Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k=1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions to the k-Yamabe problem was recently proved by Gursky and Viaclovsky for k>n/2. In this paper we prove the existence of solutions for the remaining cases k <n/2, k=n/2, assuming that the equation is variational.
In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of a complete non-compact Yamabe soliton has non-negative scalar curvature. A new proof of Kazdan-Warner condition is also presented.
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also s…
We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.
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…
We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone $Γ$ satisfies $μ^+_Γ\le 1$, which includes the $σ_k-$Yamabe problem for $k$ not smaller than half of the dimension of the manifold.
Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.
Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $γ= 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.
We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.
We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a la…
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm then a slightly modified version of the Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Hermitian manifolds, is close enough to a constant function in a Hölder norm then the Chern-Yamabe problem has a solution for generic values of the fundamental constant.
Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curvature $f$ on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.
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 study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.
This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The estimation of Ricci curvature is deduced and also an isometry theorem is found in gradient Ricci-Yamabe soliton satisfying a finite weighted Dirichlet integral. Further, it is proved that a Ricci-Yamabe soliton reduces to an Einstein manifold when the potential vector field becomes concircular. Moreover, the eigenvalue and the corresponding eigenspace of the …
We review recent compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow.
We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all radially symmetric gradient Yamabe solitons. We also show that any eternal solutions 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.
We study the Yamabe flow starting from an asymptotically flat manifold $(M^n,g_0)$. We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if $Y(M,[g_0])>0$, and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.
We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $Σ$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $Σ$ is locally volume-minimizing in a manifold $M^n$ with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally $M$ splits along $Σ$. In the case that the scalar curvature of $M$ is at least $-n(n-1)$ and $Σ$ locally minimizes a certain functional inspired by [30], a n…
We will give a simple proof that the metric of any compact Yamabe gradient soliton (M,g) is a metric of constant scalar curvature when the dimension of the manifold n>2.
This article uses the iterative schemes and perturbation methods to completely solve the Han-Li conjecture, i.e. the general boundary Yamabe problem with prescribed constant scalar curvature and constant mean curvature on compact manifolds $ (M, \partial M, g) $ with non-empty smooth boundary, $ \dim M \geqslant 3 $. It is equivalent to show the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n - 2} Δ_{g} u + R_{g} u = λu^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial ν} + \frac{n-2}{2} h_{g} u = \frac{n-2}{2} ζu^{\frac{n}{n - 2}} $ on $ \partial M $ with some…
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.
In this paper, we introduce the concept of quasi Yamabe gradient solitons, which generalizes the concept of Yamabe gradient solitons. By using some ideas in [7,8], we prove that $n$-dimensional $(n\geq3)$ complete quasi Yamabe gradient solitons with vanishing Weyl curvature tensor and positive sectional curvature must be rotationally symmetric. We also prove that any compact quasi Yamabe gradient solitons are of constant scalar curvature.
On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(α,β)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(α,β)$-Ricci-Yamabe flow by estimating its curvatures.
In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric is a constant. Existence of solutions is obtained through the combination of variational methods, second order Sobolev inequalities, and the $W^{2,2}$ blow-up theory developed by Hebey and Robert. Positivity of the solutions is obtained from a novel argument proven here for the first time that is rooted in the conformal covariance property of the Paneitz-Bra…
Let (M,g) be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar's work. Moreover, we reduce the remaining cases to the positive mass theorem.
This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov-Safonov …
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.
In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are of Yamabe metrics.
In this paper, we give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
We derive lower bounds on the scalar curvature of complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that the corresponding potential functions have at most quadratic growth in distance. We also obtain a finite topological type property on complete shrinking gradient Yamabe solitons under suitable scalar curvature assumptions.
This paper deals with the conformal deformation of the standard metric in a domain on the sphere to a complete metric with the constant scalar curvature. The problem of description of domains allowing such deformation originates in the works of Loewner and Nirenberg, and Schoen and Yau concerned with the locally conformally flat manifolds. The goal of this work is to apply ideas from the nonlinear potential theory to the problem. They allow, in particular, to solve the problem in the case of the constant negative scalar curvature.
The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal conformally invariant operators. We investigate a non-compactness property of the fractional Yamabe problem by constructing bubbling solutions to its small perturbations.
Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term with a negative smooth function, the set of solutions of Yamabe problem is still a compact set.
We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d>(n-2)/2. Our main theorem is inspired by the classical results by Aviles-McOwen and Loewner-Nirenberg …
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.
We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.
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.
We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the c…
The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE. However, the singular behavior of these solutions, which is important in describing many important questions in conformal geometry, is little u…
Let n be an integer such that 25 \leq n \leq 51. We construct a smooth metric g on S^n with the property that the set of constant scalar curvature metrics in the conformal class of g is not compact.
We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when $n \geq 25$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which is anticipated to be fundamental for a study of the nonumbilic case.
In this note we prove that a (anti-)self dual quasi Yamabe soliton with positive sectional curvature is rotationally symmetric. This generalizes a recent result of G. Huang and H. Li in dimension four. Whence, (anti-) self dual gradient Yamabe solitons with positive sectional curvature is rotationally symmetric. We also prove that half conformally flat gradient Yamabe soliton has a special warped product structure provided that the potential function has no critical point.
In this paper,we prove the following Myers-type theorem: if $(M^n,g)$, $n\geq 3$, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition $Rc\geq εRg>0$, where $ε>0$ is an uniform constant, then $M^n$ must be compact.
It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k<n, are critical points for a certain Hilbert type functional with respect to volume preserving conformal variations. This motivates the corresponding Yamabe problem: is it true that any metric on a closed manifold is conformal to a metric with constant 2k-Gauss-Bonnet curvature? Using perturbative methods we affirmatively answer this question for small perturbations of certain space forms. More precisely, i…
One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $σ_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds $(X,g)$ for which this problem admits multiple non-homoth…
We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with $g$ on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: Given a function $f$ on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equi…
In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.
