The Ricci flow for simply connected nilmanifolds
By Jorge Lauret. 2010
51 citations recorded by the source corpus.
We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))|g(t) converges smoothly to a Ricci soliton, uniformly on compact sets. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly non-isomorphic to N.
Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging
View Paper: https://www.semanticscholar.org/paper/15c62ff3bcc8e7ab6aad74413553d6f5a504bcda
How researchers see this paper
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In Applied Mathematics
- Anna Fino
- Ramiro A. Lafuente
- Luigi Vezzoni
- Michael Jablonski
- Romina M. Arroyo
- V. Manero
- M. Fernández
- Fabio Paradiso
In Geometry and Topology
- Anna Fino
- Luigi Vezzoni
- Ramiro A. Lafuente
- Michael Jablonski
- Romina M. Arroyo
- V. Manero
- M. Fernández
- Fabio Paradiso
In Radiology, Nuclear Medicine and Imaging
- Luigi Vezzoni
- Romina M. Arroyo
- V. Manero
- Anna Fino
- Michael Jablonski
- Ramiro A. Lafuente
- M. Fernández
Citations in the corpus
- Laplacian Flow of Closed G$$_2$$2-Structures Inducing Nilsolitons (2013, methodology): “4 The limit can be also computed fixing the G2-structure and changing the Lie bracket as in [23].”
- Homogeneous Ricci Solitons are Algebraic (2013, background): “Homogeneous Ricci solitons arise naturally as limits under the Ricci flow [15, 14] and, independently, hold a distinguished place apart from other homogeneous metrics.”
- The long‐time behavior of the homogeneous pluriclosed flow (2017, methodology): “Resembling the Ricci flow (see [17]), in Theorem A the ODE analysis is made possible by a close link with geometric invariant”
- Special Hermitian metrics on compact solvmanifolds (2015, methodology): “The idea in [15] consists in adapting the argument used by Lauret in [32] to study the Ricci flow regarding the pluriclosed flow in the invariant case as a flow of brackets instead of invariant metrics.”