We show that the moduli space of deformations of a compact coassociative submanifold L has a natural local embedding as a submanifold of H(L,R). We show that a G2-manifold with a T -action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R with positive induced metric where R ∼= H(T ,R). By studying minimal surfaces in quadrics we show...

A short proof of the Caratheodory conjecture about index of an isolated umbilic on the convex 2–dimensional sphere is suggested.

Abstract We give a short proof of local well-posedness for the initial boundary value problem in general relativity with sole condition requirement that is umbilic. This includes as special case totally geodesic we had previously addressed [ 8]. The based on wave coordinates and key observation momentum constraint always valid umbilic boundaries. allows greater freedom choice conditions, since ...