Related papers: On homogeneous Ricci solitons

Let $(M^n,g)$ $(n\ge 4)$ be a complete noncompact $\kappa$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}> 0$ away from a compact set $K$ of $M$. We prove that there is no any $(n-1)$-dimensional compact split limit…

We give a short, direct proof that the full holonomy group of a solution to the Ricci flow is invariant up to isomorphism using the invariance of the reduced holonomy under the flow.

Using quaternions, we give a concise derivation of the Ricci tensor for homogeneous spaces with topology of the 3-dimensional sphere. We derive explicit and numerical solutions for the Ricci flow PDE and discuss their properties. In the…

We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to…

In this paper, we investigate the nature of Einstein solitons, whether it is steady, shrinking or expanding on almost $\alpha$-cosymplectic $3$-manifolds. We also prove that a simply connected homogeneous almost $\alpha$-cosymplectic…

Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have…

A Ricci soliton is a natural generalization of an Einstein metric. On a pseudo-Riemannian manifold (M, g), it is defined by : $LX g + \r{ho} = {\lambda} g, where X is a smooth vector field on M , LX denotes the Lie derivative in the…

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1)…

We study monotonic quantities in the context of combined geometric flows. In particular, focusing on Ricci solitons as the ambient space, we consider solutions of the heat type equation integrated over embedded submanifolds evolving by mean…

Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. This paper deals with the study of conformal Ricci solitons within the framework of warped product manifolds which extends the notion of usual…

In [4], we proved that every noncompact ancient $\kappa$-solution to the Ricci flow in dimension $3$ is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that…

In this paper, we study $\kappa$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the…

Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh…

In the present work we find the Lie point symmetries of the Ricci flow on an $n$-dimensional manifold. and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this…

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution…

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the…

We show that for any solvable Lie group of real type, any homogeneous Ricci flow solution converges in Cheeger-Gromov topology to a unique non-flat solvsoliton, which is independent of the initial left-invariant metric. As an application,…

The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in…

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the…

We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the…