arXiv research

Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.

Constructs scalar-flat Kähler metrics with varying conical singularities.

The paper shows how to create scalar flat metrics with very large ADM mass.

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

We classify radial scalar flat metrics with constant third coeffcient of its TYZ expansion. As a byproduct of our analysis we provide a characterization of Simanca's scalar flat metric.

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…

We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kähler surfaces. This is used to prove that for any scalar-flat Kähler ALE surface, all small deformations of complex structure also admit scalar-flat Kähler ALE metrics. A local moduli space of scalar-flat Kähler ALE metrics is then constr…

Let $M$ be a simply-connected closed manifold of dimension $\geq 5$ which does not admit a metric with positive scalar curvature. We give necessary conditions for $M$ to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of $M$. As a consequence any simply-connected …

The paper studies scalar flat Kähler metrics on line bundles and proves their properties.

Refined asymptotics of scalar-flat ALE four-manifolds

Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.

The paper proves compactness of scalar-flat metrics on low-dimensional manifolds with umbilic boundary.

Classifies scalar-flat toric Kähler instantons in 4D.

Abreu-Sena-Dias have constructed two distinct families of scalar-flat Kähler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that these are the only J-complete scalar-flat Kähler metrics on any given strictly un…

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an ap…

In this article, we give a survey of our construction of a local moduli space of scalar-flat Kähler ALE metrics in complex dimension $2$. We also prove an explicit formula for the dimension of this moduli space on a scalar-flat Kähler ALE surface which deforms to the minimal resolution of $\mathbb{C}^2/Γ$, where $Γ$ is…

The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.

Witten and Yau (hep-th/9910245) have recently considered a generalisation of the AdS/CFT correspondence, and have shown that the relevant manifolds have certain physically desirable properties when the scalar curvature of the boundary is positive. It is natural to ask whether similar results hold when the scalar curvat…

The paper proves stability for scalar-flat metrics on manifolds with boundary.

There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in ${\rm{SU}}(2)$. The main result in this paper shows that for any non-cyclic finite subgroup $Γ\subset {\rm{U}}(2)$ containing no complex reflections, there exist scalar-flat Kähler ALE met…

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.

We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric…

A new construction is presented of scalar-flat Kaehler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new example…

The Yamabe flow on flat manifolds converges to a scalar flat metric.

Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.

The study proves an expansion theorem for scalar-flat asymptotically conical Kähler metrics.

Let $X$ be a ruled surface over a curve of genus $g$. We prove that $X$ has a scalar-flat Hermitian metric if and only if $g\geq 2$ and $m(X)>2-2g$ where $m(X)$ is an intrinsic number depends on the complex structure of $X$.

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we produce the generalized connected sum along $K$. On this third manifold we produ…

Extremal metrics lead to scalar-flat Kähler cones.

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Eu…

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.

Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some remaining cases of this …

We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold…

Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. In this paper, we study the existence of a complete scalar-flat Kähler metric on $X \setminus D$ on the assumption that $D$ has a constant positive scalar curvature Kähler metric.

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev e…

Proves curvature comparison for Riemannian bands in low dimensions.

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critica…

Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if t…

New method detects Kaehler scalar flat metrics and minimal hypersurfaces.

We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line…

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condi…

New rigidity results for critical metrics of a quadratic curvature functional.

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for wh…

Solves geodesic equations on specific metrics types.

In this article, we prove that a quotient of a K3 surface by a free Z_2+Z_2 action does not admit any metric of positive scalar curvature. This shows that the scalar flat anti self-dual metrics (SF-ASD) on this manifold can not be obtained from a family of metrics for which the scalar curvature changes sign, contrary t…