An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions
by Hongxin Guo, Robert Philipowski and Anton Thalmaier
Abstract
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.
Potential Anal. 42 (2015), 483-497
![[ORIGINAL ARTICLE]](../GRAPHICS/original_article.gif)
http://dx.doi.org/10.1007/s11118-014-9442-5
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