  We show that the Poincar\'e constant of a log-concave measure in Euclidean
space is monotone increasing along the heat flow. In fact, the entire spectrum
of the associated Laplace operator is monotone decreasing. Two proofs of these
results are given. The first proof analyzes a curvature term of a certain
time-dependent diffusion, and the second proof constructs a contracting
transport map following the approach of Kim and Milman.
