For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is strictly less than the Yamabe invariant of $S^{m+n}$ provided $n\geq 2$. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.
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…
The goal of this paper is to study conformal Yamabe soliton and $*$-Yamabe soliton, whose potential vector field is torse forming. Here, we have characterized conformal Yamabe soliton admitting potential vector field as torse forming with respect to Riemannian connection, semi-symmetric metric connection and projective semi-symmetric connection on Riemannian manifold. We have also shown the nature of $*$-Yamabe soliton with torse forming vector field on Riemannian manifold admitting Riemannian connection. Lastly we have developed an example to corroborate some theorems regarding Riemannian con…
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.
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 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 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.
Let $(M^m,g)$ be a closed Riemannian manifold $(m\geq 2)$ of positive scalar curvature and $(N^n,h)$ any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second $N-$Yamabe constant of $(M\times N,g+th)$ as $t$ goes to $+\infty$. We obtain that $\lim_{t \to +\infty}Y^2(M\times N,[g+th])=2^{\frac{2}{m+n}}Y(M\times \re^n, [g+g_e]).$ If $n\geq 2$, we show the existence of nodal solutions of the Yamabe equation on $(M\times N,g+th)$ (provided $t$ large enough). When the scalar curvature of $(M,g)$ is constant, we prove that $\lim_{t \to +\infty}Y^2_N(M\times …
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.
Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + δh)$ has at least $Cat(M) +1 $ solutions for $δ$ small enough, where $Cat(M)$ denotes the Lusternik-Schnirelmann-category of $M$. Cat(M) of the solutions obtained have energy arbitrarily close to the minimum.
Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, its Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe in…
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.
Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many t's accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive sc…
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $σ_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.
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.
The special nature of gradient Yamabe soliton equation which was first observed by Cao-Sun-Zhang\cite{CaoSunZhang} shows that a complete gradient Yamabe soliton with non-constant potential function is either defined on the Euclidean space with rotational symmetry, or on the warped product of the real line with a manifold of constant scalar curvature. In this paper we consider the classification in the latter case. We show that a complete gradient steady Yamabe soliton on warped product is necessarily isometric to the Riemannian product. In the shrinking case, we show that there is a continuous…
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…
For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of the Yamabe equation.
In this paper, we study the existence of complete Yamabe metric with zero scalar curvature on an n-dimensional complete Riemannian manifold $(M,g_0)$, $n\geq 3$. Under suitable conditions about the initial metric, we show that there is a global fine solution to the Yamabe flow. The interesting point here is that we have no curvature assumption about the initial metric. We show that on an n-dimensional complete Riemannian manifold $(M,g_0)$ with non-negative Ricci curvature, $n\geq 3$, the Yamabe flow enjoys the local $L^1$-stability property from the view-point of the porous media equation. Co…
Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to $g$ and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.
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 H^n-Yamabe constants of Riemannian products (H^n \times M^m, g_h^n +g), where (M,g) is a compact Riemannian manifold of constant scalar curvature and g_h^n is the hyperbolic metric on H^n. Numerical calculations can be carried out due to the uniqueness of (positive, finite energy) solutions of the equation Δu -λu + u^q =0 on hyperbolic space H^n under appropriate bounds on the parameters λ, q, as shown by G. Mancini and K. Sandeep. We do explicit numerical estimates in the cases (n,m)=(2,2),(2,3) and (3,2).
Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $σ_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $σ_k-$curvature. The $σ_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the Schouten tensor of a Riemannian metric. The main goal of this paper is to find solutions to the singular $σ_2-$Yamabe problem with isolated singularities in any compact non-degenerate manifold such that the Weyl tensor vanishing to sufficiently high order at the singular point. …
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.
We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2 $ we show that for any $q<\frac{ n-d_2+2 }{n - d_2 -2}$ the number of positive solutions of the equation $-Δ_g u + λu = λu^q$ tends to $\infty$ as $λ\rightarrow +\infty$. We apply this result to prove multiplicity results for positive solutions of critical and supercritical equations. In particular we prove multiplicity results …
We define a new formal Riemannian metric on a conformal class in the context of the $v_{\frac{n}{2}}$-Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover, this variational framework suggests that solutions to this problem are unique in a given conformal class.
The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the Riemannian manifold is compact, is nonlocally conformally flat and has a dimension equal to or greater than $6$. In $2015$, Case considered a Yamabe-type problem in the setting of smooth measure space in manifolds and for a parameter $m$, which generalizes the original Yamabe problem when $m=0$. Addit…
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
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…
Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -Δ_{g}u+au=0 & \text{ on }M \\ \partial_νu+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $ν$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_…
Given closed Riemannian manifold $(M^n, g)$ of positive Ricci curvature $Ricci(g) \geq (n-1)g$ we study isoperimetric regions on the spherical cone over $M$. When $g$ is Einstein we use this to compute the Yamabe constant of $(M \times {\bf R}, g + dt^2)$ and so to obtain lower bounds for the Yamabe invariant of $M\times S^1$.
We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on $\textbf{S}^n \times\textbf{ S}^n$, and study solutions which are invariant by the cohomogeneity one diagonal action of $O(n+1)$. We obtain multiplicity results for both positive and nodal solutions. In particular we prove the existence of nodal solutions of the Yamabe equation on these products which depend non-trivially on both factors
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.
In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold $(M,g)$ of dimension $n\geq 3$. Let $ψ(x)$ be any smooth function on $M$. Let $p=\frac{n+2}{n-2}$ and $c_n=\frac{4(n-1)}{n-2}$. We study the Yamabe-type flow $u=u(t)$ satisfying {u_t}=u^{1-p}(c_nΔu-ψ(x)u)+r(t)u, \ \ in \ M\times (0,T),\ T>0 with r(t)=\int_M(c_n|\nabla u|^2+ψ(x)u^2)dv/ \int_Mu^{p+1}, which preserves the $L^{p+1}(M)$-norm and we can show that for any initial metric $u_0>0$, the flow exists globally. We also show that in some cases, the global solution converges to a smooth solution to the equation c…
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 let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the Euclidean metric. We also show that the Yamabe quotient of (M \times R^n,g+g_E) is improved by Steiner symmetrization with respect to M. It follows from this last assertion that the dependence on R^n of the Yamabe minimizer of (M \times R^n,g+g_E) is radial.
Suppose $(M,g_0)$ is a compact Riemannian manifold without boundary of dimension $n\geq 3$. Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of $g_0$ with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.
In this paper we have obtained evolution of some geometric quantities on a compact Riemannian manifold $M^n$ when the metric is a Yamabe soliton. Using these quantities we have obtained bound on the soliton constant. We have proved that the commutator of two soliton vector fields with the same metric in a given conformal class produces a Killing vector field. Also it is shown that the soliton vector field becomes a geodesic vector field if and only if the manifold is of constant curvature.
On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. On a closed n-dimensional manifold, $n\ge 5$, we compare the three basic conformally covariant operators : the Branson-Paneitz, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. Equality cases are also characterized.
In this short note, exploits of constructions of $\mathcal{F}$-structures coupled with technology developed by Cheeger-Gromov and Paternain-Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature.…
In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities change, analogous to how the Yamabe constants and the Yamabe invariants do, under the connected sum operations.
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 Liouville type problem.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $φ$ solution of the following Yamabe type equation Δφ+ hφ= \tilde h φ^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n/2$ and $\tilde h\in \mathbb R$. We give the regularity of $φ$ with respect to the value of $p$. Finally, we consider the results in geometry when $g$ is a singular Riemannian metric and $h=\frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature of $g$.
