Article (Scientific journals)
An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions
Guo, Hongxin; Philipowski, Robert; Thalmaier, Anton
2015In Potential Analysis, 42 (2), p. 483-497
Peer Reviewed verified by ORBi
 

Files


Full Text
POTA_ancient_solutions.pdf
Author preprint (104.06 kB)
Download

All documents in ORBilu are protected by a user license.

Send to



Details



Keywords :
Ricci flow; Brownian motion ·; Entropy
Abstract :
[en] 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.
Disciplines :
Mathematics
Author, co-author :
Guo, Hongxin ;  Wenzhou University > School of Mathematics and Information Science
Philipowski, Robert ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Thalmaier, Anton ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
yes
Language :
English
Title :
An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions
Publication date :
February 2015
Journal title :
Potential Analysis
ISSN :
1572-929X
Publisher :
Springer, Amsterdam, Netherlands
Volume :
42
Issue :
2
Pages :
483-497
Peer reviewed :
Peer Reviewed verified by ORBi
FnR Project :
FNR7628746 - Geometry Of Random Evolutions, 2014 (01/03/2015-28/02/2018) - Anton Thalmaier
Available on ORBilu :
since 30 December 2013

Statistics


Number of views
325 (43 by Unilu)
Number of downloads
325 (27 by Unilu)

Scopus citations®
 
8
Scopus citations®
without self-citations
8
OpenCitations
 
7
WoS citations
 
8

Bibliography


Similar publications



Contact ORBilu