Prescribing Morse Scalar Curvatures: Pinching and Morse Theory

Malchiodi, Andrea; Mayer, Martin

doi:10.1002/cpa.22037

Cited by 15 publications

(21 citation statements)

References 78 publications

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“…(iii) the comparability of sublevels of different functionals are incompatible with non existence assumptions, as we argue here and in [25].…”

Section: The Euler Characteristicmentioning

confidence: 69%

“…Mountain Pass results are available under restrictive assumptions, cf. [12], [25], while an index counting formula is not available, cf. Lemma 3.2.…”

Section: The Morse Casementioning

confidence: 99%

“…(ii) let M ⊆ M +,S,k,k i1,...,im (M ) be any spread consisting of pinched functions, for which existence by Theorem 1(ii) in [25] is guaranteed, provided more than one critical point has negative Laplacian. Then, 1.)…”

Section: A Deformation Schemementioning

confidence: 99%

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Prescribing Morse scalar curvatures: incompatibility of non existence

Mayer¹

2021

Preprint

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Given a closed manifold of positive Yamabe invariant and for instance positive Morse functions upon it, the conformally prescribed scalar curvature problem raises the question, whether or not such functions can by conformally changing the metric be realised as the scalar curvature of this manifold. As we shall quantify, the answer is depending on the complexity of these functions most likely yes and, if one such Morse function happens not to be conformally prescribable, most others can and this in a controllable way.

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“…(iii) the comparability of sublevels of different functionals are incompatible with non existence assumptions, as we argue here and in [25].…”

Section: The Euler Characteristicmentioning

confidence: 69%

“…Mountain Pass results are available under restrictive assumptions, cf. [12], [25], while an index counting formula is not available, cf. Lemma 3.2.…”

Section: The Morse Casementioning

confidence: 99%

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Prescribing Morse scalar curvatures: incompatibility of non existence

Mayer¹

2021

Preprint

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“…In this way one recovers the compactness and one then studies the behavior of blowing up solution u ε of (N P ε ) as the parameter ε goes to zero. Actually it can be proved that finite energy blowing up solutions of (N P ε ) can have only isolated simple blow up points which are critical points of the function K, see [31,32,23,35]. The reason of the additional difficulty in the high dimensional case lies in the complexity of the blow up phenomenon.…”

mentioning

confidence: 99%

“…Regarding the high dimensional case n ≥ 5, A. Malchiodi and M. Mayer [35] obtained recently an interesting existence criterium under some pinching condition. Their result reads as follows:…”

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confidence: 99%

The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions

Ahmedou¹,

Ayed²

2020

Preprint

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In this paper we study the Nirenberg problem on standard half spheres (S n + , g), n ≥ 5, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent:where K ∈ C 3 (S n + ) is a positive function. This problem has a variational structure but the related Euler-Lagrange functional J K lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional J K in their neighborhood and compute their contribution to the difference of topology between the level sets of J K , hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for (P) on half spheres of dimension n ≥ 5.

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Prescribing Morse scalar curvatures: Subcritical blowing-up solutions

Malchiodi¹,

Mayer²

2020

Journal of Differential Equations

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Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical approximations for the equation, gaining compactness properties. Together with the results in [30], we completely describe the blow-up phenomenon in case of uniformly bounded energy and zero weak limit in positive Yamabe class. In particular, for dimension greater or equal to five, Morse functions and with non-zero Laplacian at each critical point, we show that subsets of critical points with negative Laplacian are in one-to-one correspondence with such subcritical blowing-up solutions.

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Prescribing Morse Scalar Curvatures: Pinching and Morse Theory

Cited by 15 publications

References 78 publications

Prescribing Morse scalar curvatures: incompatibility of non existence

Prescribing Morse scalar curvatures: incompatibility of non existence

The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions

Prescribing Morse scalar curvatures: Subcritical blowing-up solutions

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