Prescribing the Scalar Curvature Problem on Three and Four Manifolds

Hichem Chtioui

doi:10.1515/ans-2003-0404

Prescribing the Scalar Curvature Problem on Three and Four Manifolds

Advanced Nonlinear Studies ◽

10.1515/ans-2003-0404 ◽

2003 ◽

Vol 3 (4) ◽

Cited By ~ 14

Author(s):

Hichem Chtioui

Keyword(s):

Scalar Curvature ◽

Riemannian Manifolds ◽

Critical Points ◽

New Class ◽

Critical Points At Infinity ◽

Four Manifolds ◽

Curvature Problem

AbstractThis paper is devoted to the prescribed scalar curvature problem on 3 and 4- dimensional Riemannian manifolds. We give a new class of functionals which can be realized as scalar curvature. Our proof uses topological arguments and the tools of the theory of the critical points at infinity.

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Addenda to the book “Critical points at infinity in some variational problems” and to the paper “The scalar-curvature problem on the standard three-dimensional sphere”

Annales de l Institut Henri Poincaré C Analyse non linéaire ◽

10.1016/s0294-1449(16)30121-4 ◽

1996 ◽

Vol 13 (6) ◽

pp. 733-739

Author(s):

Abbes Bahri

Keyword(s):

Scalar Curvature ◽

Critical Points ◽

Three Dimensional ◽

Variational Problems ◽

Dimensional Sphere ◽

Critical Points At Infinity ◽

Curvature Problem

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Construction of solutions for a critical problem with competing potentials via local Pohozaev identities

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500716 ◽

2020 ◽

pp. 2050071

Author(s):

Qihan He ◽

Chunhua Wang ◽

Da-Bin Wang

Keyword(s):

Differential Equations ◽

Scalar Curvature ◽

Critical Points ◽

Critical Equation ◽

Finite Dimensional Reduction ◽

Finite Dimensional ◽

Bounded Functions ◽

Funct Anal ◽

Curvature Problem

In this paper, we consider the following critical equation: [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if [Formula: see text], [Formula: see text] has a stable critical point [Formula: see text] with [Formula: see text] and [Formula: see text], then the above equation has infinitely many positive solutions, where [Formula: see text] is the unique positive solution of [Formula: see text] with [Formula: see text]. Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606–2633], it implies that the role of stable critical points of [Formula: see text] in constructing bump solutions is more important than that of [Formula: see text] and that [Formula: see text] can influence the sign of [Formula: see text], i.e. [Formula: see text] can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of [Formula: see text] which include the case of a saddle point.

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β-flatness condition in CR spheres multiplicity results

International Journal of Mathematics ◽

10.1142/s0129167x20500238 ◽

2020 ◽

Vol 31 (03) ◽

pp. 2050023

Author(s):

Najoua Gamara ◽

Boutheina Hafassa ◽

Akrem Makni

Keyword(s):

Lower Bound ◽

Scalar Curvature ◽

Critical Points ◽

Number Of Solutions ◽

Multiplicity Results ◽

Type Formula ◽

Critical Points At Infinity ◽

Cauchy Riemann

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Riemann spheres under [Formula: see text]-flatness condition. To find a lower bound for the number of solutions, we use Bahri’s methods based on the theory of critical points at infinity and a Poincaré–Hopf-type formula.

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Prescribing scalar curvatures: non compactness versus critical points at infinity

Geometric Flows ◽

10.1515/geofl-2019-0004 ◽

2019 ◽

Vol 4 (1) ◽

pp. 51-82 ◽

Cited By ~ 1

Author(s):

Martin Mayer

Keyword(s):

Riemannian Manifold ◽

Scalar Curvature ◽

Critical Points ◽

Gradient Flow ◽

Slight Modification ◽

Positive Function ◽

Flow Lines ◽

Critical Points At Infinity

Abstract We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.

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Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

Advanced Nonlinear Studies ◽

10.1515/ans-2002-0201 ◽

2002 ◽

Vol 2 (2) ◽

Cited By ~ 1

Author(s):

Mohamed Ben Ayed ◽

Khalil El Mehdi ◽

Mohameden Ould Ahmedou

Keyword(s):

Boundary Conditions ◽

Variational Problem ◽

Scalar Curvature ◽

Critical Points ◽

Existence Results ◽

Topological Methods ◽

Critical Points At Infinity

AbstractThis paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

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Topological Methods for Boundary Mean Curvature Problem on Bn

Advanced Nonlinear Studies ◽

10.1515/ans-2014-0212 ◽

2014 ◽

Vol 14 (2) ◽

Author(s):

Mohammed Ali Al-Ghamdi ◽

Hichem Chtioui ◽

Khadijah Sharaf

Keyword(s):

Critical Points ◽

Mean Curvature ◽

Topological Method ◽

Topological Methods ◽

Critical Points At Infinity ◽

The Mean ◽

Curvature Problem ◽

Boundary Mean Curvature

AbstractUsing an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the n-dimensional balls.

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Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere 𝕊+3

Geometry ◽

10.1155/2014/582367 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Ridha Yacoub

Keyword(s):

Scalar Curvature ◽

Critical Points ◽

Mean Curvature ◽

Existence Results ◽

Curvature Condition ◽

Multiplicity Results ◽

3 Dimensional ◽

Existence And Multiplicity ◽

Curvature Problem ◽

Boundary Mean Curvature

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

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Almost Non-negative Scalar Curvature on Riemannian Manifolds Conformal to Tori

Journal of Geometric Analysis ◽

10.1007/s12220-021-00677-2 ◽

2021 ◽

Author(s):

Brian Allen

Keyword(s):

Scalar Curvature ◽

Riemannian Manifolds

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Topological tools for the prescribed scalar curvature problem on S n

Open Mathematics ◽

10.2478/s11533-014-0443-9 ◽

2014 ◽

Vol 12 (12) ◽

Author(s):

Dina Abuzaid ◽

Randa Ben Mahmoud ◽

Hichem Chtioui ◽

Afef Rigane

Keyword(s):

Scalar Curvature ◽

Conformal Metrics ◽

Standard Sphere ◽

Multiplicity Results ◽

Existence And Multiplicity ◽

Hopf Formula ◽

Curvature Problem ◽

Formula Type

AbstractIn this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

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