2 -structure introduced by Bryant [4] (together with the predating examples of Bryant [4] and Lauret [25]), and showed the long time existence of the flow; in these cases the flow does not converge as t → ∞. In [29], Lambert-Lotay proved the long-time existence and convergence of the G 2 -Laplacian flow in the case of semi-flat coassociative T 4 -fibrations, via the reduction of the G 2 -Laplacian flow to a spacelike mean curvature flow in H 2 (T 4 ) = R 3,3 (the connection with spacelike mean curvature flow is another observation due to Donaldson [10]).…”
Section: The G 2 -Laplacian Flowmentioning
confidence: 99%
“…It is not difficult to show that any closed G 2 -structure with a semi-flat coassociative fibration has special torsion of positive type (see §3.4). This is interesting in light of the fact that semi-flat coassociative fibrations are preserved by the Laplacian flow [20].…”
Section: Maximal Submanifoldsmentioning
confidence: 99%
“…20. (Triple root antipodal to single root ) We now suppose that Q → M is a U(2) +structure of type S such that for all points on M the polynomial S (4.11) has a triple root and an antipodal single root.…”
mentioning
confidence: 99%
“…e.g. [11,18,20,22,30]). Furthermore, the G 2 -structure ϕ = π * ω ∧ θ + π * ρ on the total space of the S 1 -bundle π : M → N is closed whenever the 2-form ω on N is symplectic and ρ satisfies the condition dρ = −ω 0 ∧ ω.…”
Section: Introductionmentioning
confidence: 99%