arXiv research

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

Study classifies Riemann solitons on specific 3D Lorentzian groups.

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 unimodular case, the Ricci soliton equation does not admit solutions in the set of l…

Left-invariant Cotton solitons on homogeneous manifolds are determined. Moreover, algebraic Cotton solitons are studied providing examples of non-invariant Cotton solitons, both in the Riemannian and Lorentzian homogeneous settings.

New Kähler solitons found that are not $U(n)$-invariant.

The paper classifies solitons in the Heisenberg space.

Classifies solitons on invariant surfaces in solvable Lie group.

New gradient Ricci solitons found for $SU(2)$ invariants.

Smooth approximations of Kähler-Ricci solitons found using quantized metrics and Futaki invariants.

Study expanding gradient Ricci solitons with Euclidean base.

The paper examines Lorentz Ricci solitons on specific Lie groups.

In this paper we show that all conformal metrics to a pseudo-euclidean space invariant under the translation group, and all the conformal metrics product manifold also invariant by translation where F m it is Ricci flat semi-Riemannian manifold, are gradient Ricci almost soliton. We also proved that all conformal metri…

New steady gradient Ricci solitons found on specific four-manifolds.

The paper examines soliton surfaces using a parallel transport frame field in 4D space.

The three-dimensional Heisenberg group $H_3$ has three left-invariant Lorentz metrics $g_1$, $g_2$ and $g_3$. They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric $g_1$ as a Lorentz Ricci soliton. This Ricci soliton $g_1$ is a shrinking non-gradient Ricci soliton. Likew…

Let $F$ be a left invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left invariant Riemannian metric ${\hat{\textbf{\textit{a}}}}$ and a vector field $X$ which is $I_{\hat{\textbf{\textit{a}}}}(M)$-invariant. If the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci so…

New steady solitons found with SO(3) symmetry.

We use the momentum construction for $S^1$-invariant Kähler metrics as developed by Hwang-Singer to construct new examples of steady Kähler-Ricci solitons. We also prove that these solitons are unique in their Kähler class, provided the vector field and the asymptotic behaviour are fixed.

As of today, there are very few known complete shrinking Ricci solitons in dimension 4, and all examples discovered so far are Kähler and/or Einstein. In this note, we prove that any four dimensional J-invariant gradient shrinking Ricci solitons satisfy a differential form identity relating Kählerity annd Einstein-ness…

Study proves structure results for homogeneous spaces supporting specific equations.

The study explores maximal symmetry in Ricci solitons on Lie groups.

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-ba…

The study explores $S^1$-invariant Laplacian flow on 6-manifolds.

New types of Ricci solitons found in 4D Lorentzian geometry.

The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…

Uniform bounds on $SO(2) imes SO(3)$-invariant Ricci solitons on $\mathbb{S}^4$.

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

The present paper deals with the study of Ricci solitons on invariant and anti-invariant submanifolds of $(LCS)_n$-manifolds with respect to Riemannian connection as well as quarter symmetric metric connection.

Study stabilizes translating solitons in hyperbolic space for MCF.

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times P, {\rm d}t^2+g_0)$ and the Killing field is $X=\partial_t$. Similarly to what h…

Study of $k$-almost Yamabe solitons in perfect fluid spacetimes.

In this paper, we investigate the relationship between algebraic soliton metrics and soliton metrics for geometric evolution equations on Lie groups. After discussing the general relationship between algebraic soliton metrics and soliton metrics, we investigate the cross curvature flow and the second order renormalizat…

New method constructs nilpotent Lie algebras from quivers.

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

The paper studies a flow on complex Lie groups, showing convergence to solitons.

Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.

The purpose of this paper is to introduce the Ricci Yang-Mills soliton equations on nilpotent Lie groups. In the 2-step nilpotent setting, we show that these equations are strictly weaker than the Ricci soliton equations. Using techniques from Geometric Invariant Theory, we develop a procedure to build many different k…

New Lie algebras from quivers lead to rigid Ricci solitons.

Study on Hermitian curvature flow on special linear groups, disproving a conjecture and finding non-algebraic solitons.

We consider the non-trivial Ricci soliton on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. It is a Kähler metric invariant by the $U(2)$ action on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. We study its Yamabe equation and prove it has exactly one $U(2)-$invariant solution up to homothecies.

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isom…

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarch…

Classifies timelike translating solitons in Minkowski space.

We construct a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over Kähler-Einstein manifolds of positive scalar curvature. They include a four-dimensional $U(2)$-invariant, non-collapsed Riemannian steady …

It is known that all left-invariant pseudo-Riemannian metrics on $H_3$ are algebraic Ricci solitons. We consider generalizations of Riemannian $H$-type, namely pseudo$H$-type and $pH$-type. We study algebraic Ricci solitons of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups of both types.

Study of pluriclosed flow on Oeljeklaus-Toma manifolds, showing convergence to a soliton.

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …