Seminar

In this talk, we will discuss some gluing techniques in calibrated geometry. We prove that for a compact manifold, there exist lots of metrics in any conformal class, with respect to which, in each nonzero de Rham current homology class of dimension smaller than half the manifold dimension, an area minimizer can be realized by a liner combination of some fixed finitely many smooth submanifolds (viewed as currents).

We can also apply similar gluing techniques to a submanifold with mild singularities. For instance, we construct examples of compact Riemannian manifolds which support (coflatly) calibrated codimension-one singular compact submanifolds. If time permits, we may discuss some connections between semi-definite calibrated geometry and optimal transportations discovered in "Pseudo-Riemannian geometry calibrates optimal transportation" by Kim, McCann and Warren.