Abstracts Riemannian 2024

Based on analogy with J-holomorphic maps, Aaron Smith proposed in 2011 a class of first-order PDEs whose solutions are mappings that conformally parametrize calibrated submanifolds in ambient spaces equipped with a vector cross product. I will report on previous and recent joint works with Spiro Karigiannis (Waterloo) and Jesse Madnick (Oregon) on the analytical properties and variational characterization of these mappings, focusing on the case of associative submanifolds in ambient spaces equipped with a G2-structure.

I will describe progress towards the construction of complete Calabi-Yau metrics on the complement of ample, simple normal crossings anti-canonical divisors. This talk will discuss joint works with Y. Li, and F. Tong and S.-T. Yau.

It is well-known that the number of J-holomorphic curves in an almost manifold can change when we vary the almost complex structure J. I will talk about an analogous phenomenon in gauge theory on surfaces, where the number of solutions to certain partial differential equations can change when the equations are perturbed. These two phenomena are closely related, as they are both caused by a wall structure in the space of Cauchy-Riemann operators on a surface. This talk is based on joint work with Thomas Walpuski.

We will discuss a family of flows of G₂-structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G₂-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 a joint work with Panagiotis Gianniotis and Spiro Karigiannis (arXiv:2311.05516) and another work in progress.

We study the geometries associated to various 3-forms on a symplectic 6-manifold of different orbital type. As an application, we show that how this can be used to find Lagrangian fibrations and other geometric structures of interest on symplectic manifolds.

A $7$-manifold with a $G_2$-structure $\varphi$ admits a $G_2$-connection with totally skew-symmetric torsion if and only if $d * \varphi = \theta \wedge * \varphi$, where $\theta$ is the Lee form of the $G_2$-structure.
In the talk I will present recent results on $7$-manifolds admitting a $G_2$-connection with closed totally skew-symmetric torsion. In particular, I will discuss the twisted $G_2$ equation, which represents the $G_2$-analogue of the twisted Calabi-Yau equation for $SU(n)$-structures introduced by Garcia-Fernández, Rubio , Shahbazi and Tipler. The talk is based on a joint work with Lucia Merchan and Alberto Raffero.