Abstract We use the geometric mean to parametrize metrics in Hassan–Rosen ghost-free bimetric theory and pose initial-value problem. The of two positive definite symmetric matrices is a well-established mathematical notion which can be under certain conditions extended quadratic forms having Lorentzian signature, say g f . In such case, null cone metric h middle cones appearing as average space...

We prove that for the mean curvature flow of closed embedded hypersurfaces, intrinsic diameter stays uniformly bounded as approaches first singular time, provided all singularities are neck or conical type. In particular, assuming Ilmanen’s multiplicity one conjecture and no cylinder conjecture, we conclude in two-dimensional case always bounded. also obtain sharp $$L^{n-1}$$ bound curvature. T...