arXiv research

Proves properties of generalized quasi Yamabe gradient solitons.

The paper classifies 3D complete gradient Yamabe solitons.

The study classifies specific types of solitons with bounded scalar curvature.

Study on warped product Yamabe solitons with constant fiber curvature.

We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we prove a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

No nontrivial harmonic 1-forms on certain gradient Ricci solitons.

We completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation. While contact Riemannian (or Lorentz\-ian) Ricci solitons are necessarily trivial, that is, $K$-contact and Einstein, the paracontact metric case allows nontrivial examples. Both homogeneous and inhom…

This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. In this paper, we obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian s…

The paper classifies special geometric shapes in 2D and 3D.

Simply-connected shrinking Kähler-Ricci solitons are proven.

We find exact solutions describing Ricci flows of four dimensional pp-waves nonlinearly deformed by two/three dimensional solitons. Such solutions are parametrized by five dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion which can be related, for instance, to antisymmetric ten…

The paper classifies expanding gradient Yamabe solitons based on scalar curvature.

The paper studies gradient Ricci-Harmonic solitons on warped product manifolds.

The study identifies special solitons on 3-manifolds.

The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.

We introduce the concept {\it $h$-almost Ricci soliton} which extends naturally the {\it almost Ricci soliton} by Pigola-Rigoli-Rimoldi-Setti and show that a compact nontrivial $h$-almost Ricci soliton of dimension no less than three with $h$ having defined signal and constant scalar curvature is isometric to a standar…

We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off-diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild me…

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…

Study gradient Yamabe solitons on warped product manifolds.

Using the `Riemann Problem with zeros' method, Ward has constructed exact solutions to a (2+1)-dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact …

Conditions for Riemannian manifolds to be Euclidean spheres or spaces.

This paper classifies solitons under specific tensor conditions.

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If $g$ is not a local maxim…

The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.

The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.

Study of solitons in Laplacian flow on 7-manifolds.

Study of 3D steady gradient Ricci solitons using level set flow.

The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.

The paper characterizes grim hyperplanes for translating solitons in mean curvature flow.

The paper proves rigidity and vanishing theorems for translating solitons.

In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that …

This paper classifies all expanding Ricci solitons on surfaces.

The paper classifies periodic solitons in curve flows on the light-cone.

We study complete Riemannian manifolds satisfying the equation $Ric+\nabla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE $Δ_f f + mμe^{2f/m}=0$ for $μ\leq 0$. By developing a gradient estimate for $f$, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ri…

The paper studies Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.

Study on geodesic flows on a two-torus for additional first integrals.

The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tan…

Novel analysis of generalized Ricci solitons leads to stability results and deformations.

Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.

We study both function theoretic and spectral properties of the weighted Laplacian $Δ_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-Émery curvature $Ric_f$ bounded from below by a constant. In particular, we establish a gradient estimate for positive $f-$harmonic functions and a sharp upper…

All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the nilpotent case). The tools from geometric invariant theory used to study Einste…

We propose an intuitive interpretation for nontrivial $L^2$-Betti numbers of compact Riemann surfaces in terms of certain loops in embedded pairs of pants. This description uses twisted homology associated to the Hurewicz map of the surface, and it satisfies a sewing property with respect to a large class of pair-of-pa…

Up to now, the only known examples of homogeneous nontrivial Ricci soliton metrics are the so called solsolitons, i.e. certain left invariant metrics on simple connected solvable Lie groups. In this paper, we describe the moduli space of solsolitons of dimension less or equal than 6, up to isomorphism and scaling. We s…

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular s…

The paper explores new Riemannian structures and their properties.

We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is nontrivial, the flow exists for all time. For higher genus surfaces the flow al…

We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of metrics, for two classes of four dimensional nonholonomic manifolds: a) tangent bundles with two dimensional base manifolds and b) pseudo-Riemannian/ Einstein manifolds. Such spacetimes are enabled with nonholonomic distributions and associated non…

In this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics. A. Nadel has defined an iteration scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial periodic points. First, we prove that no such periodic points can exist. We define…