Scalar-flat Metrics Research Articles

Singular Yamabe problem for scalar flat metrics on the sphere

Let Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} be a domain on the unit n-sphere Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb {S}}^n$$\\end{document} and g∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{{\\,}_\\circ }{g}$$\\end{document} the standard metric of Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}^n$$\\end{document}, n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document}. We show that there exists a conformal metric g with vanishing scalar curvature R(g)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(g)=0$$\\end{document} such that (Ω,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Omega , g)$$\\end{document} is complete if and only if the Bessel capacity Cα,q(Sn\\Ω)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}_{\\alpha , q}({\\mathbb {S}}^n\\setminus \\Omega )=0$$\\end{document}, where α=1+2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1+\\frac{2}{n}$$\\end{document} and q=n2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=\\frac{n}{2}$$\\end{document}. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf–Rinow theorem for the divergent curves.

Geodesics on metrics of Eguchi–Hanson type

Geodesic equations are solved when at least two of theta , phi and psi are constant, or r is constant, on scalar flat metrics of Eguchi–Hanson type. They can also be solved also on Eguchi–Hanson metrics which are Ricci flat if only phi is constant. However, the explicit solution of the geodesic equations is not available yet if only psi is constant.

Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with umbilic boundary

Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that ∂M has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.

A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

Let (M, g) be a compact Riemannian n-dimensional manifold with umbilic boundary It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have \(\partial M\) as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is \(n=6,7,8\), provided that the Weyl tensor is always not vanishing on the boundary.

Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> a compact Riemannian <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of <inline-formula><tex-math id="M3">\begin{document}$ g $\end{document}</tex-math></inline-formula> there are scalar-flat metrics that have <inline-formula><tex-math id="M4">\begin{document}$ \partial M $\end{document}</tex-math></inline-formula> as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term <inline-formula><tex-math id="M5">\begin{document}$ h_{g} $\end{document}</tex-math></inline-formula> with a negative smooth function <inline-formula><tex-math id="M6">\begin{document}$ \alpha, $\end{document}</tex-math></inline-formula> the set of solutions of Yamabe problem is still a compact set.

Compactness for conformal scalar-flat metrics on umbilic boundary manifolds

Let (M,g) a compact Riemannian n-dimensional manifold with umbilic boundary. It is well known that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have ∂M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set, provided n=8 and the Weyl tensor of the boundary is always different from zero, or if n>8 and the Weyl tensor of M is always different from zero on the boundary.

A compactness theorem for scalar-flat metrics on 3-manifolds with boundary

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 exponent on the boundary.

Invariant scalar-flat Kähler metrics on $$\mathcal {O} (- \ell )$$ O ( - ℓ )

This paper aims at giving a unified framework for a number of well-known extremal Kahler metrics with a big group of symmetries, with a special emphasis on the case of scalar-flat Kahler metrics, all stemming from the general construction of \(\mathrm{U} (m)\)-invariant extremal Kahler metrics on the space \(\mathbb {C}^m {\setminus } \{0\}\), based on the momentum profile introduced by Hwang and Singer (Trans Am Math Soc 354:2285–2325, 2002).

Metrics in Projective Differential Geometry: The Geometry of Solutions to the Metrizability Equation

Pseudo-Riemannian metrics with Levi-Civita connection in the projective class of a given torsion-free affine connection are equivalent to the maximal rank solutions of a certain overdetermined projectively invariant differential equation often called the metrizability equation. Dropping this rank assumption, we study the solutions to this equation, given less restrictive generic conditions on its prolonged system. In this setting, we find that the solution stratifies the manifold according to the strict signature (pointwise) of the solution and does this in way that locally generalizes the stratification of a model, where the model is, in each case, a corresponding Lie group orbit decomposition of the sphere. Thus the solutions give curved generalizations of such embedded orbit structures. We describe the smooth nature of the strata and determine the geometries of each of the different strata types; this includes a metric on the open strata that becomes singular at the strata boundary, with the latter a type of projective infinity for the given metric. The work also provides new results for the projective compactification of scalar-flat metrics.

On the third coefficient of TYZ expansion for radial scalar flat metrics

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

Convergence of scalar-flat metrics on manifolds with boundary under a Yamabe-type flow

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.

Blow-up phenomena for scalar-flat metrics on manifolds with boundary

Let ( M n , g ) be a compact Riemannian manifold with boundary ∂ M. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have ∂ M as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball B n , in dimensions n ⩾ 25 , for which this set is non-compact. These manifolds have umbilic boundary, but they are not conformally equivalent to B n .

An existence theorem of conformal scalar-flat metrics on manifolds with boundary

Let.M; g/ be a compact Riemannian manifold with boundary. We address the Yamabe-type problem of finding a conformal scalar-flat metric on M whose boundary is a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some of the remaining cases of this problem.

A compactness theorem for scalar-flat metrics on manifolds with boundary

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 condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.

Generalized connected sum construction for scalar flat metrics

In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M1 � K M2 of two compact Riemannian scalar flat manifolds (M1,g1) and (M2,g2) along a common Riemannian submanifold (K ,gK ) 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 M1 and M2. 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 clas- sification, 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.

A Morse theoretical approach for the boundary mean curvature problem on [formula omitted

In this paper we investigate existence as well as multiplicity of scalar flat metric of prescribed boundary mean curvature on the standard 4-dimensional ball. Due to the existence of critical point at infinity, the standard variational methods cannot be applied. To overcome this difficulty, we prove that in a neighborhood of critical points at infinity, a Morse lemmas at infinity reduction holds, then develop a whole Morse theory of this noncompact variational problem. In particular we establish, under generic boundary condition Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As further application of this Morse theoretical approach, we prove more existence results and extend a topological invariant introduced by A. Bahri.

Singular Hyperbolic Monopoles

We study the twistor theory of singular hyperbolic SU(2) monopoles following the approach taken by Kronheimer [9] in the Euclidean case. We use our results to show that the moduli space of charge 1 monopoles possesses a natural 2-sphere of scalar flat Kahler metrics. In the zero mass limit, the metrics reduce to a class of metrics first studied by LeBrun in [10].

Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary

Let (M, g) be a compact manifold with nonempty boundary and finite Sobolev quotient Q(M, ∂M). We prove that there exists a conformal deformation which is scalar-flat and has constant boundary mean curvature, if n = 4 or 5 and the boundary is not umbilic. In particular we prove such existence for any smooth and bounded open set of the Euclidean space, finishing the remaining cases of a theorem of J. F. Escobar.

Scalar Flat Metrics of Eguchi- Hanson Type**The project partially supported by National Natural Science Foundation of China under Grant No. 10231050 and the Innovation Project of the Chinese Academy of Sciences

We use a new method to construct a class of asymptotically locally flat, scalar flat metrics. These metrics were constructed via algebraic geometry method by LeBrun before and provide counterexamples to the generalized positive action conjecture of Hawking and Pope.

Asymptotically flat and scalar flat metrics on $\mathbb {R}^3$ admitting a horizon

We give a new construction of asymptotically flat and scalar flat metrics on $\mathbb {R}^3$ with a stable minimal sphere. The existence of such a metric gives an affirmative answer to a question raised by R. Bartnik (1989).