Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An entropy formula for the heat equation on manifolds with
  time-dependent metric, application to ancient solutions   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.

13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn