  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 and codimensions, we
classify ancient mean curvature flows in S^n with low area: they are steady or
shrinking equatorial spheres. In the mean curvature flow case in S^3, we
classify ancient flows with more relaxed area bounds: they are steady or
shrinking equators or Clifford tori. In the embedded curve shortening case in
S^2, we completely classify ancient flows of bounded length: they are steady or
shrinking circles.
