  We give a simple, direct proof of the backward uniqueness of solutions to a
class of second-order geometric evolution equations including the Ricci and
cross-curvature flows. The proof, based on a classical argument of
Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in
the place of Carleman inequalities. We also demonstrate the applicability of
the technique to the $L^2$-curvature flow and other higher-order equations.
