Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big (M^n,\,g,\,X,\,\lambda \big )$ with constant scalar curvature is isometric to a Euclidean sphere $\mathbb {S}^{n}$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.     

On complete Finslerian Yamabe solitons

Here it is shown that any Finslerian compact Yamabe soliton with bounded above scalar curvature is of constant scalar curvature. Furthermore, this extension of Yamabe solitons is developed for inequalities and among the others, it is proved that a forward complete non-compact shrinking Yamabe soliton has finite fundamental group and its first cohomology group vanishes, providing the scalar curvature is strictly bounded above.     

Three-dimensional homogeneous Lorentzian Yamabe solitons

A geometric characterization of Yamabe solitons on homogeneous Lorentzian manifolds of dimension three is given. As a consequence, Lorentzian Yamabe solitons and left-invariant Lorentzian Yamabe solitons are classified in this setting, showing the existence of Yamabe solitons which are not left-invariant.     

Properties of complete non-compact Yamabe solitons

In this article, we first study the local volume estimate of the complete non-compact Yamabe soliton. Then we study the behavior of the potential function of the steady Yamabe soliton with positive Ricci curvature. We also study the scalar curvature decay of steady and expanding Yamabe solitons with Ricci pinching condition.     

On noncompact quasi Yamabe gradient solitons

We study τ-quasi Yamabe gradient solitons on complete noncompact Riemannian manifolds. We prove several scalar curvature estimates under some conditions and get a non-local collapsing result based on the gradient estimate of the potential function. We also derive a decay theorem and a finite topological type result.     

The classification of locally conformally flat Yamabe solitons

This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the first part it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvature are rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabe solitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation on Rⁿ${\mathbb{R}}^n$ is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space–time maximum must be a steady Yamabe soliton.     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

The scalar curvature equation on

We give existence results for solutions of the prescribed scalar curvature equation on S³, when the curvature function is a positive Morse function and satisfies an index-count condition.     

Non-negative scalar curvature

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C ^* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.     

Prescribed scalar curvature on then-sphere

This paper considers the prescribed scalar curvature problem onS ⁿ forn>-3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We then show that forn=3 this is the only blow up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed scalar curvature problem onS ³.This article was processed by the author using the style filepljourlm from Springer-Verlag.     

Remarks on non-compact gradient Ricci solitons

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L ^p -Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L ^p conditions on the relevant quantities.