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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 ) .

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

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] .

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 .

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.

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 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.

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

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.

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.

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.

We use bifurcation theory to prove the existence of uncountably many unit volume constant scalar curvature metrics in manifolds of the form T k × M , k ≥ 2 , with M a compact manifold (without boundary), that are conformal, but not isometric, to a product metric g flat ⊕ g , where g flat is a flat metric on the k -torus T k , and g is a fixed constant scalar curvature metric on M .

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study and characterize the ( α , β ) -metrics of scalar flag curvature. 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 and classify ( α , β ) -metrics of scalar flag curvature with isotropic S-curvature. Finally, we conclude that the non-trivial regular ( α , β ) -metrics of scalar flag curvature with isotropic S-curvature on an n-dimensional manifold M ( n ≥ 3 ) must be Randers metrics.

For the harmonic function on two dimensional Riemannian manifolds with constant Gaussian curvature, some differential equalities and inequalities are established to study the curvature estimate of the steepest descents by the maximum principle.

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.

This work deals with braneworld scenarios with generalized gravity. We investigate models where the potential of the scalar field is polynomial or nonpolynomial. We obtain exact and approximated solutions for the scalar field, warp factor and energy density, in the complex scenario with no restriction on the scalar curvature. In particular, we describe the case where the brane may split, engendering internal structure, with the splitting caused by the same parameter that controls deviation from standard gravity.

Let (M,g) be an n -dimensional Riemannian manifold and T * M be its cotangent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compatible paracomplex structures on T * M . Second, we construct locally decomposable Golden Riemannian structures on T * M . Finally we investigate curvature properties of T * M .

In this paper we shall show that a compact gradient generalized m -quasi-Einstein metric ( M n , g , ∇ f , λ ) with constant scalar curvature must be isometric to a standard Euclidean sphere S n with the potential f well determined.

We consider the problem of conformal metrics equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball n , n ≥ 4. By variational methods, we prove the existence of two peak solutions that concentrate around a strict local maximum points of the mean curvature under certain conditions.

The purpose of this work is to discuss some new results on the scalar curvature problem in dimension n ≥ 3 . We give precise estimates on the losses of compactness and we prove the existence of at least a solution under the assumption that the prescribed function is flat near its critical points.

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorms for a general corank 1 sub-Riemannian structure with symmetries done by C.Li and I.Zelenko., Comment: 6 pages. arXiv admin note: substantial text overlap with arXiv:0908.4397

Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case.

In this paper we consider the equiform motion of a circle by studying the scalar curvature for the corresponding two-dimensional surface locally. We prove that if the scalar curvature K is constant, then K = 0. We describe the equations that govern such surfaces.

Riemannian cubics are curves that generalise cubic polynomials to arbitrary Riemannian manifolds, in the same way that geodesics generalise straight lines. Considering that geodesics can be extended indefinitely in any complete manifold, we ask whether Riemannian cubics can also be extended indefinitely. We find that there are always exceptions in Riemannian manifolds with strictly negative sectional curvature. On the other hand, we show that Riemannian cubics can always be extended in complete locally symmetric Riemannian manifolds of non-negative curvature.

Let ( M n , g ) be a complete noncompact Bach-flat n -manifold with the positive Yamabe constant and constant scalar curvature. Assume that the L 2 -norm of the trace-free Riemannian curvature tensor R ∘ m is finite. In this paper, we prove that ( M n , g ) is a constant curvature space if the L n 2 -norm of R ∘ m is sufficiently small. Moreover, we get a gap theorem for ( M n , g ) with positive scalar curvature. This can be viewed as a generalization of our earlier results of 4-dimensional Bach-flat manifolds with constant scalar curvature R ≥ 0 [Y.W. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011) 516–521]. Furthermore, when n > 9 , we derive a rigidity result for R 0 .

Solving a second order, quasilinear initial value problem, we prove existence and uniqueness of both space-like and time-like immersions of prescribed mean curvature vector in general semi-Riemannian manifolds. Such surfaces arise for example as critical points of generalized area functionals as well as self-similar solutions of the mean curvature flow.

We deal with the Q -curvature problem on a 4-dimensional compact Riemannian manifold ( M , g ) with ∫ M Q g d V g = 8 π 2 and positive Paneitz operator P g . Let Q be a positive smooth function. The question we consider is, when can we find a metric g which is conformal to g , such that Q is just the Q -curvature of g . A sufficient condition to this question is given in this paper.

In this Note, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere Sn, 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 the well known results due to Y. Li (1995) [10].

We let ( M m , g ) be a closed smooth Riemannian manifold with positive scalar curvature S g , and prove that the Yamabe constant of ( M × R n , g + g E ) ( n , m ≥ 2 ) is achieved by a metric in the conformal class of ( g + g E ) , where g E is the Euclidean metric. We do this by showing that the Yamabe functional of ( M × R n , g + g E ) is improved under Steiner symmetrization with respect to M , and so, the dependence on R n of the Yamabe minimizer of ( M × R n , g + g E ) is radial.

In this paper we consider the problem of prescribing the Webster scalar curvature on the three CR sphere of C 2 . We use techniques related to the theory of critical points at infinity, and obtain multiplicity results for curvature satisfying a CR “flatness condition”.

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for which this set is noncompact. These manifolds have umbilic boundary, but they are not conformally equivalent to the unit ball., 44 pages. Some typos corrected. This is a more detailed version of a paper accepted to Journal of Differential Equations

In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres., minor and final typo corrections

We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extends the Corvino gluing near infinity to non-constant scalar curvature metrics.

A relative Lorentzian volume comparison estimate between spacelike hypersurfaces is studied with the integral curvature bound in terms of Ricci and Scalar curvature which generalize the Bishop–Gromov volume comparison theorem.

Let ( M 4 , g ) be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that ( M 4 , g ) has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small L 2 -norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for ( M 4 , g ) with positive scalar curvature.

In a recent paper [C. Atindogbe, Scalar curvature on lightlike hypersurfaces, Appl. Sci. 11 (2009) 9–18], the present author considered the concept of extrinsic (induced) scalar curvature on lightlike hypersurfaces. This scalar quantity has been studied on lightlike hypersurfaces equipped with a given normalization. But a very important problem was left open: How to characterize the set of all normalizations admitting a prescribed extrinsic scalar curvature? In this paper, we provide various responses to this question, supported by examples.

Let (Mn,g), n⩾3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg⩽n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched conditionn2(n−2)n−1C

Let ( M , g ) be a noncompact, connected, orientable smooth N -dimensional Riemannian manifold without boundary. We consider the existence of solutions of problem (P) Δ g u + a u log u = 0 on M , where Δ g stands for the Laplacian in M and a > 0 .

In this Note we prescribe a fourth order curvature – the Q-curvature on the standard n-sphere, n ⩾ 5 . Under the “flatness condition” of order β, n − 4 ⩽ β n near each critical point of the prescribed Q-curvature function, we prove new existence result through an Euler–Hopf type formula. Our argument gives a lower bound on the number of conformal metrics having the same Q-curvature.

We consider the existence of contact forms of prescribed Webster scalar curvature on a (2n+1)-dimensional CR compact manifold locally conformally CR equivalent to the standard unit sphere S2n+1 of Cn+1. We give some existence results, using dynamical and topological methods involving the study of the critical points at infinity of the associated noncompact variational problem.

We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvatures in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L 2 -norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C 1 , α -close to the original metric on a smaller ball but still of definite size.