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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

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.

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 $(P)$.

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

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.

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 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.

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

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 note is to investigate the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. In this note, we give a necessary and sufficient condition on the norm of the gradient of the potential function for a CPE metric to be Einstein.

We prove that there exist no complete vertical graphs in M 2 × R with extrinsic curvature bounded from above by a negative constant, when M 2 is a Riemannian surface with a pole and non-negative curvature.

This note is a study of nonnegativity conditions on curvature preserved by the Ricci flow. We focus on a specific class of curvature conditions which we call non-coercive: These are the conditions for which nonnegative curvature and vanishing scalar curvature does not imply flatness. We show, in dimensions greater than 4, that if a Ricci flow invariant nonnegativity condition is satisfied by all Einstein curvature operators with nonnegative scalar curvature, then this condition is just the nonnegativity of scalar curvature. As a corollary, we obtain that a Ricci flow invariant curvature condition, which is stronger than a nonnegative scalar curvature, cannot be strictly satisfied by curvature operators (other than multiples of the identity) of compact Einstein symmetric spaces. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces., 120 pages, 12 figures, (v2) minor revision; (v3) new sections on Finsler manifolds, slow growth distributions, Heisenberg group; (v4) major revision, new extended section on 3D contact structures, constant curvature, improved results about existence of ample geodesics on SR structures, 2 new appendices, many minor revisions; (v5) 1 new appendix, minor revisions

We classify the conformally flat hypersurfaces f : M 3 → R 4 with three distinct principal curvatures and constant scalar curvature.

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.

In this paper we consider the homothetic motion of Lorentzian circle by studying the scalar curvature for the corresponding cyclic surface locally. We prove that if the scalar curvature is constant, then . We describe the equations that govern such surfaces.

We establish a sharp inequality for a doubly warped product submanifold in a Riemannian manifold of quasi-constant curvature.

We analyze an upper bound on the curvature of a Riemannian manifold, using "root-Ricci" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our root-Ricci bound implies Gunther's inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.

При исследовании римановых многообразий важное значение имеет установление связи между различными типами кривизны и топологией риманова пространства. Одной из особых кривизн при этом является секционная кривизна. Наиболее наглядными примерами этого являются теоремы Адамара – Картана, М. Громова, теорема о сфере, теорема сравнения углов треугольника А.Д. Александрова − В.А. Топоногова, уравнения теории относительности А. Эйнштейна и ряд других результатов. В общем случае задача исследования римановых многообразий с ограничениями на секционную кривизну представляется достаточно сложной. Естественно поэтому рассматривать ее в классе однородных римановых пространств, в частности в классе групп Ли с левоинвариантной римановой метрикой. В данном направлении хорошо известны результаты М. Берже, С. Аллофа – Н. Уоллача, ряда других математиков по исследованию однородных римановых многообразий положительной секционной кривизны. Другим естественным ограничением является изучение секционной кривизны, а также ее оператора в классе конформно плоских римановых метрик. Данный класс метрик допускает удобное аналитическое представление, а спектр оператора секционной кривизны тесно связан с секционной кривизной. Исследован спектр оператора секционной кривизны конформно плоских римановых многообразий. Кроме того, изучен спектр оператора секционной кривизны в случае конформно полуплоских метрических групп Ли.DOI 10.14258/izvasu(2015)1.1-19, An establishment of communication between various types of curvature and topology of a Riemannian space is important for the research of Riemannian manifolds. The sectional curvature is one of the special types of curvatures. Some of the most known examples are Hadamard – Cartan’s theorem, M. Gromov’s theorem, the sphere theorem, the A.D. Alexandrov – V.A. Toponogov’s theorem of comparison of a triangle corners, the equations of A. Einstein’s theory of relativity, and some other results. Generally, a study of Riemannian manifolds with restrictions on the sectional curvature is assumed to be complicated. Therefore, it would appear reasonable to consider the study in a class of homogeneous Riemannian spaces, and, in particular, in a class of Lie groups with left invariant Riemannian metrics. In this direction there are well-known M. Berger’s, S. Alloff – N. Wallach’s research results and research results of some other mathematicians. Another natural restriction is a study of the sectional curvature and its operator in a class of conformally flat Riemannian metrics. This class of metrics allows convenient analytical representation, and the spectrum of the sectional curvature operator of such metrics is closely connected with the sectional curvature. In this paper, the sectional curvature operator spectrum of conformally flat Riemannian manifolds is investigated. Besides, the spectrum of the sectional curvature operator is investigated for the case of half conformally flat metric Lie groups.DOI 10.14258/izvasu(2015)1.1-19

Let $(M^{n},g_{0})$ be a $n=3,4,5$ dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function $K>0$ on $M$ we consider a scalar curvature flow, that tends to prescribe $K$ as the scalar curvature of a metric $g$ conformal to $g_{0}$. We show global existence and in case $M$ is not conformally equivalent to the standard sphere smooth flow convergence and solubility of the prescribed scalar curvature problem under suitable conditions on $K$.

The Schouten tensor A of a Riemannian manifold ( M , g ) provides the important σ k -scalar curvature invariants, that are the symmetric functions in the eigenvalues of A, where, in particular, σ 1 coincides with the standard scalar curvature Scal ( g ) . Our goal here is to study compact manifolds with positive Γ 2 -curvature, i.e., when σ 1 ( g ) > 0 and σ 2 ( g ) > 0 . In particular, we prove that a 3-connected non-string manifold M admits a positive Γ 2 -curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group π can always be realised as the fundamental group of a closed manifold of positive Γ 2 -curvature and of arbitrary dimension greater than or equal to six.

In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

Motivated by Lohkamp's conjecture on curvature deformation in [13] , we present a local smooth decrease of scalar curvature by big scale on a sphere as follows. Given any positive numbers N, a , b with a b π , we obtain a C ∞ -continuous path of Riemannian metrics g t , 0 ≤ t ≤ 1 , on the 4-dimensional sphere S 4 , with g 0 being the round metric of constant curvature 1, such that the scalar curvatures s ( g t ) are strictly decreasing in t on the open ball B b g 0 ( p ) of g 0 -radius b centered at a point p, s ( g 1 ) − N on B a g 0 ( p ) and g t = g 0 on the complement of the ball B b g 0 ( p ) . This result goes beyond what can be done with Corvino's local first-order deformation theory of scalar curvature [5] . Albeit done on a sphere, the argument here seems generalizable to a larger class of metrics.

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold ( M n , g ) endowed with the complementary orthogonal distributions D 1 and D 2 . We provide conditions for symmetric ( 0 , 2 ) -tensors T of a simple form (defined on M ) to admit metrics g , conformal to g , that solve the partial Ricci equations. The solutions are given explicitly. Using above solutions, we also give examples to the problem of prescribing the mixed scalar curvature related to D i . In aim to find “optimally placed” distributions, we calculate the variations of the total mixed scalar curvature (where again the partial Ricci curvature plays a key role), and give examples concerning minimization of a total energy and bending of a distribution.

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