A counterexample to the Yamabe problem for complete noncompact manifolds

Jin, Zhiren

doi:10.1007/bfb0082927

Cited by 34 publications

(15 citation statements)

References 7 publications

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“…We thus have a bounded sequence {u k } in H 2 (Ω, g) of solutions to (19); equivalently, {ũ k } is a bounded sequence of solutions in…”

Section: The Yamabe Problem On a Riemannian Domainmentioning

confidence: 99%

“…By (20), u k converges to some u ∈ H 1 (Ω, g). Taking the limit on both sides of (19), it follows that…”

Section: The Yamabe Problem On a Riemannian Domainmentioning

confidence: 99%

“…Finally, Schoen used the positive mass theorem to prove that (2) has a solution if M has dimension 3, 4, 5 or is locally conformally flat, and M is not conformal to the standard sphere. There are also results for manifolds with boundary [4,5,6,14] and open manifolds [3,8,19] with certain restrictions.…”

Section: Introductionmentioning

confidence: 99%

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Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain

Rosenberg¹,

Xu²

2021

Preprint

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We introduce an iterative scheme to solve the Yamabe equation −a∆gu + Su = λu p−1 on small domains (Ω, g) ⊂ R n equipped with a Riemannian metric g. Thus g admits a conformal change to a constant scalar curvature metric. The proof does not use the traditional functional minimization. Applications to the Yamabe problem on closed manifolds, manifolds with boundary, and noncompact manifolds are given in forthcoming papers.

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“…We thus have a bounded sequence {u k } in H 2 (Ω, g) of solutions to (19); equivalently, {ũ k } is a bounded sequence of solutions in…”

Section: The Yamabe Problem On a Riemannian Domainmentioning

confidence: 99%

“…By (20), u k converges to some u ∈ H 1 (Ω, g). Taking the limit on both sides of (19), it follows that…”

Section: The Yamabe Problem On a Riemannian Domainmentioning

confidence: 99%

See 1 more Smart Citation

Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain

Rosenberg¹,

Xu²

2021

Preprint

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“…The problem of existence of solutions of (1) when R(g) and h(g) are constants is referred as the Yamabe problem which was completely solved when ∂M = ∅ in a sequence of works, beginning with H. Yamabe himself [34], followed by N. Trudinger [33] and T. Aubin [1], and finally by R. Schoen [29]. When the manifold (M, g 0 ) is complete but not compact, the existence of a conformal metric solving the Yamabe Problem does not hold in general, as we can see in the work of Zhiren [35].…”

Section: Introductionmentioning

confidence: 99%

Escobar type theorems for elliptic fully nonlinear degenerate equations

Abanto¹,

Espinar²

2019

American Journal of Mathematics

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In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are the hemisphere (or a geodesic ball in S m ), punctured balls and annular domains.Our results extend those of Escobar in [8] when m ≥ 3, and Hang-Wang in [18] and Jimenez in [19] when m = 2.

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“…The Yamabe problem was also studied on complete noncompact Riemannian manifolds. In this case, there is a simple counterexample such that the Yamabe problem does not have a solution (see [Jin 1988]). See also [Aviles and McOwen 1988;Bland and Kalka 1989;Große and Nardmann 2014;Kim 1997;2000;Zhang 2003] and references therein for results related to the Yamabe problem on noncompact Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2016

Pacific J. Math.

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We introduce the SU(N) Casson-Lin invariants for links L in S 3 with more than one component. Writing L = 1 ∪ • • • ∪ n , we require as input an n-tuple (a 1 , . . . , a n ) ∈ ‫ޚ‬ n of labels, where a j is associated with j . The SU(N) Casson-Lin invariant, denoted h N,a (L), gives an algebraic count of certain projective SU(N) representations of the link group π 1 (S 3 L), and the family h N,a of link invariants gives a natural extension of the SU(2) Casson-Lin invariant, which was defined for knots by X.-S. Lin and for 2-component links by Harper and Saveliev. We compute the invariants for the Hopf link and more generally for chain links, and we show that, under mild conditions on the labels (a 1 , . . . , a n ), the invariants h N,a (L) vanish whenever L is a split link.

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A counterexample to the Yamabe problem for complete noncompact manifolds

Cited by 34 publications

References 7 publications

Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain

Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain

Escobar type theorems for elliptic fully nonlinear degenerate equations

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