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]. The first author has also worked on boundary conditions for this flow [21] [22]. The elliptic counterpart was studied by Bartnik [3] and Bartnik and Simon [4]. For higher codimensions, less is known. The flow of compact manifolds was investigated by G. Li and Salavessa [24]. The higher codimensional maximal surface equation was recently studied by Y. Li [25].Entire graphs. There are several well-known explicit long-time solutions to spacelike MCF. Throughout the article we let ·, · denote the standard quadratic form with signature (n, m) on R n,m and let |x| 2 = x, x for x ∈ R n,m . Recall that x = 0 is spacelike if |x| 2 > 0, lightlike or null if |x| 2 = 0 and timelike if |x| 2 < 0. The light cone is the set of lightlike vectors.Example 1.1 (Grim Reaper). The Grim Reaper is the unique translating solution to (1.3) in R 1,1 (up to translations, dilations and rotations), given bŷis a self-expander for (1.2) (i.e. X ⊥ = tH) coming out of the light cone. For each t, M t is an embedded hyperbolic space in R n,1 .Explicit solutions may be constructed from Examples 1.1 and 1.2 in higher codimension, simply by evolving in R n,1 ⊂ R n,m .All the examples described thus far are entire graphs, and so it is natural to study this setting, where we have the following existence theorem. Theorem 1.3. Let Ω = R n , so the initial spacelike submanifold M 0 is an entire graph. There exists a spacelike solution M t of mean curvature flow starting at M 0 which is smooth and exists for all t > 0. Furthermore, if the mean curvature of M 0 is bounded, then M t attains the initial data M 0 smoothly as t → 0. See Theorem 5.2 for further details. Notice that we make no assumption on the spacelike condition at infinity, so we can start with initial data that asymptotically develops lightlike directions (like the Grim Reaper), and that we obtain long-time existence even without an initial bound on the mean curvature. This theorem is an extension and improvement of the codimension 1 result proven in [11, Theorem 4.2] (see also Remark 4.4).Remark 1.4. As is to be expected for entire flows we make no statement about uniqueness in Theorem 1.3, and solutions to (1.3) are not unique in general. For example, if we take M 0 to be the Grim Reaper in Example 1.1 at t = 0, which is a translating solution, then any solution constructed by our proof of Theorem 1.3 cannot remain a translator (since it would satisfy |û(x, t) −û 0 (x)| ≤ √ 2nt).We are also prove results on the qualitative behaviour of entire flows. In Section 8 we develop further e...

A report on the hypersymplectic flow

“…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]).…”

Exact Filters and Joins of Closed Sublocales

“…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].…”

“…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.…”

Closed G$_2$-structures on unimodular Lie algebras with non-trivial center

“…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.…”