Geometry and Analysis Seminar

I will describe an extremal problem for the fundamental tone of submanifolds in the hyperbolic space, and will show that singular minimal submanifolds occur as natural maximisers for it. I will also discuss a closely related rigidity phenomenon for the Laplacian spectra of minimal submanifolds.

$3$-$(\alpha,\delta)$-Sasaki manifolds are a natural generalisation of $3$-Sasaki manifolds, which in dimension $7$ are intricately related to $G_2$ geometry. We show how these are closely related to various types of quaternionic Kähler orbifolds via connections with skew-torsion and an interesting canonical submersion. Making use of this relation we discuss curvature operators and show that in dimension 7 many such manifolds have strongly positive curvature, a notion originally introduced by Thorpe.