A spinorial energy functional: critical points and gradient flow

Ammann, Bernd; Weiß, Hartmut; Witt, Frederik

doi:10.1007/s00208-015-1315-8

Cited by 24 publications

(57 citation statements)

References 43 publications

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“…The Laplacian flow for co-closed G 2 -structures was introduced by Karigiannis-McKay-Tsui in [KMT12] and a modified co-flow was studied by Grigorian [Gri13]. An approach via gradient flow of energy-type functionals was introduced by Weiss-Witt [WW12] and Ammann-Weiss-Witt in [AWW16].…”

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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Section: Introductionmentioning

confidence: 99%

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

2019

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“…This energy functional has several important symmetries. We restrict here to the invariance under so-called spin diffeomorphisms and refer for the other symmetries to [1]. A spin diffeomorphism is a diffeomorphism f : M → M for which the induced map df : P → P lifts to the spin structureP .…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Spinor Flow

Freibert

Schiemanowski

Weiß

2019

The Quarterly Journal of Mathematics

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“…Let M be a compact spin manifold and N the union of all pairs (g, ϕ) where g is a Riemannian metric on M and ϕ ∈ Γ(Σ(M, g)) is a spinor of the spin manifold (M, g) whose pointwise norm is constant and equal to 1. The spinorial energy functional E, introduced in [2], is defined by…”

Section: Introductionmentioning

confidence: 99%

“…Then, if the fibers are sufficiently short (i.e. ε is small enough), the spinor flow converges to a 2-dimensional sphere in infinite time.This theorem can be seen as a special case of the conjecture that S 1 -principal bundles with suitable Riemannian metrics and sufficiently short fibers together with S 1 -invariant spinors collapse to the base manifold under the spinor flow.In [2] it was observed that the volume-normalized standard metric on S 3 together with a Killing spinor is a critical point of the volume-normalized spinor flow. It is not clear whether this critical point is stable.…”

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The spinorial energy functional: solutions of the gradient flow on Berger spheres

Wittmann

2016

Ann Glob Anal Geom

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We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Weiß, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volumenormalized standard 3-sphere together with a Killing spinor is a stable critical point of the volume-normalized version of the flow. Our results also include an example of a critical point of the volume-normalized flow on the 3-sphere, which is not a Killing spinor.

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A spinorial energy functional: critical points and gradient flow

Cited by 24 publications

References 43 publications

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

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

Homogeneous Spinor Flow

The spinorial energy functional: solutions of the gradient flow on Berger spheres

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