We develop a new approach to prove the
-entropy
formula for the Witten Laplacian via warped product on Riemannian manifolds,
giving a natural geometric interpretation of the quantities appearing in the
-entropy formula. We
also prove the
-entropy
formula for the Witten Laplacian on compact Riemannian manifolds with time
dependent metrics and potentials, as well as for the backward heat equation associated
with the Witten Laplacian on compact Riemannian manifolds equipped with Lott’s
modified Ricci flow. Our results extend to complete Riemannian manifolds with negative
-dimensional
Bakry–Émery Ricci curvature, and to compact Riemannian manifolds with
-super
-dimensional
Bakry–Émery Ricci flow. As an application, we prove that the optimal
logarithmic Sobolev constant on compact manifolds equipped with the
-super
-dimensional
Bakry–Émery Ricci flow is decreasing in time.