Related papers: Some Geometry and Analysis on Ricci Solitons

In this paper, we study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of K\"{a}hler-Einstein metrics on complete K\"{a}hler manifolds with negative Ricci curvature,…

In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under…

We study some properties of a $3$-dimensional manifold with a diagonal Riemannian metric as an almost $\eta$-Ricci soliton from the following points of view: under certain assumptions, we determine the potential vector field if $\eta$ is…

In this research article, initially, we prove some sharp inequalities on statistical submersions involving Ricci and scalar curvatures of the statistical manifolds. In addition, we establish the geometrical bearing on statistical…

This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $Ad_K$-invariant irreducible summands, the existence of parameter families of non-homothetic…

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow…

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any…

This undergraduate thesis is focused on introducing the reader to concepts related to the search for topological obstructions to the existence of compact gradient shrinking Ricci soliton metrics in dimension four. It contains a discussion…

Let $(M^4, g, f)$ be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. If its scalar curvature is $1$, Cheng-Zhou \cite{Cheng-Zhou} proved that it is a finite quotient…

In Eddington gravity, the action principle involves only the symmetric parts of the connection and the Ricci tensor, with a metric that emerges proportionally to the latter. Here, we relax this symmetric character, prolong the action with…

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive…

We introduce a flow of K\"ahler structures over Fano manifolds with formal limit at infinite time a K\"ahler-Ricci soliton. This flow correspond to a Perelman's modified backward K\"ahler-Ricci type flow that we call Soliton-K\"ahler-Ricci…

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the…

We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In…

We consider almost Einstein solitons $(V,\lambda)$ in a Riemannian manifold when $V$ is a gradient, a solenoidal or a concircular vector field. We explicitly express the function $\lambda$ by means of the gradient vector field $V$ and…

We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a $C^{0}$ orbifold at any finite-time singularity, so has an…

We establish a uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and a uniform curvature bound. Additionally, we extend the non-collapsing result to a broader class of smooth metric measure…

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry…

For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol}…

A weak metric $f$-structure $(f,Q,\xi_i,\eta^i,g)\ (i=1,\ldots,s)$, generalizes the metric $f$-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We…