Short-time behaviour of a modified Laplacian coflow of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-structures

Grigorian, Sergey

doi:10.1016/j.aim.2013.08.013

Cited by 51 publications

(100 citation statements)

References 16 publications

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“…Several foundational results for the Laplacian flow for closed G 2 -structures were established in a series of papers [LW17,LW19b,LW19a] by Lotay-Wei. 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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“…with respect to the adapted frame (f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ). Now we claim that χ t , given by (11), with…”

Section: For Anymentioning

confidence: 93%

The Laplacian coflow on almost-abelian Lie groups

Bagaglini

Fino

2018

Annali di Matematica

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“…Unlike the Laplacian flow, a gauge fixing cannot be used to make the Laplacian coflow parabolic, thus a modification is necessary. In [9] Grigorian proposed the following modification…”

Section: From Theorem 21 To the Stability Of The Modified Laplacian mentioning

confidence: 99%

“…

We prove a general result about the stability of geometric flows of "closed"sections of vector bundles on compact manifolds. Our theorem allows to prove a stability result for the modified Laplacian coflow in G2-geometry introduced by Grigorian in [9] and for the balanced flow introduced by the authors in [2].

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confidence: 92%