Entropy, stability and harmonic map flow

Boling, Jess; Kelleher, Casey Lynn; Streets, Jeffrey

doi:10.1090/tran/6949

Cited by 8 publications

(9 citation statements)

References 22 publications

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“…This mirrors similar entropy concepts defined for the mean curvature flow, Yang-Mills flow, and the harmonic map heat flow, in [7], [16], and [2], respectively. The quantity λ (ϕ, σ) is shown in [8] to be invariant under the scaling (ϕ, σ) → c 3 ϕ, c 2 σ .…”

Section: Spacesupporting

confidence: 61%

Isometric flows of $G_2$-structures

Grigorian¹

2020

Preprint

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We survey recent progress in the study of flows of isometric G2-structures on 7-dimensional manifolds, that is, flows that preserve the metric, while modifying the G2-structure. In particular, heat flows of isometric G2structures have been recently studied from several different perspectives, in particular in terms of 3-forms, octonions, vector fields, and geometric structures. We will give an overview of each approach, the results obtained, and compare the different perspectives.

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Section: Spacesupporting

confidence: 61%

Isometric flows of $G_2$-structures

Grigorian¹

2020

Preprint

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“…As a result, it is not strong enough to control the small scale behaviour of a G 2 -structure ϕ. In this section, motivated by analogous functionals for the mean curvature flow [CM12], the high dimensional Yang-Mills flow [KS16] and the Harmonic map heat flow [BKS17], we introduce an entropy functional, and use it and the almost monotonicity of Section 5.1 to establish an ǫ-regularity result, as well as to show that small entropy controls torsion.…”

Section: Entropy and ε-Regularitymentioning

confidence: 99%

“…Inspired by work of Colding-Minicozzi in [CM12] and Boling-Kelleher-Streets on the harmonic map heat flow [BKS17] and work of Kelleher-Streets on the Yang-Mills flow [KS16] we define an entropy functional and use it in Theorem 5.15 to establish that, if we have sufficiently small entropy, then we have long time existence and convergence of the flow to a G 2 -structure ϕ ∞ with small divergence-free torsion.…”

Section: Introductionmentioning

confidence: 99%

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

2019

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We study a flow of G2-structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the underlying connection yields a nice reaction-diffusion equation for the torsion along the flow. We define a scale-invariant quantity Θ for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi [CM12] on the mean curvature flow, we define an entropy functional and after proving an ε-regularity theorem, we show that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a G2-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth G2-structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.

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“…Thus, we require a quantity sturdier than Θ. Motivated by analogous functionals for the mean curvature flow [CM12], the high dimensional Yang-Mills flow [BKS17], the harmonic map heat flow [KS16] and the isometric flow of G 2 -structures [DGK21], we define the following entropy functional. Definition 5.3.…”

Section: Singularity Structurementioning

confidence: 99%

Harmonic flow of $\mathrm{Spin}(7)$-structures

Dwivedi¹,

Loubeau²,

Earp³

2021

Preprint

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We formulate and study the isometric flow of Spin(7)-structures on compact 8-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of H-structures. We then specialise to H = Spin(7) ⊂ SO(8), obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an ε-regularity theorem. Moreover, we prove Cheeger-Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-I singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric Spin(7)-structures, based on squares of spinors, which may be of independent interest.

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Entropy, stability and harmonic map flow

Cited by 8 publications

References 22 publications

Isometric flows of $G_2$-structures

Isometric flows of $G_2$-structures

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

Harmonic flow of $\mathrm{Spin}(7)$-structures

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