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In this paper, we give existence and multiplicity results for the problem of prescribing the Webster scalar curvature on the three CR sphere of C 2 under mixed conditions: non-degenerancy and flatness.

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker , which generalizes the previous partial results.

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.

A linear transformation of velocity for a mechanical system is said to quasi-linearize the equations of motion of the system if it eliminates all terms quadratic in the velocity. It is well-known that controller/observer synthesis becomes tractable when the dynamics of a mechanical system are in quasi-linearized form. In this technical note, we show that the quasi-linearization property is equivalent to the property that the Lie algebra of Killing vector fields is pointwise equal to the tangent space to the configuration manifold with the Riemannian metric induced by the mass tensor of the mechanical system. A sufficient condition for this property is that the Riemannian manifold be locally symmetric. We further show that a necessary and sufficient condition for quasi-linearizability on 2-D Riemannian manifolds is that the scalar curvature is constant. The above results extend the zero Riemannian curvature condition that has been extensively applied since its introduction in 1992. Moreover, the local symmetricity condition and the constant scalar curvature condition can be easily verified using differentiation.

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

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo’s conjecture such as that of Brendle–Marques–Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

Finite volume scheme for regularised Riemannian mean curvature flow equation is discussed. Stability estimates and the uniqueness of the numerical solution are listed. Convergence of the numerical scheme to the discrete solution is listed as well. Numerical results are presented in the final section.

Some basic inequalities, involving the scalar curvature and the mean curvature, for a pseudo-Riemannian submanifold of a pseudo-Riemannian manifold are obtained. We also find inequalities for spacelike submanifolds. Equality cases are also discussed. (C) 2016 Elsevier B.V. All rights reserved.

This chapter focuses on geometric and spectral consequences of curvature bounds. Several of the results presented here have analogues in Riemannian geometry but in some cases one can go even beyond the Riemannian results and there also striking differences. The geometric setting of this chapter are tessellations and the curvature notion arises as a combinatorial quantity which can be interpreted as an angular defect and goes back to Descartes. First, we study the geometric consequences of curvature bounds. Here, a discrete Gauss–Bonnet theorem provides a starting point from which various directions shall be explored. These directions include analogues of a theorem of Myers, a Hadamard–Cartan theorem, volume growth bounds, strong isoperimetric inequalities and Gromov hyperbolicity. Secondly, we investigate spectral properties of the Laplacian which are often consequences of the geometric properties established before. For example we present analogues to a theorem of McKean about the spectral gap, a theorem by Donnelly-Li about discrete spectrum, we discuss the phenomena of compactly supported eigenfunctions and briefly elaborate on stability of the l2 spectrum for the Laplacian on l p .

We give a classification for connected complete, locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing L 2 -forms. In addition, we prove vanishing theorems for these forms on some complete Riemannian manifolds. Our proofs are based on the Bochner technique that is a most elegant and important analytical method in differential geometry “in the large”.

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.

For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data sets approach sets for which no solutions exist. We also prove that there are AE seed data sets which include a Yamabe nonpositive metric and lead to solutions of the conformal constraints. These data sets allow the mean curvature function to have zeroes., 27 pages

In this work, we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to 6 and arbitrarily large widths. The search for metrics with such properties is motivated by the rigidity result of min-max minimal spheres in three-manifolds obtained by Marques and Neves (Duke Math J 161(14):2725–2752, 2012).

The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibred manifold, we reduce the problem to solution of a leafwise elliptic equation, which has three stable solutions -- only one of them corresponds to the case of a foliated Riemannian manifold., 28 pages

In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [15]., Comment: 14 pages, new references added, a few typo fixed

Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant scalar curvature. We form the connected sum M' # M'' of M' and M'' by removing small balls from M' and M'' and joining the S^{n-1} boundaries together. In this paper we use analysis to construct metrics with constant scalar curvature on M' # M''. Our description is quite explicit, in contrast to the general Yamabe case when one knows little about what the metric looks like. There are 9 cases, depending on the signs of the scalar curvature on M' and M'' (positive, negative, or zero). We show that the constant scalar curvature metrics either develop small "necks" separating M' and M'', or one of M', M'' is crushed small by the conformal factor. When both have positive scalar curvature, we construct three different metrics with scalar curvature 1 in the same conformal class., 45 pages, LaTeX. (v2) Rewritten, shorter, references added

$f(R)$ modification of Einstein's gravity is an interesting possibility to explain the late-time acceleration of the Universe. In this work we explore the cosmological viability of one such $f(R)$ modification proposed by Kruglov [Phys. Rev. D 89, 064004 (2014)]. We show that the model violates fifth-force constraints. The model is also plagued with the issue of a curvature singularity in a spherically collapsing object, where the effective scalar field reaches the point of diverging scalar curvature.

It has been realized that the proof of Theorem 5.1 in Section 5 is imcomplete. It was pointed out by Professor Jongsu Kim and Israel Evangelista. Here we give a correct proof of Theorem 5.1

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 paper we study the closedness properties of generalized cur- vature 2-forms, which are said to be Riemannian, Conformal, Projective, Concircular and Conharmonic curvature 2-forms, associated to each generalized curvature tensors on a Riemannian manifold. Corresponding to each curvature tensors, such generalized curvature 2-forms are the associated curvature 2-forms. In particular, we focus on the closedness of difierential 2-forms associated to the divergence of generalized curvature tensors, which is weaker than the notion of harmonic curvature. In this case, we give an algebraic condition involving the Riemann curvature tensor and the Ricci tensor arising from an old identity due to Lovelock.

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a ‘friction term’ which permits the usual analytic gluing construction to be carried out.

We investigate the mean curvature of semi-Riemannian graphs in the semi-Riemannian warped product M×fℝe , where M is a semi-Riemannian manifold, ℝe is the real line ℝ with metric edt2 (e = ±1), and f: M→ℝ + is the warping function. We obtain an integral formula for mean curvature and some results dealing with estimates of mean curvature, among these results is a Heinz–Chern type inequality.

In this paper, we obtain the mcci curvature and the scalar curvature on SU (3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU (3) to be positive.

In this paper we consider the problem of finding closed hypersurfaces with prescribed Weingarten curvature in a large class of Riemannian manifolds. Under some sufficient conditions, we obtain an existence result by establishing a priori estimates and standard degree theory for the admissible solutions to the prescribed curvature equation. We mainly show that the primary technique developed in Urbas, Wang and the second author’s work in 2004 is flexible enough to be used in more general ambient geometry.

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].

In this paper, in the special setting of a Riemannian foliation en- dowed with a bundle-like metric, we obtain conditions that force the vanishing of the kernel of the basic Dirac operator associated to the metric; this way we extend the traditional setting of Riemannian foli- ations with basic-harmonic mean curvature, where Bochner technique and vanishing results are known to work. Beside classical conditions concerning the positivity of some curvature terms we obtain new rela- tions between the mean curvature form and the kernel of the basic Dirac operator

In this paper, we study a complete -Riemannian manifold whose curvature bounded from below. Let be a compact totally geodesic submanifold of . Then, for any , we can make use of the first variation formula and the second variation formula of distance to prove that is bounded.

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold M. We prove that if the criti- cal point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an n-dimensional Rie- mannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

We find a one-parameter family of Riemannian metrics on for for some number with the following property: is the Euclidean metric on , the scalar curvatures of are strictly decreasing in t in the open unit ball and is isometric to the Euclidean metric in the complement of the ball.

In this paper, we consider an n-dimensional compact Riemannian manifold (M,g) of constant scalar curvature and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ together with a condition on Ricci curvature of M, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forces M to be isometric to the n-sphere Sn(λ).

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations onRiemannian warped product manifold M =(a, ∞ )× f N with prescribed scalar curvature functions.Key words : Warped Product, Scalar Curvature, Partial Differential Equation 1. Introduction In a recent study [7-9] , M.C. Leung has studied theproblem of scalar curvature functions on Riemannianwarped product manifolds and obtained partial resultsabout the existence and nonexistence of Riemannianwarped metric with some prescribed scalar curvaturefunction. He has studied the uniqueness of positive solu-tion to equation(1.1)where is the Laplacian operator for an n-dimen-sional Riemannian manifold (N, g 0 ) and d n = n−2/4(n−1). Equation (1.1) is derived from the conformal defor-mation of Riemannian metric [1,4-6,8,9] .Similarly, let (N, g 0 ) be a compact Riemanniandimensional manifold. We consider the (n+1)−dimen-sional Riemannian warped product manifold M =(a, ∞)× f N with the metric g = dt

We consider minimal closed hypersurfaces \({M \subset \mathbb{S}^7(1)}\) with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.

@@ t gi jD 2Ri j; t2T0; T/; on a smooth, compact n-dimensional Riemannian manifold M. If the flow has uniformly bounded scalar curvature and develops Type I singularities at T , we show that suitable blow-ups of the evolving metrics converge in the pointed Cheeger‐Gromov sense to a Gaussian shrinker by using Perelman’s W-functional. If the flow has uniformly bounded scalar curvature and develops Type II singularities at T , we show that suitable scalings of the potential functions in Perelman’s entropy functional converge to a positive constant on a complete, Ricci flat manifold. We also show that if the scalar curvature is uniformly bounded along the flow in certain integral sense then the flow either develops a Type II singularity at T or it can be smoothly extended past time T .

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K = 3ċ + σ where σ and c are scalar functions on M. In this paper, we establish the intrinsic relation between scalar functions c(x) and σ(x). More general, by invoking the Ricci identities for a one form, we investigate Finsler metric of weakly isotropic flag curvature K = 3θ/F + σ and show that F has constant flag curvature if θ is horizontally parallel.

This paper is devoted to the study of the nonlinear elliptic problem with supercritical critical exponent (Pε) : −Δu = K|u|4/(n−2)+εu in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 3, K is a C3 positive function and ε is a positive real parameter. We show first that in dimension 3, for ε small, (Pε) has no sign-changing solutions with low energy which blow up at two points. For n ≥ 4, we prove that there are no sign-changing solutions which blow up at two nearby points. We also show that (Pε) has no bubble-tower sign-changing solutions.

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 existence of conformal metrics with prescribed scalar curvature on n-dimensional Riemannian manifolds, $n \geq 5 $. Using precise estimates on the losses of compactness, we characterize the critical points at infinity of the associated variational problem and we prove existence results for curvatures satisfying an assumption of Bahri-Coron type.

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”.

Using the Lyapunov–Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on Sn (n ≥ 6), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have Cn - 1, β smoothness.