On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.     

Critical metrics of the scalar curvature functional satisfying a vanishing condition on the Weyl tensor

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.     

Riemannian metrics of positive scalar curvature on compact manifolds with boundary

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S ⁿ is a group.     

A note on extremal metrics of non-constant scalar curvature

By working in ? ⁿ with potentials of the forma logu + s(u), u the square of the distant to the origin, we obtain extremal Kähler metrics of nonconstant scalar curvature on the blow-up of ? ⁿ at \(\vec 0\) . We then show that these metrics can be completed at ∞ by adding a ?? ^n?1, and reobtain the extremal Kähler metrics of non-constant scalar curvature constructed by Calabi on the blow-up of ?? ⁿ at one point. A similar construction produces this type of metrics on other bundles over ?? ^{n ? 1}.     

Critical metrics of the total scalar curvature functional on 4‐manifolds

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      

Hypersurfaces of the hyperbolic space with constant scalar curvature

We classify hypersurfaces of the hyperbolic space ?ⁿ⁺¹(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mⁿ is a complete hypersurfaces with constant scalar curvature n(n ? 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n? 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.     

On Square metrics of scalar flag curvature

In this paper, we study the problem whether a Finsler metric of scalar flag curvature is locally projectively flat. We consider a special class of Finsler metrics — square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that in dimension n ≥ 3, any square metric of scalar flag curvature is locally projectively flat.     

Ricci curvature, minimal volumes, and Seiberg-Witten theory

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with non-trivial Seiberg-Witten invariants. These allow one, for example, to exactly compute the infimum of the L ²-norm of Ricci curvature for any complex surface of general type. We are also able to show that the standard metric on any complex-hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results for Einstein metrics. Oblatum 14-III-2000 & 8-II-2001?Published online: 4 May 2001     

Prescribed scalar curvature on then-sphere

This paper considers the prescribed scalar curvature problem onS ⁿ forn>-3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We then show that forn=3 this is the only blow up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed scalar curvature problem onS ³.This article was processed by the author using the style filepljourlm from Springer-Verlag.     

Non-negative scalar curvature

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C ^* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

Bach-flat critical metrics for quadratic curvature functionals

In this paper, we study the critical metrics for quadratic curvature functionals involving the Ricci curvature and scalar curvature in the space of Riemannian metrics with unit volume. For these functionals, Einstein metrics are always critical metrics. However, a converse problem is not always true. The purpose of this paper is to show that, under the condition that the critical metrics are Bach-flat, a partial converse is true.     

Variational properties of the Gauss–Bonnet curvatures

The Gauss–Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss–Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss–Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss–Bonnet curvature.     

Prescribing scalar curvature on Sn and related problems,part II: Existence and compactness

This is a sequel to [30], which studies the prescribing scalar curvature problem on Sⁿ. First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lower-dimensional cases n = 2, 3. Next we study the problem for n ≥ 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is β ϵ (n − 2, n). The key point there is that for the class of K mentioned above we have completed L^∞ apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is β = n − 2, the L^∞ estimates for solutions fail in general. In fact, two or more blowup points occur. On the other hand, we provide some existence and compactness results when the order of flatness at critical points of K(x) is β ϵ [n − 2,n). With this result, we can easily deduce that C^∞ scalar curvature functions are dense in C^1,α (0 < α < 1) norm among positive functions, although this is generally not true in the C² norm. We also give a simpler proof to a Sobolev-Aubin-type inequality established in [16]. Some of the results in this paper as well as that of [30] have been announced in [29]. © 1996 John Wiley & Sons, Inc.     

Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert–Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n + 1), Sp(n + 1) and Spin(9)-homogeneous metrics.     

Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n?3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n?4, we give conditions on the function h to guarantee there is only one simple blow-up point.     

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.     

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.     

Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties

In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.      

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.