We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal complete metrics.
This also gives rise to a new proof of the well-known result for closed manifolds and positive Yamabe invariant.
In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.
The object of the present paper is to study some properties of 3-dimensional trans-Sasakian manifold whose metric is η-Yamabe soliton. We have studied here some certain curvature conditions of 3-dimensional trans-Sasakian manifold admitting η-Yamabe soliton. Lastly we construct a 3-dimensional trans-Sasakian manifold satisfying η-Yamabe soliton.
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
In this work we determine bifurcation instants for 1-parameter families of solutions to the Yamabe problem defined on maximal flag manifolds. We also study the local rigidity points, namely, a isolated solution of the Yamabe problem.
We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold.
The object of the present paper is to study some properties of (LCS)$_n$-manifolds whose metric is Yamabe soliton. We establish some characterization of (LCS)$_n$-manifolds when the soliton becomes steady. Next we have studied some certain curvature conditions of (LCS)$_n$-manifolds admitting Yamabe solitons. Lastly we construct a 3-dimensional (LCS)$_n$-manifold satisfying the results.
For a Poincare-Einstein manifold under certain restrictions, X. Chen, M. Lai and F. Wang proved a sharp inequality relating Yamabe invariants. We show that the inequality is true without any restriction.
Suppose $M_{1}$ and $M_{2}$ are $3$-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a CR structure with positive CR Yamabe constant.
On a compact strictly pseudoconvex CR manifold $(M,þ)$, we consider the CR Yamabe constant of its infinite conformal covering. By using the maximum principles, we then prove a uniqueness theorem for the CR Yamabe flow on a complete noncompact CR manifold. Finally we obtain some properties of the CR Yamabe soliton on complete noncompact CR manifolds.
We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.
We prove that the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is non-negative. Equivalently that the infimum of the L^{n/2} norm of the scalar curvature, over the space of all Riemannian metrics on the manifold, is zero.
We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.
For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.
Suppose $M_{1}$ and $M_{2}$ are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a spherical CR structure with positive CR Yamabe constant.
In this research article, we establish the geometrical bearing on Riemannian submersions in terms of $η$-Ricci-Yamabe Soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton and $η$-Yamabe soliton. We also discuss the various conditions for which the target manifold of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton, $η$-Yamabe soliton and quasi-Yamabe soliton. In a particular case when the potential filed $V$ of the $η$-Ricci-Yamabe soliton is of gradient type, we derive a La…
We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe constant at infinity is even locally constant with respect to this topology. We also discuss to which extent the Yamabe constant is continuous with respect to coarser topologies on the space of Riemannian metrics.
In this paper, we study the existence of global Yamabe flow on asymptotically flat (in short, AF or ALE) manifolds. Note that the ADM mass is preserved in dimensions 3,4 and 5. We present a new general local existence of Yamabe flow on a complete Riemannian manifold with the initial metric quasi-isometric to a background metric of bounded scalar curvature. Asymptotic behaviour of the Yamabe flow on ALE manifolds is also addressed provided the initial scalar curvature is non-negative and there is a bounded subsolution to the corresponding Poisson equation. We also present a maximum principle fo…
The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that *If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature -1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold. *If the semi-Riemannian metric of a three-dimensional paraco…
This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a conformally invariant energy generalizing the Willmore energy of a surface whose variational derivative with respect to variations of the boundary hypersurface is a multiple of the obstruction to smoothness of the singular Yamabe metric itself. The existence of such an energy answers a…
We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.
It is shown that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Namely, it is proved that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature
Let $(M,g)$ be a compact Riemannian manifold with non-empty boundary. Provided $f$ an isoparametric function of $(M,g)$ we prove existence results for positive solutions of the Yamabe equation that are constant along the level sets of $f$. If $(M,g)$ has positive constant scalar curvature, minimal boundary and admits an isoparametric function we also prove multiplicity results for positive solutions of the Yamabe equation on $(M \times N,g+th) $ where $(N,h)$ is any closed Riemannian manifold with positive constant scalar curvature.
We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hijazi inequality for the underlying spin manifold.
The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow $g(t)$. The considered flow in covariant symmetric $2$-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of $g(t)$. Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most handled situation we express…
We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular, given a compact smooth manifold M which does not admit metrics of positive scalar curvature, we prove that the Yamabe invariant of M is an upper bound for the Yamabe invariant of any manifold obtained by performing surgery in M on spheres of codimension greater than 2 .
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 points of pairwise non homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds.
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only if the conformal infinity of $(X,g)$ is of positive Yamabe type. If this positivity is satisfied, we also show that the Green function of the fractional conformal Laplacian $P(α)$ on the conformal infinity is non-negative for all $α\in [0, 2]$.
In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.
This manuscript examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci-Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $η$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes Ricci-Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional $(κ, μ)'$-AKM equipped with a gradient ARYS is either locally isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of $3$-dimensional non-K…
The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\times B where T^m$ is the m-dimensional torus, and B is a closed spin manifold of nonzero \hat{A}-genus has zero Yamabe invariant. We generalize it to various T-structured manifolds, for example T^m-bundles over such B whose transition functions take values in Sp(m,Z) (or Sp(m-1,Z)\oplus \pm 1 for odd m).
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at least two non-homothetic Riemannian metrics of null scalar curvature and constant mean curvature of the boundary on the product manifold.
The purpose of this article is to study the geometry of gradient almost Yamabe solitons immersed into warped product manifolds $I\times_{f}M^{n}$ whose potential is given by the height function from the immersion. First, we present some geometric rigidity on compact solitons due to a curvature condition on the warped product manifold. In the sequel, we investigate conditions for the existence of totally geodesic, totally umbilical and minimal solitons. Furthermore, in the scope of constant angle immersions, a classification of rotational gradient almost Yamabe soliton immersed into $\mathbb{R}…
We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(α,β)$-Ricci-Yamabe soliton with $α\neq 0$ and potential vector field collinear with the Reeb vector field $ξ$ on a complete contact metric manifold with the Reeb vector field $ξ$ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $ξ$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $α\neq 0$, then …
The goal of the present paper is to deliberate certain types of metric such as $*$-$η$-Ricci-Yamabe soliton on $α$-Cosymplectic manifolds with respect to quarter-symmetric metric connection. Further, we have proved some curvature properties of $α$-Cosymplectic manifolds admitting quarter-symmetric metric connection. Here, we have shown the characteristics of the soliton when the manifold satisfies quarter-symmetric metric connection on $α$-Cosymplectic manifolds. Later, we have acquired Laplace equation from $*$-$η$-Ricci-Yamabe soliton equation when the potential vector field $ξ$ of the solit…
An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator, and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positiv…
We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If $(M^m,g)$ is a closed manifold of constant positive scalar curvature, which we normalize to be $m(m-1)$, we consider the Riemannian product with the $n$-dimensional Euclidean space: $(M^m \times \mathbf{R}^n, g+ g_E)$. And study the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant $λ(m,n)$ such that the solution is stable if and only if $λ_1 \geq λ(m,n)$, where $λ_1$ is the first positive eigenvalue o…
We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.
In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar curvature and surface area of a geodesic ball in a Riemannian manifold with a pole satisfying steady Yamabe soliton.
The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,μ)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,μ)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space…
We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant $Y\leq 0$. Previous work by the second and third named authors \cite{ChenWang} showed that while the Yamabe flow always converges in a global weighted sense when $Y>0$, the flow must diverge when $Y\leq 0$. We show here in the $Y\leq 0$ case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.
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 equipped with the warped product structure. Then we study three important applications in the Lorentzian and the neutral settings for the particular class, called as gradient Yamabe soliton: We proved the existence of the non-trivial gradient Yamabe soliton on generalized Robertson-Walker spacetimes, standard static spacetimes and Walker manifolds.
We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality. We prove some classification theorems of four manifolds according to some conformal invariants (see Theorems 1.3 and 1.6), which generalize the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}.
On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological obstruction to the existence of a $G$-invariant nonsingular solution to the normalized Ricci flow on $M$. In particular, for certain $m$ and $n$, $m\Bbb CP_2 # n\bar{\Bbb CP}_2$ admits an infinite family of topologically equivalent but smoothly distinct non-free actions of $\Bbb Z_d$ such that it admits no nonsingular solution to the normalized Ricci flow for any in…
