I will discuss the subject of real Kaehler submanifolds, that is, isometric immersions \(f\colon M^{2n}\to\mathbb{R}^{2n+p}\) of a Kaehler manifold \((M^{2n},J)\) of complex dimension \(n\geq 2\) into Euclidean space with codimension \(p\). In particular, I will present a recent result that shows that for codimension \(2p\leq 2n-1\) generic rank conditions on the second fundamental form of \(f\) imply that the submanifold has to be minimal. In fact, for codimension \(p\leq 11\) we have a stronger conclusion, namely, that \(f\) must be holomorphic with respect to some complex structure in the ambient space.

This is joint work with A. de Carvalho and M. Dajczer.

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in absence of predictability and hence compatibly with quantum field theory and black hole evaporation.