Soliton Metrics Research Articles

Constraints on Kaluza–Klein gravity from Gravity Probe B

Using measurements of geodetic precession from Gravity Probe B, we constrain possible departures from Einstein's General Relativity for a spinning test body in Kaluza-Klein gravity with one additional space dimension. We consider the two known static and spherically symmetric solutions of the 5D field equations (the soliton and canonical metrics) and obtain new limits on the free parameters associated with each. The theory is consistent with observation but must be "close to 4D" in both cases.

Classification of Ordered Type Soliton Metric Lie Algebras by a Computational Approach

We classify some soliton nilpotent Lie algebras and possible candidates in dimensions 8 and 9 up toisomorphy.We focus on type of derivations, where is the dimension of the Lie algebras. We present algorithms to generate possible algebraic structures.

Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determine the existence of such a metric by analyzing algebraic properties of the Lie algebra in question and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics. We finish with a similar result for locally left-invariant metrics on compact nilmanifolds.

Explicit Ricci Solitons on Nilpotent Lie Groups

The primary purpose of this paper is to obtain explicit, coordinate-based descriptions of Ricci flow solutions—especially those corresponding to Ricci solitons—on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott’s blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz.

The space of solvsolitons in low dimensions

Up to now, the only known examples of homogeneous nontrivial Ricci soliton metrics are the so-called solvsolitons, i.e., certain left-invariant metrics on simple connected solvable Lie groups. In this article, we describe the moduli space of solvsolitons of dimension ≤ 6 up to isomorphism and scaling. We start with the already known classification of nilsolitons and, following the characterization given by Lauret in (J. Reine Angew. Math. 650:1–21, 2011), we describe the subspace of solvsolitons associated to a given nilsoliton, as the quotient of a Grassmanian by a finite group.

Computing Perelmanʼs ν-functional

This is a short note in which we show how to calculate the value of Perelmanʼs ν-functional for a variety of metrics. In particular we complete the calculation of values for the known 4-dimensional Einstein and shrinking Ricci soliton metrics.

The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kahler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kahler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kahler Ricci flow must converge to a Kahler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kahler Ricci soliton metrics on toric Fano surfaces.

Moduli of Einstein and non-Einstein nilradicals

A nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras.

Riemannian Groupoids and Solitons for Three-Dimensional Homogeneous Ricci and Cross-Curvature Flows

In this paper, we investigate the behavior of three-dimensional homogeneous solutions of the cross-curvature flow using Riemannian groupoids. The Riemannian groupoid technique, originally introduced by J. Lott, allows us to investigate the long-term behavior of collapsing solutions of the flow, producing solitons in the limit. We also review Lott's results on the long-term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross-curvature soliton metrics on Sol and Nil, and show that the cross-curvature flow of SL(2,R) limits to Sol.

Ricci Flow on Surfaces with Conical Singularities

This paper studies the normalized Ricci flow on surfaces with conical singularities. It’s proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear drop and the football, it’s shown that they admit a Ricci soliton metric.

The Ricci flow for nilmanifolds

We consider the Ricci flow for simply connectednilmanifolds. This translates to a Ricci flow on the space ofnilpotent metric Lie algebras. We consider the evolution of the innerproduct and the evolution of structure constants, as well as theevolution of these quantities modulo rescaling. We set up systems ofO.D.E.'s for some of these flows and describe their qualitativeproperties. We also present some explicit solutions for the evolutionof soliton metrics under the Ricci flow.

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.

Compactness theorems for gradient Ricci solitons

In this paper, we prove a compactness theorem for gradient Ricci solitons. Let ( M α , g α ) be a sequence of compact gradient Ricci solitons of dimension n ≥ 4 , whose curvatures have uniformly bounded L n 2 norms, whose Ricci curvatures are uniformly bounded from below, with uniformly lower bounded volume and with uniformly upper bounded diameter; then there must exist a subsequence ( M α , g α ) converging to a compact orbifold ( M ∞ , g ∞ ) with finitely many isolated singularities, where g ∞ is a gradient Ricci soliton metric in an orbifold sense.

Rank-one Einstein solvmanifolds of dimension 7

We use a variational approach to determine explicitly all the 7-dimensional rank-one Einstein solvmanifolds. We prove that each one of the 34 nilpotent Lie algebras of dimension 6 admits a rank-one solvable extension which can be endowed with an Einstein left-invariant Riemannian metric. This also produces 34 Ricci soliton metrics on R 6 which are homogeneous.

The geodesics of the Kaluza-Klein wormhole soliton

The Kaluza-Klein wormhole soliton metric is a regular localized solution with Minkowskian signature, to the sourceless five-dimensional Einstein equations. We apply five-to-three dimensional reduction to convert the problem of geodesic motion of neutral or charged test particles in this metric to a non-relativistic potential problem, which we discuss in detail, studying bound and scattering states. We show that there is no observable difference between scattering of a spinless test particle by a point charge and by a wormhole soliton.

Smooth extensions of a class of 2n soliton metrics

A particular class of 2n soliton solutions of Einstein's equations, obtained from a flat background by the inverse scattering method of Belinsky and Zakharov, (1978), is considered. Straightforward application of the method leads to metrics whose coefficients are singular along certain hypersurfaces. The geometrical nature of the singularities is analysed. It is found that in some cases they represent curvature singularities. In the other cases they originate only on the choice of coordinates and one can find explicitly smooth extensions through these hypersurfaces. Part of the discussion is restricted to the two soliton metrics, but the results are immediately applicable in the general case.