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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

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 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).

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.

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.

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.

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.

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.

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

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.

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.

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

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.

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

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.

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.

Motivated by the strong maximum principle for the Paneitz operator in dimension 5 or higher found in a preprint by Gursky and Malchiodi and the calculation of the second variation of the Green's function pole's value on 3 in our preprint, we study the Riemannian metric on 3-manifolds with positive scalar and Q curvature. Among other things, we show it is always possible to find a constant Q curvature metric in the conformal class. Moreover, the Green's function is always negative away from the pole, and the pole's value vanishes if and only if the Riemannian manifold is conformal diffeomorphic to the standard 3. Compactness of constant Q curvature metrics in a conformal class and the associated Sobolev inequality are also discussed.© 2014 Wiley Periodicals, Inc.

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].

In this paper, we prove Chen’s general inequalities and Chen–Ricci inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature. We also establish inequalities between the warping function and the squared mean curvature for warped product submanifolds in a Riemannian manifold of nearly quasi-constant curvature.

We find a formula for the derivative of the mean integral curvature of a surface in a three-dimensional Riemannian space with respect to infinitesimal deformations.

We find a convenient expression for the value of the covariant curvature 4-tensor of an arbitrary Riemannian manifold on a quadruple of its Killing vector fields. With its use, we in particular obtain a simple deduction of the well-known formula to calculate the sectional curvature of a homogeneous Riemannian space.

The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.

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.

If $\pi :M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.

In this paper, a complete description of all translation hypersurfaces with constant scalar curvature in the Euclidean space is presented.

In this paper, the authors consider a family of smooth immersions Ft: Mn → Nn+1 of closed hypersurfaces in Riemannian manifold Nn+1 with bounded geometry, moving by the Hk mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the Hk mean curvature flow solution with finite total mean curvature on a finite time interval [0, Tmax) can be extended over Tmax. This result generalizes the extension theorems in the paper of Li (see “On an extension of the Hk mean curvature flow, Sci. China Math., 55, 2012, 99–118”).

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M given by R-n/2(g) := integral(M) vertical bar R(g)vertical bar(n//2) dv(g) where R(g), dv(g) denote the Riemannian curvature and volume form corresponding to g. We show that there are locally symmetric spaces which are unstable critical points for this functional.

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 .

This paper is about prescribing the Webster scalar curvature on a compact CR manifold of dimension 2n+1 ≥ 5, which is locally CR equivalent to the standard CR unit sphere S2n+1. The associated variational problem being noncompact we approach this issue with techniques from the critical points at infinity theory, combined with topological tools. Our existence results are based on a generalized Bahri-Coron type criteria. We also give an upper bound on the Morse index of the obtained solutions.

We find a C ∞ -continuous path of Riemannian metrics gt on Rk, k � 3, for 0 � t � " for some number " > 0 with the following property: g0 is the Euclidean metric on Rk, the scalar curvatures of gt are strictly decreasing in t in the open unit ball and gt is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

In [16], Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for Riemannian submersions from nonnegatively curved spaces even without the strict positive curvature assumption and irrespective of the particular metric.

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.

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 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.