A gradient flow of isometric $\mathrm{G}_2$ structures
| Autor: | Spiro Karigiannis, Shubham Dwivedi, Panagiotis Gianniotis |
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| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Mean curvature flow 010102 general mathematics 01 natural sciences Physics::Fluid Dynamics Singularity Torsion tensor Differential Geometry (math.DG) 0103 physical sciences Compactness theorem Subsequence FOS: Mathematics Gravitational singularity Hausdorff measure 010307 mathematical physics Geometry and Topology 0101 mathematics Balanced flow Mathematics |
| DOI: | 10.48550/arxiv.1904.10068 |
| Popis: | We study a flow of $G_2$ structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists 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 connection gives a nice diffusion-reaction equation for the torsion along the flow. We define a quantity for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi on the mean curvature flow, we define an entropy functional and after proving an $\epsilon$-regularity theorem, we show that low entropy initial data lead to solutions of the flow which exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We also study the finite-time singularities and show that at the singular time the flow converges to a smooth $G_2$ 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 has a subsequence which converges to a shrinking soliton for the flow. Comment: Final revised version following referee's suggestions. Only minor changes. To appear in "Journal of Geometric Analysis". 62 pages |
| Databáze: | OpenAIRE |
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