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We consider the following prescribed curvature problem involving polyharmonic operator: $$\begin{aligned} D_mu=Q(|y'|,y'')u^{m^*-1}, \;u>0, \; u \in {\mathcal {H}}^{m}({\mathbb {S}}^{N}), \end{aligned}$$ where $$m^*=\frac{2N}{N-2m},\; N\ge 4m+1$$ , $$m \in {\mathbb {N}}_+$$ , $$(y',y'') \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{N-2}$$ , and $$Q(|y'|,y'')$$ is a bounded nonnegative function in $${\mathbb {R}}^{+} \times {\mathbb {R}}^{N-2}$$ . $${\mathbb {S}}^N$$ is the unit sphere with induced Riemannian metric g, $$D_m$$ is the polyharmonic operator given by $$D_m=\prod _{k=1}^m(-\Delta _g+\frac{1}{4}(N-2k)(N+2k-2)),$$ where $$\Delta _g$$ is the Laplace–Beltrami operator on $${\mathbb {S}}^N$$ . By using a finite reduction argument and local Pohozaev-type identities for polyharmonic operator, we prove that if $$N \ge 4m+1$$ and $$Q(r,y'')$$ has a stable critical point $$(r_0,y_0'')$$ , then the above problem has infinitely many solutions, whose energy can be arbitrarily large.

We study the problem of existence of conformal metrics with prescribed scalar curvatures on a closed Riemannian \begin{document}$ 4 $\end{document} -manifold not conformally diffeomorphic to the standard sphere \begin{document}$ S^{4} $\end{document} . Using the critical points at infinity theory of A.Bahri [ 6 ] and the positive mass theorem of R.Schoen and S.T.Yau [ 32 ], we prove compactness and existence results under the assumption that the prescribed function is flat near its critical points. These are the first results on the prescribed scalar curvature problem where no upper-bound condition on the flatness order is assumed.

This paper deals with the following prescribed scalar curvature problem − Δ u = Q ( | y ′ | , y ″ ) u N + 2 N − 2 , u > 0 , y = ( y ′ , y ″ ) ∈ R 2 × R N − 2 , where Q ( y ) is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if N ≥ 5 and Q ( r , y ″ ) has a stable critical point ( r 0 , y 0 ″ ) with r 0 > 0 and Q ( r 0 , y 0 ″ ) > 0 , then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bump solutions. Moreover, the concentration points of the bump solutions include a saddle point of Q ( y ) .

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.

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes., Comment: To appear in Inventiones Mathematicae. We have made a minor editing change

For complete noncompact Riemannian manifolds ( M n , g ) with harmonic curvature, we prove that g is Einstein under an inequality involving L n 2 -norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant. Furthermore, we achieve that M n is a constant curvature space under such inequality and finite L 2 -norm of the Weyl curvature.

In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L 2 norm of the mean curvature vector in a Riemannian manifold.

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic classes appeared in the Atiyah-Singer local index formula, namely, it is roughly a product of the $j$-function (which defines the $\hat A$-class of a manifold) and an exponential function (which defines the Chern character of a bundle). By performing two different computations for the variation of the Einstein-Hilbert action, we obtain a deep internal relations between two local curvature functions. Straightforward verification for those relations gives a strong conceptual confirmation for the whole computational machinery we have developed so far, especially the Mathematica code hidden behind the paper., Comment: 44 pages, 11 figures, some minor updates from the previous version

We obtain a sharp estimate to the scalar curvature of stochastically complete hypersurfaces immersed with constant mean curvature in a locally symmetric Riemannian space obeying standard curvature constraints (which includes, in particular, a Riemannian space with constant sectional curvature). For this, we suppose that these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Melendez (Bull Braz Math Soc 45:385–404, 2014), which is a weaker hypothesis than to assume that they have two distinct principal curvatures. Our approach is based on the equivalence between stochastic completeness and the validity of the weak version of the Omori–Yau’s generalized maximum principle, which was established by Pigola et al. (Proc Am Math Soc 131:1283–1288, 2002; Mem Am Math Soc 174:822, 2005).

A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the $$L^2$$ -norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kahler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds.

In this paper we analyzed the problem of studying locally the scalar curvature S of the two dimensional kinematic surfaces obtained by the homothetic motion of a helix in Lorentz-Minkowski space E17 $\text{E}^7_1$ . We express the scalar curvature S of the corresponding kinematic surfaces as the quotient of hyperbolic functions {cosh mϕ, sinh mϕ}, and we derive the necessary and sufficient conditions for the coefficients to vanishes identically. Finally, an example is given to show two dimensional kinematic surfaces with zero scalar curvature.

Let ( M n , g ) ( n ≥ 4 ) be an n-dimensional complete Einstein manifold. Denote by W the Weyl curvature tensor of M. We prove that ( M n , g ) is isometric to a spherical space form if ( M n , g ) has positive scalar curvature and unit volume, and the L p ( p ≥ n 2 ) -norm of W is pinched in [ 0 , C ) , where C is an explicit positive constant depending only on n, p and S, which improves the isolation theorems given by [24] , [14] , [17] . This paper also states that W goes to zero uniformly at infinity if for p ≥ n 2 , the L p -norm of W of M with non-positive scalar curvature and positive Yamabe constant is finite. Assume that M has negative scalar curvature and the L α -norm of W is finite. As application, we prove that M is a hyperbolic space form if the L p -norm of W is sufficiently small, which generalizes an L n 2 -norm of W pinching theorem in [19] .

In this paper, we consider the problem of existence and multiplicity of conformal metrics on a Riemannian compact 4-dimensional manifold $$(M^4,g_0)$$ with positive scalar curvature. We prove a new existence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri–Coron Euler–Poincare type criterium. Our argument gives estimates of the Morse index of the founded solutions and has the advantage to extend known existence results. Moreover, it provides, for generic K Morse Inequalities at Infinity, which give a lower bound on the number of metrics with prescribed scalar curvature in terms of the topological contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional.

We use optimal mass transport to provide a new proof and a dual formula to the Moser-Onofri inequality on S in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani [4] to the Sobolev inequality and of Agueh-Ghoussoub-Kang [1] to more general settings. There are however many hurdles to overcome once a stereographic projection on R is performed: Functions are not necessarily of compact support, hence boundary terms need to be evaluated. Moreover, the corresponding dual free energy of the reference probability density μ2(x) = 1 π(1+|x|2)2 is not finite on the whole space, which requires the introduction of a renormalized free energy into the dual formula. We also extend this duality to higher dimensions and establish an extension of the Onofri inequality to spheres S with n ≥ 2. What is remarkable is that the corresponding free energy is again given by F (ρ) = −nρ1− 1 n , which means that both the prescribed scalar curvature problem and the prescribed Gaussian curvature problem lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations.

It was conjectured in the 80s that every critical metric of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume must be Einstein. We prove that such a conjecture is true under a second-order vanishing condition on the Weyl tensor.

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.

We investigate curvatures of normal almost contact Riemannian 3-manifolds. In particular, we show that Kenmotsu 3-manifolds of constant scalar curvature are of constant curvature −1.

In this paper, we derive a DDVV-type inequality for submianifolds in a Riemannian manifold of nearly quasi-constant curvature. Moreover, two inequalities involving the Casorati curvature and the scalar curvature are obtained.

In this paper, we obtain sharp inequalities on Riemannian manifolds admitting a Riemannian submersion and give some characterizations using these inequalities. We improve Chen-Ricci inequality for Riemannian submersion and present some examples which satisfy this inequality.

There have been many works on the problem of finding a conformal metric on the standard sphere S n , n ≥ 3 , when the prescribed scalar curvature function is flat near its critical points with order of flatness β . All the existence results up to this research concern the case 1 β n . The present paper deals with the case β ≥ n . We provide a complete analysis of the asymptotic expansion of the associated gradient vector field for any β > 1 . Moreover, we establish (in this first part) existence results concerning some cases of β ≥ n .

We consider the following problem: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=K(y)u^{p-1} \hbox { in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right. &nbsp &nbsp &nbsp &nbsp &nbsp(P) \end{equation*} where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $p=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under the condition that the function $K(y)$ has a local maximum point, we prove the existence of infinitely many non-radial solutions for the problem $(P)$.

In this paper, we discuss some important properties of the Riemannian curvature of (α, β)-metrics. When the dimension of the manifold is greater than 2, we classify Randers metrics of weakly isotropic flag curvature (that is, Randers metrics of scalar flag curvature with isotropic S-curvature). Further, we characterize (α, β)-metrics of scalar flag curvature with isotropic S-curvature. We also characterize Einstein (α, β)-metrics and determine completely the local structure of Ricci-flat Douglas (α, β)-metrics when the dimension dim M ≥ 3.

This study investigates entire radially symmetric graphs in the warped product M × f R , where M is a Riemannian manifold with a pole and f : M → R + is the warping function. The main results demonstrate the existence and uniqueness of entire radially symmetric graphs with prescribed mean curvature.

Let ( M , g ) be a Riemannian manifold and N a C 2 submanifold without boundary. If we multiply the metric g by the inverse of the squared distance to N , we obtain a new metric structure on M ∖ N called the condition metric. A question about the behavior of this new metric arises from the works of Beltran, Dedieu, Malajovich and Shub: is it true that for every geodesic segment in the condition metric its closest point to N is one of its endpoints? Previous works show that the answer to this question is positive (under some smoothness hypotheses) when M is the Euclidean space R n . Here we find that there is a strong relation between this property and the curvature of M . In particular, we prove that the answer is also positive if M has non-negative curvature and that this property does not hold when the curvature of M is strictly negative at some point.

We discuss Levi-Civita connections on Courant algebroids. We define an appropriate generalization of the curvature tensor and compute the corresponding scalar curvatures in the exact and heterotic case, leading to generalized (bosonic) Einstein-Hilbert type of actions known from supergravity. In particular, we carefully analyze the process of the reduction for the generalized metric, connection, curvature tensor and the scalar curvature., New section and several references added based on the journal review

We study the P-scalar curvature operator used by Perelman in the context of manifolds with density. We verify that the Gromov–Lawson surgery for positive scalar curvature extends naturally to positive P-scalar curvature. By studying the first variation of the P-scalar curvature operator, we obtain local perturbation results for the P-scalar curvature. As a corollary, we construct small perturbations (in the category of manifolds with density) of the round unit hemisphere $$\mathbb {S}^n_+$$ which are compactly supported away from the boundary and yield manifolds with density with non-constant P-scalar curvature at least $$n(n-1)$$ .

Let M n (n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an (n + p)-dimensional locally symmetric Riemannian manifold N n+p . We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu [15].

In this paper, we prove an existence result of prescribing Webster scalar curvature on the CR sphere in cases where the prescribed function exhibits reflection or rotation symmetry.

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems on Riemannian metrics in a same conformal class that fix the boundary and satisfy certain integral condition on the scalar curvature and on the mean curvature along the boundary. As an application, we will state some rigidity theorems on the conformal class of static metrics.

We study complete noncompact 1-minimal stable hypersurfaces M in Sn+1. We obtain that there is no stable complete noncompact 1-minimal hypersurface in S4 with nonzero Gauss-Kronecker curvature and _nite total curvature. We show that there is no stable complete noncompact 1-minimal hypersurface in S4 with polynomial volume growth, nonzero Gauss-Kronecker curvature and supM H2 < 5/2. At last, we discuss stable complete noncompact hypersurfaces M with constant scalar curvature R > 1 in Sn+1 and show another nonexistence result. These results are generalised versions of results of Alencar, Santos, Zhou and Silva Neto on stable hypersurfaces in Rn+1 with constant scalar curvature.

In this paper, we obtain that every biharmonic non-degenerate hypersurfaces in semi-Euclidean space E 5 with constant scalar curvature of diagonal shape operator has zero mean curvature.

In this article, we find a connection between Brown–York mass and the first Dirichlet eigenvalue of a Schrodinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly supported conformal deformations which either increase or decrease scalar curvature. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question that arises naturally in (Corvino in Commun Math Phys 214:137–189, 2000; Lohkamp in Math Ann 313:385–407, 1999).

In this paper, we consider the problem of existence of confor- mal metrics with prescribed mean curvature on the unit ball of Rn, n � 3. Under the assumption that the order of flatness at critical points of pre- scribed mean curvature function H(x) is � 2)1,n 2), we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula. 1. Introduction and main result In this article, we consider the problem of existence of conformal scalar flat metric with prescribed boundary mean curvature on the standard n-dim- ensional ball. Let B n be the unit ball in R n , n ≥ 3, with Euclidean metric g0. Its boundary will be denoted by S n−1 and will be endowed with the standard metric still denoted by g0. Let H : S n−1 → R be a given function, we study the problem of finding a conformal metric g = u 4 n−2g 0 such that Rg = 0 in B n and hg = H on S n−1 . Here Rg is the scalar curvature of the metric g in B n and hg is the mean curvature of g on S n−1 . This problem is equivalent to solving the following nonlinear boundary value equation: (P) ( �u = 0 in B n ∂u ∂ν + n − 2 2 u = n − 2 2 Hu n n−2 on S n−1

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

We present in this note a lower bound for the Calabi functional in a given K\"ahler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality., Comment: 10 pages

We introduce a conformal Sasakian manifold and we find the inequality involving Ricci curvature and the squared mean curvature for semi-invariant, almost semi-invariant, $\theta $-slant, invariant and anti-invariant submanifolds tangent to the Reeb vector field and the equality cases are also discussed. Also the inequality involving scalar curvature and the squared mean curvature of some submanifolds of a conformal Sasakian space form are obtained.

In this paper, we prove the existence of warping functions on Riemannian warped product manifolds with some prescribed scalar curvatures according to the fiber manifolds of class (B).

We obtain a local uniqueness result for bubbling solutions of the prescribed scalar curvature problem in R N . Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K ( y ) . In particular, we prove in this paper that if K ( y ) is periodic in y 1 with period 1 and has a local maximum at 0, then a bubbling solution whose blow-up set is { ( j L , 0 , ⋯ , 0 ) : j = 0 , ± 1 , ± 2 , ⋯ } must be periodic in y 1 provided the positive integer L is large enough.

The technique of covariant differentiation is being implemented into the single integral variational calculus with second derivatives in Riemannian manifolds and into the Grässer-Rund-Weyssenhoff homogeneous generalized canonical framework for the parameter-invariant variational problem. As an example of physical nature we relate to each other the equations of motion with higher derivatives that describe the physical phenomena known as ‘Zitterbewegung’, the electromagnetic self-interaction, and spherical top. Although two of these phenomena, the quiver (the ‘Zitterbewegung’) and the radiation friction, have been known only within the framework of Special Relativity, our approach allows to generalize the corresponding equations of motion to the curved spacetime. From the geometrical point of view these generalizations depend neither on the signature of the Riemannian space nor on its dimension.

In this paper, we prove the existence and the nonex-istence of warping functions on Riemannian warped product mani-folds with some prescribed scalar curvatures according to the bermanifolds of class (A). 1. IntroductionOne of the basic problems in the di erential geometry is studying theset of curvature functions which a given manifold possesses.The well-known problem in di erential geometry is that of whetherthere exists a warping function of warped metric with some prescribedscalar curvature function. One of the main methods of studying di er-ential geometry is by the existence and the nonexistence of a warpedmetric with prescribed scalar curvature functions on some Riemannianwarped product manifolds. In order to study these kinds of problems,we need some analytic methods in di erential geometry.For Riemannian manifolds, warped products have been useful in pro-ducing examples of spectral behavior, examples of manifolds of negativecurvature (cf. [2, 3, 4, 5, 7, 13, 14]), and also in studying L

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition., Comment: Title slightly changed. Introduction improved and changed to include the recent results (arXiv:1407.0673) on Conjecture 1.5. Appendix C removed and published as a separate paper (arXiv:1508.01058). Although Appendix B was shortened for publication, this arxiv version remains complete and unchanged. Some typos corrected. 71 pages

This paper focuses on the study of the prescribed mean curvature problem on the unit ball. If the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small, then the existence of positive solutions of conformal mean curvature equation has been known. The purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method.

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980's that every CPE metric must be Einstein. We prove that a 4-dimensional CPE metric with harmonic tensor must be isometric to a round sphere

In this paper, we are interested in solving the following partial differential equation − Δ θ u + R u = f u 1 + 2 / n on a compact strictly pseudo-convex CR manifold ( M , θ ) of dimension 2 n + 1 with n ⩾ 1 . This problem naturally arises when solving the prescribing Webster scalar curvature problem on M with the prescribed function f . Using variational techniques, we prove several non-existence, existence, and multiplicity results when the function f is sign-changing.

The aim of this short note is to study the space of metrics with constant scalar curvature satisfying the critical point equation on compact manifolds, for simplicity, CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we prove that a CPE metric must be isometric to a round sphere under suitable integral conditions inherited from the standard sphere.