On the prescribed scalar curvature problem on 4-manifolds

Ayed, Mohamed Ben; Chen, Yansong; Chtioui, Hichem; Hammami, Mokhles

doi:10.1215/s0012-7094-96-08420-3

Cited by 95 publications

(173 citation statements)

References 19 publications

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“…, y js ) (we remark that all the points can be in the boundary or in S 4 + ). Arguing as in [3] and [7], we can find a change of variable …”

Section: Proof Of Our Resultsmentioning

confidence: 99%

“…Ben Ayed-El Mehdi-Ould Ahmedou [9], [10] gave some topological conditions on K to prescribe the scalar curvature under minimal boundary conditions on half spheres of dimension n ≥ 4. Note that problem (P ) is in some sense related to the well known scalar curvature problem −Δ g u + (n(n − 2)/4)u = Ku (n+2)/(n−2) , u > 0 in S n (3) to which much work has been devoted (see [3,4,7,13,14,[19][20][21] and the references therein). As for (3), there are topological obstructions of Kazdan-Warner type to solve (P ) (see [11]) and so a natural question arises: under which conditions on K, (P ) has a positive solution?…”

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 99%

“…In the four dimensional case, we have a balance phenomenon, that is, the self interaction and the interaction are of the same size (see [7]). In the case of (P ), Ben Ayed et al proved that the balance phenomenon appears if we assume that (∂K/∂ν)(y) < 0 where y is any critical point of K 1 = K |∂S 4 + .…”

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 99%

“…Thus, we can apply the vector field (denoted here Z 1 ) defined in Proposition 3.1 of [10] (see also [7]). Then, we get For the indices i ≤ q, we introduce the sets…”

Section: Construction Ofmentioning

confidence: 99%

“…(we will apply Z 11 only to u 3 ). Then, we get We can obtain the estimate (ii), arguing as in Appendix B of [7], using the estimates (i), ∇J 0 (u + v) and v . Observe that in V 2 (p, ε) and V 3 (p, ε) the maximum of λ i 's is always a decreasing function along the flow lines.…”

Section: Construction Ofmentioning

confidence: 99%

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Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Ayed

Ghoudi

Bouh

2012

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.

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“…, y js ) (we remark that all the points can be in the boundary or in S 4 + ). Arguing as in [3] and [7], we can find a change of variable …”

Section: Proof Of Our Resultsmentioning

confidence: 99%

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 99%

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 99%

“…Thus, we can apply the vector field (denoted here Z 1 ) defined in Proposition 3.1 of [10] (see also [7]). Then, we get For the indices i ≤ q, we introduce the sets…”

Section: Construction Ofmentioning

confidence: 99%

Section: Construction Ofmentioning

confidence: 99%

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Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Ayed

Ghoudi

Bouh

2012

Nonlinear Differ. Equ. Appl.

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Prescribing scalar curvature on Sn and related problems, part II: Existence and compactness

1998

Comm. Pure Appl. Math.

177

157

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This is a sequel to [30], which studies the prescribing scalar curvature problem on S". First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lower-dimensional cases n = 2,3.Next we study the problem for n 2 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is p E (n -2,n). The key point there is that for the class of K mentioned above we have completed Lm apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is = n -2, the L" estimates for solutions fail in general. In fact, two or more blowup points occur.On the other hand, we provide some existence and compactness results when the order of flatness at critical points of K ( x ) is p E [n -2,n). With this result, we can easily deduce that C" scalar curvature functions are dense in CIa (0 < a < 1) norm among positive functions, although this is generally not true in the C2 norm.We also give a simpler proof to a Sobolev-Aubin-type inequality established in [ 161.Some of the results in this paper as well as that of [30] have been announced in I291.

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Prescribing Morse Scalar Curvatures: Pinching and Morse Theory

Malchiodi¹,

Mayer²

2021

Comm Pure Appl Math

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We consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension n≥5. We prove new existence results using Morse theory and some analysis on blowing‐up solutions under suitable pinching conditions on the curvature function. We also provide new nonexistence results showing the sharpness of some of our assumptions, both in terms of the dimension and of the Morse structure of the prescribed function. © 2021 Wiley Periodicals, Inc.

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On the prescribed scalar curvature problem on 4-manifolds

Cited by 95 publications

References 19 publications

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Prescribing scalar curvature on Sn and related problems, part II: Existence and compactness

Prescribing Morse Scalar Curvatures: Pinching and Morse Theory

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