Related papers: A note on compact gradient Yamabe solitons

We show that in an arbitrarily fixed conformal class on a closed manifold, the upper bound condition of the total scalar curvature is $C^{0}$-closed if its Yamabe constant is nonpositive. Moreover, we show that if a conformal class on a…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For a metric $g$ on $M$, we let $\la_2(g)$ be the second eigenvalue of the Yamabe operator $L_g:= \frac{4(n-1)}{n-2} \Delta_g + \scal_g$. Then, the second Yamabe invariant…

Let $M^n$, $n\ge3$, be a compact differentiable manifold with nonpositive Yamabe invariant $\sigma(M)$. Suppose $g_0$ is a continuous metric with $V(M, g_0)=1$, smooth outside a compact set $\Sigma$, and is in $W^{1,p}_{loc}$ for some…

For a compact manifold $M$ of $\dim M =n\geq 4$, we study two conformal invariants of a conformal class $C$ on $M$. These are the Yamabe constant $Y_C(M)$ and the $L^{\frac{n}{2}}$-norm $W_C(M)$ of the Weyl curvature. We prove that for any…

We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the Einstein constraint…

Let $(M^n,g,e^{-\phi}dV_g,e^{-\phi}dA_g,m)$ be a compact smooth metric measure space with boundary with $n\geqslant 3$. In this article, we consider several Yamabe-type problems on a compact smooth metric measure space with or without…

In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the…

In this short note, using G\"unther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature…

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation…

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant…

We show that there are infinitely many pairwise nonhomothetic, complete, periodic metrics with constant scalar curvature that are conformal to the round metric on $S^n\setminus S^k$, where $k < \frac{n-2}{2}$. These metrics are obtained by…

A classical theorem in conformal geometry states that on a manifold with non-positive Yamabe invariant, a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all…

The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding…

In this paper we have investigated some aspects of gradient $\rho$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $\rho$-Einstein soliton is isometric to the Euclidean sphere by…

On a compact stratified space (X, g) there exists a metric of constant scalar curvature in the conformal class of g, if the scalar curvature satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the…

In this paper, we study the uniqueness of type II Yamabe metrics in conformal classes on a compact connected manifold with boundary, and we investigate Obata-type theorems for type II Yamabe metrics. In particular, we establish a theorem…

In this paper, we study structures of almost Yamabe solitons which are not necessarily gradient. First, we investigate conditions that both compact and noncompact almost Yamabe solitons become trivial solitons which means the given vector…

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$\gamma$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is…

In this paper, we consider a closed 3-manifold $M$ with flat conformal structure $C$. We will prove that, if the Yamabe constant of $(M, C)$ is positive, then $(M, C)$ is Kleinian.

The purpose of this article is to study generalized quasi Yamabe gradient solitons on warped product manifolds. First, we obtain some necessary and sufficient conditions for the existence of generalized quasi Yamabe gradient solitons…