Scalar Curvature Equation Research Articles

Gauge-Fixing Constant Scalar Curvature Equations on Ruled Manifolds and the Futaki Invariants

In this article we introduce and prove the solvability of the gauge-fixing constant scalar curvature equations on ruled Kaehler manifolds. We prove that when some lifting conditions for holomorphic vector fields on the base manifold are satisfied the solutions for the gauge-fixing constant scalar curvature equations are actually solutions for the constant scalar curvature equations provided the corresponding Futaki invariants vanish.

On positive solutions of the scalar curvature equation when the curvature has variable sign

On positive solutions of the scalar curvature equation when the curvature has variable sign

Exotic solutions of the conformal scalar curvature equation in ℝn

We construct global exotic solutions of the conformal scalar curvature equation Δu+[n(n−2)/4]Ku(n+2)/(n−2)=0 in ℝn, with K(x) approaching 1 near infinity in order as close to the critical exponent as possible.

The scalar curvature equation on [formula omitted

The scalar curvature equation on [formula omitted

Problème de Dirichlet pour la courbure scalaire dans [formula omitted

We study the Dirichlet problem in a bounded domain for the fully nonlinear elliptic partial differential equation of second order expressing the prescription of the m th symmetric function of the principal curvatures of a spacelike hypersurface in the Minkowski space R n,1 . We solve the prescribed scalar curvature equation ( m=2) in R 3,1 , when the domain Ω and the Dirichlet data are strictly convex.

BOUNDEDNESS OF POSITIVE SOLUTIONS OF THE CONFORMAL SCALAR CURVATURE EQUATION AND POSITIVE SCALAR CURVATURE

We study the boundedness of positive solutions of the conformal scalar curvature equation in a complete Riemannian manifold, in connection with conformal metrics of strictly positive scalar curvature.

Singular ground states for the scalar curvature equation in $\mathbbR^N$

We use dynamical-systems techniques to study the positive solutions of the scalar curvature equation in $R^n$, $n \geq 3$. In certain cases we prove the existence of a Cantor-like set of singular ground states with slow decay.

Structural change of solutions for a scalar curvature equation

A semilinear elliptic equation \[ \Delta u + \{1 + {\varepsilon} k(|x|) \} u^p=0, \quad x \in {\bf R}^n, \] is studied, where $n>2$ and ${\varepsilon}$ is a small parameter. It is known that for $p=(n+2)/(n-2)$ fixed, the structure of radial solutions drastically changes under the perturbation ${\varepsilon} k(|x|)$. In this paper it is shown that such a structural change can be understood in a natural way if the exponent $p$ also is taken as a parameter. The Pohozaev identity plays an important role in the perturbation analysis.

Generalized scalar curvature type equations on compact Riemannian manifolds

The paper is concerned with nonlinear equations of critical Sobolev growth involving the p-Laplace operator. These equations generalize the more classical scalar curvature equation.

Estimates of the scalar curvature equation via the method of moving planes III

Estimates of the scalar curvature equation via the method of moving planes III

Concentration of Solutions for the Scalar Curvature Equation on RN

Concentration of Solutions for the Scalar Curvature Equation on RN

Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

In this paper, we consider the problem of the existence of non-negative weak solution u of $$\left\{ \begin{gathered} \Delta u + u^p = 0 in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ having a given closed set S as its singular set. We prove that when\(\frac{n}{{n - 2}}< p< \frac{{n + 2\sqrt {n - 1} }}{{n - 4 + 2\sqrt {n - 1} }}\) and S is a closed subset of Ω, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal scalar curvature equation for n ≥ 9, we construct a weak solution\(u \in L^{\frac{{n + 2}}{{n - 2}}} \left( {S^n } \right)\) of\(L_0 u + L^{\frac{{n + 2}}{{n - 2}}} = 0\) such that Sn is the singular set of u where L0 is the conformal Laplacian with respect to the standard metric of Sn. When n = 4 or 6, this kind of solution has been constructed by Pacard.

On the asymptotic symmetry of singular solutions of the scalar curvature equations

On the asymptotic symmetry of singular solutions of the scalar curvature equations

Constant Hermitian scalar curvature equations on ruled manifolds

Constant Hermitian scalar curvature equations on ruled manifolds

Refined asymptotics for constant scalar curvature metrics with isolated singularities

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, \Delta u + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas and Spruck, we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohozaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function\(\bar S\) on Cartan-Hadamard manifoldMn(n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for\(\bar S\) near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation.

Estimate of the conformal scalar curvature equation via the method of moving planes. II

Estimate of the conformal scalar curvature equation via the method of moving planes. II

Estimates of the conformal scalar curvature equation via the method of moving planes

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4). © 1997 John Wiley & Sons, Inc.

An application of the isoperimetric inequality to the conformal scalar curvature equation

An application of the isoperimetric inequality to the conformal scalar curvature equation

Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold

Throughout thispaper,we refertothisequation as equation(/,M). Now, we areinterestedin thestructureofthe moduli spaceof(complete) conformal metricson M with scalarcurvature/.In thiswork, we study the equation(f,M) in the case when (M,g) isa subdomain ofa compact Rie- mannian manifold (M,g). More precisely,we considermainly the case when h(Lg) > 0,(M,g) isthecomplement Af\Sofa compact submanifold E, and/ isnonpositive. Under this assumption, Mazzeo (12) proved that, when d = dimS < (n- 2)/2 and / = 0 on M, fullsolutionspace of scalarflatcomplete conformal metrics on M is parametrized by the space of strictlypositive measures on S. This fact means that ￡is the Martin boundary of the Laplacian with respectto a scalarflatcomplete conformal metricon M. When / has a compact support,any conformal metricuq~1gon M with scalarcurvature/is bounded above by some scalarflatconformal metric<pq~lg