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Curvature flows for almost-hermitian Lie groups


Author: Jorge Lauret
Journal: Trans. Amer. Math. Soc. 367 (2015), 7453-7480
MSC (2010): Primary 53C30, 53C44
DOI: https://doi.org/10.1090/S0002-9947-2014-06476-3
Published electronically: December 11, 2014
MathSciNet review: 3378836
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Abstract: We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of $ 2n$-dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to better visualize the possible pointed limits of solutions, under diverse rescalings, as well as to address regularity issues. Immortal, ancient and self-similar solutions arise naturally from the qualitative analysis of the bracket flow. The Chern-Ricci flow and the symplectic curvature flow are considered in more detail.


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Additional Information

Jorge Lauret
Affiliation: Universidad Nacional de Córdoba, FaMAF and CIEM, 5000 Córdoba, Argentina
Email: lauret@famaf.unc.edu.ar

DOI: https://doi.org/10.1090/S0002-9947-2014-06476-3
Received by editor(s): September 28, 2013
Received by editor(s) in revised form: March 19, 2014
Published electronically: December 11, 2014
Additional Notes: The author’s research was partially supported by grants from CONICET, FONCYT and SeCyT (Universidad Nacional de Córdoba)
Article copyright: © Copyright 2014 American Mathematical Society