Constant Scalar Curvature Metrics Research Articles

Convexity of the 𝐾-energy on the space of Kähler metrics and uniqueness of extremal metrics

We establish the convexity of Mabuchi’s K K -energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Elliptic Function Solutions in Jackiw-Teitelboim Dilaton Gravity

We present a new family of solutions for the Jackiw-Teitelboim model of two-dimensional gravity with a negative cosmological constant. Here, a metric of constant Ricci scalar curvature is constructed, and explicit linearly independent solutions of the corresponding dilaton field equations are determined. The metric is transformed to a black hole metric, and the dilaton solutions are expressed in terms of Jacobi elliptic functions. Using these solutions, we compute, for example, Killing vectors for the metric.

Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics

Well-known conjectures of Tian predict that the existence of canonical Kähler metrics should be equivalent to various notions of properness of Mabuchi’s K-energy functional. First, we provide counterexamples to Tian’s first conjecture in the presence of continuous automorphisms. Second, we resolve Tian’s second conjecture, confirming the Moser–Trudinger inequality for Fano manifolds. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal replacements for Tian’s first conjecture in the setting of smooth and singular Kähler–Einstein metrics, with or without automorphisms, as well as for Kähler–Ricci solitons. Moreover, we reduce both Tian’s original first conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion.

Killing fields generated by multiple solutions to the Fischer–Marsden equation II

In the process of finding Einstein metrics in dimension [Formula: see text], we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space [Formula: see text] can be at most [Formula: see text], with equality implying that [Formula: see text] is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor [Formula: see text]. In this second paper, we apply our results in the first paper to show that either [Formula: see text] is a standard sphere or the dimension of the space of Fischer–Marsden solutions can be at most [Formula: see text].

Bifurcation of periodic solutions to the singular Yamabe problem on spheres

We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $\mathbb{S}^1$ inside $\mathbb{S}^m , m \geq 5$, that are conformal to the round (incomplete) metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of $\mathbb{S}^m \setminus \mathbb{S}^1 \cong \mathbb{S}^{m-2} \times \mathbb{H}^2$.

Metrics of constant scalar curvature on sphere bundles

Let G/H be a Riemannian homogeneous space. For an orthogonal representation ϕ of H on the Euclidean space Rk+1, there corresponds the vector bundle E=G×ϕRk+1→G/H with fiberwise inner product. Provided that ϕ is the direct sum of at most two representations which are either trivial or irreducible, we construct metrics of constant scalar curvature on the unit sphere bundle UE of E. When G/H is the round sphere, we study the number of constant scalar curvature metrics in the conformal classes of these metrics.

Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces

This paper produces explicit strongly Hermitian Einstein–Maxwell solutions on the smooth compact 4-manifolds that are $$S^2$$ -bundles over compact Riemann surfaces of any genus. This generalizes the existence results by LeBrun (J Geom Phys 91:163–171, 2015, The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016). Moreover, by calculating the (normalized) Einstein–Hilbert functional of our examples, we generalize Theorem E of LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), which speaks to the abundance of Hermitian Einstein–Maxwell solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly Hermitian Einstein–Maxwell solutions, first found in LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common Kahler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.

The local Yamabe constant of Einstein stratified spaces

On a compact stratified space (X,g), a metric of constant scalar curvature exists in the conformal class of g if the scalar curvature Sg satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant Yℓ(X). This latter is a conformal invariant introduced in the recent work of K. Akutagawa, G. Carron and R. Mazzeo. It depends on the local structure of X, in particular on its links, but its explicit value is unknown. We show that if the links satisfy a Ricci positive lower bound, then we can compute Yℓ(X). In order to achieve this, we prove a lower bound for the spectrum of the Laplacian, by extending a well-known theorem by A. Lichnerowicz, and a Sobolev inequality, inspired by a result due to D. Bakry. A particular stratified space, with one stratum of codimension 2 and cone angle bigger than 2π, must be handled separately – in this case we prove the existence of an Euclidean isoperimetric inequality.

An integral inequality for constant scalar curvature metrics on Kähler manifolds

We present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refinement of S.-T. Yau's Chern number inequality from the Einstein metric case to constant scalar curvature metric case.

On the structure of linearization of the scalar curvature

For a compact $n$-dimensional manifold a critical point metric of the total scalar curvature functional satisfies the critical point equation (1) below, if the functional is restricted to the space of constant scalar curvature metrics of unit volume. The right-hand side in this equation is nothing but the adjoint operator of the linearization of the total scalar curvature acting on functions. The structure of the kernel space of the adjoint operator plays an important role in the geometry of the underlying manifold. In this paper, we study some geometric structure of a given manifold when the kernel space of the adjoint operator is nontrivial. As an application, we show that if there are two distinct solutions satisfying the critical point equation mentioned above, then the metric should be Einstein. This generalizes a main result in [6] to arbitrary dimension.

Killing fields generated by multiple solutions to the Fischer–Marsden equation

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

The Sasaki Join, Hamiltonian 2-Forms, and Constant Scalar Curvature

We describe a general procedure for constructing new explicit Sasaki metrics of constant scalar curvature (CSC), including Sasaki–Einstein metrics, from old ones. We begin by taking the join of a regular Sasaki manifold of dimension \(2n+1\) and constant scalar curvature with a weighted Sasakian 3-sphere. Then by deforming in the Sasaki cone we obtain CSC Sasaki metrics on compact Sasaki manifolds \(M_{l_1,l_2,\mathbf{w}}\) of dimension \(2n+3\) which depend on four integer parameters \(l_1,l_2,w_1,w_2\). Most of the CSC Sasaki metrics are irregular. We give examples which show that the CSC rays are often not unique on the underlying fixed strictly pseudoconvex CR manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two-dimensional subcone of Sasaki–Ricci solitons in the Sasaki cone, and a unique Sasaki–Einstein metric in each of the two-dimensional subcones.

Multiplicity of constant scalar curvature metrics in [formula omitted

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

Erratum to: Constant scalar curvature metrics on Hirzebruch surfaces

Erratum to: Constant scalar curvature metrics on Hirzebruch surfaces

Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in three, four and six dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in three-dimensional spaces because of the link to General Relativity. In particular, all three-dimensional constant negative scalar curvature spherical manifolds can be embedded as constant mean curvature surfaces in appropriate Schwarzschild solutions. Our approach, although not the simplest one, is linked to the Lichnerowicz–York method of finding initial data for Einstein equations.

On the transverse Scalar Curvature of a Compact Sasaki Manifold

Abstract We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson [10, 11], which involves the geometry of infinitedimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the strict contactomophism group

Isoparametric hypersurfaces and metrics of constant scalar curvature

We showed the existence of non-radial solutions of the equation $\Delta u -\lambda u + \lambda u^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $\lambda$. We show that for any isoparametric hypersurface $M\subset S^m$ there are solutions such that $M$ is a regular level set (and the number of such solutions increases with $\lambda$). We also show similar results for isoparametric hypersurfaces in general Riemannian manifolds. These solutions give multiplicity results for metrics of constant scalar curvature on conformal classes of Riemannian products.

Stability under deformations of Hermite-Einstein almost Kähler metrics

On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of compatible almost-complex structures, diffeomorphic at each time to the initial one, and inducing constant Hermitian scalar curvature metrics.

Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many t's accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.

THREE DIMENSIONAL CRITICAL POINT OF THE TOTAL SCALAR CURVATURE

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.