Seminario Vidal Abascal

I will give an overview over the recent developments in positive sectional curvature. In particular, over the course of this summer negative solutions to both famous Hopf conjectures (existence on products and positive Euler characteristic) have been announced (and only one of those announcements is unreadable AI slop!). On the other hand, under mild symmetry assumptions (an isometric action of rank ≥ 3), one can show that the Euler characteristic conjecture is true.

This is joint work with Dennis Wulle.

Singular Riemannian foliations generalize isometric group actions and encode rich transverse geometry. In the polar case, the leaf space carries a natural orbifold structure.

We introduce and study a class of singular Riemannian foliations with sections that we call Coxeter polar foliations. We obtain two main reconstruction results. First, we prove that, for a singular Riemannian foliation admitting sections, every Riemannian orbifold metric on the leaf space is induced by a Riemannian metric on the ambient manifold adapted to the foliation. Second, we show that the leaf space of a Coxeter polar foliation admits a complete system of weights that uniquely determines both the foliation and the ambient manifold up to foliated diffeomorphism.

This is part of an ongoing collaboration with Diego Corro.

Nearly G₂ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, Spin(7). We study nearly parallel G₂-structures invariant under a cohomogeneity-one action of SO(4), and construct the first inhomogeneous nearly parallel G₂-structure on the Berger space SO(5)/SO(3). The cone over this metric has full holonomy Spin(7). The existence of the solution is established by a rigorous computer-assisted shooting argument.

We also establish a local desingularisation of a finite quotient of the sine cone over S³ × S³ by gluing in the asymptotically conical C₇ bubble constructed by Foscolo–Haskins–Nordström. We show that as the singular orbit of a family of nearly parallel G₂ manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free G₂-metric of C₇. The talk is based on a joint work with Simon Salamon.

In this talk, we will establish that a compact Riemannian manifold admitting a nontrivial closed conformal vector field is locally conformally flat under a positive directional Ricci curvature condition.

Complex projective space ℂPⁿ, which parametrizes the complex lines in ℂⁿ⁺¹, is one of the most extensively studied objects in algebraic and complex geometry. The first Einstein metric discovered on ℂPⁿ was the Fubini–Study metric, and in 1965 Berger asked whether it was the only one. In 1982, Ziller constructed a second homogeneous Einstein metric in odd complex dimensions. Berger's question remained open in even complex dimensions, where Fubini–Study was the only known Einstein metric.

Together with Gonzalo Cao Labora (EPFL, Switzerland), we have constructed new Einstein metrics on ℂPⁿ for n ∈ {3, 4, 5, 6, 7}. These are the first known non-homogeneous examples and, together with the Fubini–Study and Ziller metrics, the only ones known to date. Our construction settles Berger's question in the previously open cases n = 4 and n = 6.

The construction relies on a cohomogeneity-one action by the Lie group SO_n+1, which allows us to reduce the Einstein equations to a system of ODEs, reparametrized by the mean curvature of the principal orbits. One of the novel aspects of the work is the use of a computer-assisted proof, a methodology that has so far been relatively unexplored in differential geometry. We use interval arithmetic and Krawczyk's fixed-point theorem, a generalization of Newton's method that makes it possible to turn numerical analysis into a rigorous proof.