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A gradient flow of isometric $G_2$ structures

Banff International Research Station for Mathematical Innovation and Discovery

We will talk about a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various an- alytic aspects of the flow including global and local derivative estimates, a compactness theorem and a local monotonicity formula for the solutions. We also study the evolution equation of the torsion and show that under a modification of the gauge and of the relevant connection, it satisfies a nice reaction-diffusion equation. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss 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 will prove that if the singularity is Type I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton of the flow. This is a joint work with Pana- giotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).

Mathematics; Differential geometry; Geometry

video/mp4

Author affiliation: University of Waterloo

2019-11-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72219

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/