Ancient gradient flows of elliptic functionals and Morse index

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions codimensions, we classify ${\bf S}^n$ with low area: they are steady or shrinking equatorial spheres. the case S}^3$, more relaxed area bounds: equators Clifford tori. embedded curve shortening S}^2$, completely bounded length: circles.