Journal ArticleDOI
Ricci soliton homogeneous nilmanifolds
TLDR
In this paper, the Ricci soliton left invariant Riemannian metric is shown to be unique up to isometry and scaling, if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein.Abstract:
We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric
$_g=cI+D$
for some
$c\in{mathbb R}$
and some derivationD of the Lie algebra
${\mathfrak n}$
ofN, where Ric
$_g$
denotes the Ricci operator of
$(N,g)$
. This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold
$(S,\tilde{g})$
is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set.read more
Citations
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Recent Progress on Ricci Solitons
TL;DR: Ricci solitons are natural generalizations of Einstein metrics and play important roles in the singularity study of the Ricci flow as discussed by the authors, and they are also special solutions to Hamilton's Ricci Flow.
Journal ArticleDOI
Rigidity of gradient ricci solitons
Peter Petersen,William Wylie +1 more
TL;DR: In this paper, the authors define a gradient Ricci soliton to be rigid if it is a flat bundle N × GRk where N is the number of vertices in the bundle.
Journal ArticleDOI
On the classification of gradient Ricci solitons
Peter Petersen,William Wylie +1 more
TL;DR: In this article, it was shown that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n, S n 1 R and R n.
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Rigidity of gradient Ricci Solitons
Peter Petersen,William Wylie +1 more
TL;DR: In this paper, the authors define a gradient Ricci soliton to be rigid if it is a flat bundle with curvature O(n 2 ) where n is the number of vertices.
Journal ArticleDOI
On Gradient Ricci Solitons with Symmetry
TL;DR: In this paper, it was shown that there are no non-compact cohomogeneity one shrinking gradient Ricci solitons with nonnegative curvature, and that the most symmetry one can expect is an isometric cohomogeneous one group action.
References
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Journal ArticleDOI
Three-manifolds with positive Ricci curvature
Book
Geometry of Nonpositively Curved Manifolds
TL;DR: In this article, a self-contained treatment of differentiable spaces of nonpositive curvature is presented, focusing on the symmetric spaces in which every geodesic lies in a flat Euclidean space of dimension at least two.
Journal ArticleDOI
Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms
TL;DR: In this paper, the authors introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.
Journal ArticleDOI
Ricci solitons on compact three-manifolds
TL;DR: In this article, it was shown that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature, which generalizes a result obtained for surfaces by Hamilton.