# Calibrations on compact manifolds (UNUSUAL TIME!!!)

Type:
PhD Student Seminar
Audience:
Open to the Public
Building:
Zrinyi u. 14
Room:
310/A
Wednesday, October 4, 2017 - 12:00pm
Let $M$ be an $n$-dimensional compact Riemannian manifold with $n\geq 5$. As is well known, a unit-length vectorfield $V$ on $M$ exists if and only if the Euler characteristic $\chi(M)$ vanishes. Thurston showed that under this condition one can also find a codimension-1 foliation on $M$: that is, a unit-length vectorfield $V$ whose perpendicular hyperplane-distribution is integrable.
In the talk we discuss the possibility that the vectorfield is in addition divergence-free. If the perpendicular hyperplane-distribution is integrable, such vectorfields $V$ correspond to codimension-1 foliations by minimal surfaces and in this case $V$ is called a calibration. Such foliations are rather special and their existence depends very much on the metric, not just the topology. It turns out however, that in the opposite contact'' case, when the perpendicular hyperplane-distribution is non-integrable, the existence of unit-length divergence-free vectorfields is determined solely by the Euler characteristic, provided $n\geq 5$. This can be proved using a method of construction very similar to Nash's construction of smooth isometric embeddings.