Reference : An entropy formula for the heat equation on manifolds with time-dependent metric, app...
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/13907
An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions
English
Guo, Hongxin [Wenzhou University > School of Mathematics and Information Science]
Philipowski, Robert mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Thalmaier, Anton mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Feb-2015
Potential Analysis
Springer
42
2
483-497
Yes (verified by ORBilu)
International
0926-2601
1572-929X
Amsterdam
The Netherlands
[en] Ricci flow ; Brownian motion · ; Entropy
[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.
Researchers ; Professionals
http://hdl.handle.net/10993/13907
10.1007/s11118-014-9442-5
http://dx.doi.org/10.1007/s11118-014-9442-5
FnR ; FNR7628746 > Anton Thalmaier > GEOMREV > Geometry of random evolutions > 01/03/2015 > 28/02/2018 > 2014

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