Scalar Curvature Equation Research Articles

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three

abstract: We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in $\mathbb{R}^{3}$. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a ``Lagrangian'' graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.

Some existence results for a generalization of the prescribed scalar curvature equation on compact manifolds

In this paper, we prove some existence results of a critical [Formula: see text]-Laplacian equation involving critical and subcritical Hardy potential on compact Riemannian manifolds.

$\sup \times \inf$ Inequalities for the Scalar Curvature Equation in Dimensions 4 and 5

$\sup \times \inf$ Inequalities for the Scalar Curvature Equation in Dimensions 4 and 5

Einstein-Klein-Gordon System in Higher Dimensional

In this present work, we study the Einstein equation coupled with the nonlinear Klein-Gordon equation. We obtain Ricci tensor, scalar curvature, and Einstein equation of the Einstein-Klein-Gordon system in higher dimensional. If we put D=4, our formulations reduce to the four dimensional Einstein-Klein-Gordon system.

Remarks on Scalar Curvature and Concircular Field Equation

We show that the scalar curvature of a Riemannian manifold $M$ is constant if it satisfies (i) the concircular field equation and $M$ is compact, (ii) the special concircular field equation. Finally, we show that, if a complete connected Riemannian manifold admits a concircular non-isometric vector field leaving the scalar curvature invariant, and the conformal function is special concircular, then the scalar curvature is a constant.

Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry

We study existence and multiplicity of positive ground states for the scalar curvature equation Δu+K(|x|)un+2n-2=0,x∈Rn,n>2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\varDelta u+ K(|x|)\\, u^{\\frac{n+2}{n-2}}=0, \\quad x\\in {{\\mathbb {R}}}^n\\,, \\quad n>2, \\end{aligned}$$\\end{document}when the function K:{{mathbb {R}}}^+rightarrow {{mathbb {R}}}^+ is bounded above and below by two positive constants, i.e. 0<underline{K} le K(r) le overline{K} for every r > 0, it is decreasing in (0,{{{mathcal {R}}}}) and increasing in ({{{mathcal {R}}}},+infty ) for a certain {{{mathcal {R}}}}>0. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio overline{K}/underline{K} which guarantees the existence of a large number of ground states with fast decay, i.e. such that u(|x|) sim |x|^{2-n} as |x| rightarrow +infty , which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique.

Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of “lower energy” solutions to the above equation fails when the dimension of the manifold is not less than 62.

Non-degeneracy of multi-bubbling solutions for the prescribed scalar curvature equations and applications

We consider the following prescribed scalar curvature equations in RN(0.1)−Δu=K(|y|)u2⁎−1,u>0inRN,u∈D1,2(RN), where K(r) is a positive function, 2⁎=2NN−2. We first prove a non-degeneracy result for the positive multi-bubbling solutions constructed in [26] by using the local Pohozaev identities. Then we use this non-degeneracy result to glue together bubbles with different concentration rate to obtain new solutions.

Solutions to Donaldson’s Hyperkähler Reduction on a Curve

We study an infinite-dimensional hyperkahler reduction introduced by Donaldson and associated with the constant scalar curvature equation on a Riemann surface. It is known that the corresponding moment map equations admit special solutions constructed from holomorphic quadratic differentials. Here we obtain a more general existence result and so a larger hyperkahler moduli space.

The concentrated solutions on the nonhomogeneous scalar curvature problem

We consider the following scalar curvature equation−Δu=(1+εK(x))u2⁎−1,inRN. If K(x) is nonhomogeneous at the critical points, we give the precise description on the concentrated solutions, which contains the necessary and existence on the two peak solutions. Here we generalize the results of Cao-Noussair-Yan in [6] to the nonhomogeneous case.

Some uniform estimates for scalar curvature type equations

We consider the prescribed scalar curvature equation on an open setΩ of Rn,∆u = V u(n+2)/(n−2) + un/(n−2) with V ∈ C1,α (0 < α ≤ 1) and we prove the inequality supK u× infΩ u ≤ c where K is a compact set of Ω. In dimension 4, we have an idea on the supremum of the solution of the prescribed scalar curvature if we control the infimum. For this case we suppose the scalar curvature C1,α, (0 < α ≤ 1).

Bifurcation for the Constant Scalar Curvature Equation and Harmonic Riemannian Submersions

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation necessarily occurs at every point where the linearized equation has a nontrivial solution. In the model case of quaternionic Hopf fibrations, we show that $${{\,\mathrm{SU}\,}}(2)$$ -symmetry-breaking bifurcation does not occur except at the round metric.

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $\sigma_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate for these equations obtained under a weakened condition.

Multiplicity of ground states for the scalar curvature equation

We study existence and multiplicity of radial ground states for the scalar curvature equation $$\begin{aligned} \Delta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n, \quad n>2, \end{aligned}$$when the function $$K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+$$ is bounded above and below by two positive constants, i.e. $$0 0$$, it is decreasing in (0, 1) and increasing in $$(1,+\infty )$$. Chen and Lin (Commun Partial Differ Equ 24:785–799, 1999) had shown the existence of a large number of bubble tower solutions if K is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio $${\overline{K}}/{\underline{K}}$$ is smaller than some computable values.

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

AbstractIn this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth‐order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K‐energy minimizers. The new technical ingredient is a W2, 2 regularity result for the Laplacian equation Δgu=f on Kähler manifolds, where the metric has only L∞ coefficients. It is well‐known that such a W2, 2 regularity (W2, p regularity for any p &gt; 1) fails in general (except for dimension 2) for uniform elliptic equations of the form for aij ∊ L∞ without certain smallness assumptions on the local oscillation of aij. We observe that the Kähler condition plays an essential role in obtaining a W2, 2 regularity for elliptic equations with only L∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.

Conformal scalar curvature equation on Sn: Functions with two close critical points (Twin Pseudo-Peaks)

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [Formula: see text] ([Formula: see text]) when the prescribed function (after being projected to [Formula: see text]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [Formula: see text]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

An Extension Procedure for the Constraint Equations

Let $$(\bar{g}, \bar{k})$$ be a solution to the maximal constraint equations of general relativity on the unit ball $$B_1$$ of $${\mathbb R}^3$$ . We prove that if $$(\bar{g},\bar{k})$$ is sufficiently close to the initial data for Minkowski space, then there exists an asymptotically flat solution (g, k) on $${\mathbb R}^3$$ that extends $$(\bar{g}, \bar{k})$$ . Moreover, (g, k) is bounded by $$(\bar{g}, \bar{k})$$ and has the same regularity. Our proof uses a new method of solving the prescribed divergence equation for a tracefree symmetric 2-tensor, and a geometric variant of the conformal method to solve the prescribed scalar curvature equation for a metric. Both methods are based on the implicit function theorem and an expansion of tensors based on spherical harmonics. They are combined to define an iterative scheme that is shown to converge to a global solution (g, k) of the maximal constraint equations which extends $$(\bar{g},\bar{k})$$ .

Refined estimates for simple blow-ups of the scalar curvature equation on $S^n$

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the balance formula as expressed in the Pohozaev identity in an essential way.

On the conformal scalar curvature equations on Finsler manifolds

In this paper, we study conformal deformations and [Formula: see text]-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to [Formula: see text]-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

Energy estimates for a class of semilinear elliptic equations on half Euclidean balls

For a class of semi-linear elliptic equations with critical Sobolev exponents and boundary conditions, we prove pointwise estimates for blowup solutions and energy estimates. A special case of this class of equations is a locally defined prescribing scalar curvature and mean curvature type equation.