Workshop

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In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green’s functions associated with certain Kähler metrics. In these lectures, we will explain their approach, broaden the scope of their techniques, and apply these results to study the asymptotic behavior of the Kähler-Ricci flow.

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In this talk I will construct more complete Calabi-Yau metrics of Calabi type. They are higher-dimensional analogues of ALH* gravitational instantons in two dimensions. This work builds on and generalizes the results of Tian-Yau and Hein-Sun-Viaclovsky-Zhang, creating Calabi-Yau metrics that are only polynomially close to the model space. I will also show the uniqueness of such metrics in a given cohomology class with fixed asymptotic behavior.

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In this talk, we will introduce old and recent developments in the studies of collapsing manifolds with sectional and Ricci curvature bounds. 

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In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green’s functions associated with certain Kähler metrics. In these lectures, we will explain their approach, broaden the scope of their techniques, and apply these results to study the asymptotic behavior of the Kähler-Ricci flow.

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I will explain a general and flexible method originally introduced by Richard Melrose for estimating the decay of L^2-harmonic forms at infinity, but focusing on the specific example of L^2-harmonic forms of fibered boundary metrics (e.g. most types of gravitational instantons), in which case the result is due to Hausel-Hunsicker-Mazzeo in 2004 and heavily relies on the parametrix construction of Vaillant.  After outlining the method, I will introduce the pseudodifferential calculus needed and describe  the important steps of the parametrix construction.  I will also indicate other important results that can be obtained with this method.

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We will review classical estimates for elliptic PDEs, and the notions of stability and minimality for hypersurfaces. The interplay among these aspects underlies a vast number of geometric results, via the regularity and compactness theory for stable minimal hypersurfaces. We will discuss recent progress and open questions.

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We will discuss a family of flows of G_2-structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G_2-structures. We will prove short-time existence and uniqueness of solutions to the flows and a priori estimates for some specific flows in the family. We will discuss analogous flows of Spin(7)-structures. This talk is based on arXiv:2311.05516 (joint work with P. Gianniotis and S. Karigiannis) and arXiv:2404.00870.

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Closed G_2-structures on 7-manifolds are defined by a  closed  positive 3-form.  Although linear, the closed condition for a G_2-structure is very restrictive, and no general results on the existence of closed G_2-structures on compact 7-manifolds are known.  In this lecture after an introduction on closed G_2-structures, I will review known results on  the  G_2-Laplacian flow