Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8. Received: 12 July 1999     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Topological rigidity theorems for open Riemannian manifolds

In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?ⁿ if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?ⁿ if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the variation of a metric and its application

Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system, In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then （M, g） is either scalar flat or a space form.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

Einstein structures: Existence versus uniqueness

Two well-known questions in differential geometry are “Does every compact manifold of dimension greater than four admit an Einstein metric?” and “Does an Einstein metric of a negative scalar curvature exist on a sphere?” We demonstrate that these questions are related: For everyn≥5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimensionn (or even on every smooth homology sphere of dimensionn) implies the existence of metrics of negative Ricci curvature on the sphereS ⁿ for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the Palais-Smale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for anyn≥5 that: 1) If everyn-dimensional smooth homology sphere admits an Einstein metric thenS ⁿ admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compactn-dimensional manifold admits an Einstein metric then every compactn-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature.     

Une hypothèse topologique pour le problème de la courbure scalaire prescrite

By topological arguments, we set forth sufficient hypotheses for a given function K, on a compact Riemannian manifold (M, g), to be the scalar curvature of a metric conformal to g.     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler.     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

Uniqueness and non-uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary

Let (Mⁿ,g) be a compact manifold with boundary with n?2. In this paper we discuss uniqueness and non-uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of M.     

Some examples of Hermitian surfaces

The Riemannian version of the Goldberg-Sachs theorem says that a compact Einstein Hermitian surface is locally conformal Kähler. In contrast to the compact case, we show that there exists an Einstein Hermitian surface which is not locally conformal Kähler. On the other hand, it is known that on a compact Hermitian surface M ⁴, the zero scalar curvature defect implies that M ⁴ is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.     

Characterizations and integral formulae for generalized m-quasi-Einstein metrics

The aim of this paper is to present some structural equations for generalized m-quasi-Einstein metrics (M ⁿ , g, ? f, λ), which was defined recently by Catino in [11]. In addition, supposing that M ⁿ is an Einstein manifold we shall show that it is a space form with a well defined potential f. Finally, we shall derive a formula for the Laplacian of its scalar curvature which will give some integral formulae for such a class of compact manifolds that permit to obtain some rigidity results.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

The nonvacuum Einstein flow on surfaces of nonnegative curvature

We prove future nonlinear stability of homogeneous solutions to the Einstein–Vlasov system with massive particles on manifolds with topology M = ?×Σ, where Σ is either 𝕊² or 𝕋². For the sphere this implies the existence of an open subset of the initial data manifold with elements of strictly positive scalar curvature, whose developments are future geodesically complete. In combination with an earlier result for hyperbolic surfaces we conclude future completeness for the Einstein–Vlasov system in 2+1 dimensions independent of the compact spatial topology for an open set of initial data.     

Harmonic two-forms on manifolds with non-negative isotropic curvature

We prove that L ² harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L ² harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic manifold can not admit any almost K?hler structure of positive isotropic curvature.     

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT _pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.Partially supported by the grants from TGRC.     

Generalized connected sum construction for scalar flat metrics

In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M₁  \sharp_K  M₂{M = M_1 \, \sharp_K \, M_2} of two compact Riemannian scalar flat manifolds (M ₁, g ₁) and (M ₂, g ₂) along a common Riemannian submanifold (K, g _K ) whose codimension is ≥3. Here we present two constructions: the first one produces a family of “small” (in general nonzero) constant scalar curvature metrics on the generalized connected sum of M ₁ and M ₂. It yields an extension of Joyce’s result for point-wise connected sums in the spirit of our previous issues for nonzero constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1₊) class in the Kazdan–Warner classification, we refine the first construction in order to produce a family of scalar flat metrics on M. As a consequence we get new solutions to the Einstein constraint equations on the generalized connected sum of two compact time symmetric initial data sets, extending the Isenberg–Mazzeo–Pollack gluing construction.     

Submanifolds with Flat Normal Bundle

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.