arXiv research

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

Study classifies 4D Ricci solitons with specific curvature conditions.

New degree theory proves existence of solitons on 4D manifolds.

Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.

We first investigate the asymptotics of conical expanding gradient Ricci solitons by proving sharp decay rates to the asymptotic cone both in the generic and the asymptotically Ricci flat case. We then establish a compactness theorem concerning nonnegatively curved expanding gradient Ricci solitons.

New example of non-Kähler soliton with Kähler-like behavior at infinity.

Constructs expanding gradient Ricci solitons with unique properties.

No expanding breathers found in noncompact Ricci flows with certain curvature conditions.

Gradient flow expands curves to round shapes.

The paper pinches curvature in expanding Ricci solitons.

New expanding Ricci solitons found starting in dimension four.

We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expa…

We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive…

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…

Study expanding gradient Ricci solitons with Euclidean base.

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear stability of such gradient Ricci solitons.

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

The paper shows how to stabilize perturbed Kähler-Ricci solitons.

Study of ends of complete gradient Schouten solitons, showing finitely many ends for shrinking and connected infinity for expanding ones.

The study provides estimates for curvature of gradient Ricci solitons.

Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

We show that an expanding gradient Ricci solitons which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.

New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.

We produce new examples of non-Kähler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einstein manifolds with positive scalar curvature.

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $Rc=cg+D^2f,$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curva…

In this paper, we prove that expanding gradient Ricci solitons with (positively) pinched Ricci curvature are trivial ones. Namely, they are either compact or flat.

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.

We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymp…

Unique continuation result for expanding Ricci solitons.

The study finds a lower bound for the diameter of gradient ρ-Einstein solitons.

In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature sho…

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and…

The paper estimates curvature for specific 4D Ricci solitons.

We construct an expanding gradient Ricci soliton in dimension three over the topological manifold R x T^2 (the product of a line and a torus) that aproaches asymptotically a constant curvature cusp at one end, and a flat manifold on the other end. We prove that this is the only gradient soliton with this topology, prov…

Unique steady and expanding solitons with spherical links identified.

Let $(M,g,f)$ be a 3-dimensional complete steady gradient Ricci soliton. Assume that $M$ is rectifiable, that is, the potential function can be written as $f=f(r)$, where $r$ is a distance function. Then, we prove that $M$ is isometric to (1) a quotient of $\mathbb{R}^3$, or (2) the Bryant soliton. In particular, we sh…

Paper proves rigidity for Ricci solitons with specific conditions.

Let $(M^n,g,\nabla f)$, $n\geq 3$, be an expanding gradient Ricci soliton with nonnegative sectional curvature whose asymptotic cone is isometric to $C(\mathbb{S}^{n-1}(c))$ where $\mathbb{S}^{n-1}(c)$ is the standard $(n-1)$-sphere of curvature $1/c^2$, with $c\in(0,1)$. We prove that if the convergence to the asympto…

We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form $\R^2 \times M_2 \times \cdots \times M_r$, where $r \geq 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for…

Study cohomogeneity one expanding Ricci solitons on specific topologies.

In this paper we show that an expanding or steady gradient Ricci soliton warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and sufficient condition for constructing a gradient Ricci soliton warped product. …

In this paper we give some results on the topology of manifolds with $\infty$-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifold…

We study the asymptotic volume ratio of non-steady gradient Ricci solitons. Moreover, a local estimate of the volume ratio is obtained for expanding solitons which satisfy $\lim_{dist(O,x)\rightarrow\infty} |Sect|\cdot dist(O,x)^2=0$. Therefore, for such a soliton, we can show that it must have $\mathbb{R}^n$ as one of…

Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing solutions, both in the applicable range of parameters and in the complexity of the invol…

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

Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expan…

Study shows expanding Ricci solitons from specific metric cones.