  We introduce a new entropy functional for nonnegative solutions of the heat
equation on a manifold with time-dependent Riemannian metric. Under certain
integral assumptions, we show that this entropy is non-decreasing, and moreover
convex if the metric evolves under super Ricci flow (which includes Ricci flow
and fixed metrics with nonnegative Ricci curvature). As applications, we
classify nonnegative ancient solutions to the heat equation according to their
entropies. In particular, we show that a nonnegative ancient solution whose
entropy grows sublinearly on a manifold evolving under super Ricci flow must be
constant. The assumption is sharp in the sense that there do exist nonconstant
positive eternal solutions whose entropies grow exactly linearly in time. Some
other results are also obtained.
