Constant Scalar Curvature Metrics Research Articles

The Einstein-harmonic equations and constant scalar curvature Kähler metrics

Let (M,J) be a compact complex surface. In his paper [9], LeBrun showed that J-invariant solutions of the Einstein-Maxwell equations correspond to conformally Kähler constant scalar curvature metrics whose Ricci tensors are J-invariant. In the present paper, we prove that constant scalar curvature Kähler manifolds of even complex dimension give solutions of Einstein equations with matter fields which we call the Einstein-harmonic equations.

Fractional Yamabe Problem on Locally Flat Conformal Infinities of Poincaré-Einstein Manifolds

Abstract We study the fractional Yamabe problem first considered by Gonzalez-Qing [36] on the conformal infinity $(M^{n}, \;[h])$ of a Poincaré-Einstein manifold $(X^{n+1}, \;g^{+})$ with either $n=2$ or $n\geq 3$ and $(M^{n}, \;[h])$ locally flat, namely $(M, h),$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits a local situation and also a global one. The latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either $n=2$ or of dimension $n\geq 3$ and which are locally flat, and hence the minimizing technique of Aubin [4] and Schoen [48] in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau [49], which is not known to hold. Using the algebraic topological argument of Bahri-Coron [8], we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n=2$ or of dimension $n\geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.

Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

A product model for generalizing Poincaré-type Kähler metrics

We begin by defining a type of Kähler metric near the zero section of a trivial holomorphic open disk bundle N N over a compact Kähler manifold X X by incorporating flows generated by holomorphic vector fields on X X . These metrics are then shown to deviate exponentially from Poincaré-type metrics on N ∖ X N\setminus X in terms of the log-polar distance from X X in N N . Lastly we see that they arise naturally when perturbing classes containing Poincaré-type Kähler metrics of constant scalar curvature to obtain nearby cscK metrics even when the perturbed class on X X does not admit a cscK metric.

Optimal symplectic connections and deformations of holomorphic submersions

We give a general construction of extremal Kähler metrics on the total space of certain holomorphic submersions, extending results of Dervan-Sektnan, Fine, and Hong. We consider submersions whose fibres admit a degeneration to Kähler manifolds with constant scalar curvature, in a way that is compatible with the fibration structure. Thus we allow fibres that are K-semistable, rather than K-polystable; this is crucial to moduli theory. On these fibrations we phrase a partial differential equation whose solutions, called optimal symplectic connections, represent a canonical choice of a relatively Kähler metric. We expect this to be the most general construction of a canonical relatively Kähler metric provided all input is smooth. We use the notion of an optimal symplectic connection and the geometry related to it to construct Kähler metrics with constant scalar curvature and extremal metrics on the total space, in adiabatic classes.

The $S^3_{\boldsymbol{w}}$ Sasaki join construction

The main purpose of this work is to generalize the $S^{3}_{\boldsymbol{w}}$ Sasaki join construction $M \star_{\boldsymbol{l}} S^{3}_{\boldsymbol{w}}$ described in the authors' 2016 paper when the Sasakian structure on $M$ is regular, to the general case where the Sasakian structure is only quasi-regular. This gives one of the main results, Theorem 3.2, which describes an inductive procedure for constructing Sasakian metrics of constant scalar curvature. In the Gorenstein case ($c_{1}(\mathscr{D}) = 0$) we construct a polynomial whose coeffients are linear in the components of $\boldsymbol{w}$ and whose unique root in the interval $(1, \infty)$ completely determines the Sasaki–Einstein metric. In the more general case we apply our results to prove that there exists infinitely many smooth 7-manifolds each of which admit infinitely many inequivalent contact structures of Sasaki type admitting constant scalar curvature Sasaki metrics (see Corollary 6.15). We also discuss the relationship with a recent paper of Apostolov and Calderbank as well as the relation with K-stability.

Besse conjecture with positive isotropic curvature

The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function f on the manifold that satisfies the following $$\begin{aligned} (1+f)\mathrm{Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. \end{aligned}$$ It has been conjectured that if (g, f) is a solution of the critical point equation, then g is Einstein and so (M, g) is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.

On scalar curvature of r-almost Yamabe solitons

The aim of this work is to study constant scalar curvature metrics of r-almost Yamabe solitons immersed into a Riemannian manifold. We deal with the compact and complete cases to obtain that an r-Yamabe soliton must have constant scalar curvature and some triviality results. Moreover, we also prove a Schur type inequality.

Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles

We study the existence of weighted extremal K\ahler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge K\ahler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal K\ahler metrics pioneered by a seminal construction of Calabi. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally K\ahler, Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent constructions of LeBrun and Koca-T{\o}nnesen-Friedman to higher dimensions. For each admissible K\ahler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval $(-1,1)$, then there exists a weighted extremal K\ahler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no K\ahler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau-Tian-Donaldson type correspondence for the existence of K\ahler metrics of constant weighted scalar curvature in the rational admissible K\ahler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

Estimates for metrics of constant Chern scalar curvature

We prove a priori estimates for constant Chern scalar curvature metrics on a compact complex manifold conditional on an upper bound on the entropy, extending a recent result by Chen-Cheng in the Kahler setting.

Attractors of the ‘n + 1’ dimensional Einstein-Λ flow

Here we prove a global existence theorem for sufficiently small however fully nonlinear perturbations of a family of background solutions of the ‘n + 1’ dimensional vacuum Einstein equations in the presence of a positive cosmological constant Λ. The future stability of vacuum solutions in the small data and zero cosmological constant limit has been studied previously for both ‘3 + 1’ and higher dimensional spacetimes. However, with the advent of dark energy driven accelerated expansion of the Universe, it is of fundamental importance in mathematical cosmology to include a positive cosmological constant, the simplest form of the dark energy for the vacuum Einstein equations. Such Einsteinian evolution is here designated as the ‘Einstein-Λ’ flow. We study the background solutions of this ‘Einstein-Λ’ flow in ‘n + 1’ dimensional spacetimes in constant mean curvature spatial harmonic gauge, n ⩾ 3 and establish both linear and non-linear stability of such solutions. In the cases of number of spatial dimensions being strictly greater than 3, the finite dimensional Einstein moduli spaces form the centre manifolds of the dynamics. A suitable shadow gauge condition [Andersson L and Moncrief V 2011 J. Differ. Geom. 89 1–47] is implemented in order to treat these cases. In addition, the autonomous character of the suitably re-scaled Einstein flow breaks down as a consequence of including Λ(>0). We construct a Lyapunov functional (controlling a suitable norm of the small data) similar to a wave equation type energy for the non-linear non-autonomous evolution of the data and prove its decay in the direction of cosmological expansion utilizing the structure of the non-autonomous terms and smallness assumption on the data. Our results demonstrate the future stability and geodesic completeness of the perturbed spacetimes, and show that the scale-free geometry converges to an element of the space of constant negative scalar curvature metrics sufficiently close to and containing the Einstein moduli space (a point for n = 3 and a finite dimensional space for n > 3), which has significant consequences for the cosmic topology while restricting to the case of n = 3.

Hermitian metrics of constant Chern scalar curvature on ruled surfaces

It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric [6, 22, 38]. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page-Bérard-Bergery's ansatz [41, 14]. We also construct the interesting case of Hermitian metrics of zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the problem of the existence in a conformal class of critical metrics of the total Chern scalar curvature, studied by Gauduchon in [26, 27].

Conformal vector fields and Yamabe solitons

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.

Yamabe flow and metrics of constant scalar curvature on a complete manifold

In this paper, we give a condition on the initial metric which makes the global existence of Yamabe flow and we study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. As application we study the ADM mass monotonicity of Yamabe flow in AF manifolds. We use the variational method to study the existence problem of metrics with constant scalar curvature on complete non-compact Riemannian manifolds and we can give a partial affirmative answer to a question posed by Kazdan (Math Ann 261(2):227–234, 1982).

A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

AbstractWe define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

The smallest Laplace eigenvalue of homogeneous 3-spheres

We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace–Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of three-dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non-zero eigenvalue in terms of the diameter for homogeneous 3-spheres. Another application shows that the spectrum of the Laplace–Beltrami operator distinguishes up to isometry any left-invariant metric on SU(2). It is also proved the non-existence of constant scalar curvature metrics conformal and arbitrarily close to any non-round homogeneous metric on the 3-sphere. All of the above results are extended to left-invariant metrics on SO(3), that is, homogeneous metrics on the three-dimensional real projective space.

Almost-Kähler smoothings of compact complex surfaces with $A_1$ singularities

This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kahler manifolds obtained as smoothings of a constant scalar curvature Kahler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kahler smoothing (Me, ωe) admits an almost-Kahler structure (Je, ge) of constant Hermitian curvature. Moreover, we show that for e > 0 small enough, the (Me, ωe) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for ge.

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.

Complex Deformation of Critical Kahler Metrics

In this paper, we use Pacard-Xu's methods to discuss the complex deformation of constant scalar curvature metrics in the case of fixed and varying complex structures. Moreover, we also discuss the complex deformation of K\"ahler Ricci solitons.