Complete conformal metrics with negative scalar curvature in
              compact Riemannian manifolds

Avilés, Patricio; McOwen, Robert

doi:10.1215/s0012-7094-88-05616-5

Cited by 85 publications

(67 citation statements)

References 3 publications

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“…A similar result has been obtained by Loewner and Nirenberg [9] under simplifying assumptions on (M, h) and the existence part has been shown in general by Aviles and McOwen [3,4]. These authors also studied situations which in various respects were more general than the one described above.…”

Section: The Conformal Class Of Ft Is Such That the Conditionsupporting

confidence: 80%

“…on W for all values of C, ε in the range allowed above. Furthermore, we have w > υ near the boundary of W. It follows now from the maximum principle, see [9] and [3], that υ < w on W. Multiplying this inequality by ρ( n ~2)/ 2 and considering first the limit ε -> 0 and then the limit C -> 1 gives the result. D where the integration is performed over jB 3Λ (p^).…”

Section: Thenmentioning

confidence: 79%

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On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

1992

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Section: The Conformal Class Of Ft Is Such That the Conditionsupporting

confidence: 80%

Section: Thenmentioning

confidence: 79%

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

1992

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“…We present a well-known existence result for the Dirichlet boundary value problem of the Yamabe equation (A.3), see [3,34,35,52]. The proof of which is by a direct variational method, in seek of a minimizer of the corresponding energy functional.…”

Section: Appendix Amentioning

confidence: 97%

“…In [3] and [4] We call (1.1)-(1.2) the Loewner-Nirenberg problem on (M, g). Using classical variational method they obtained a sequence of solutions to (1.1) with enlarging Dirichlet boundary data that go to infinity, and using maximum principle and an integral type weak Harnack inequality, they obtained the existence of the unique solution u to (1.1)-(1.2) and analyzed the asymptotic behavior of u near the boundary.…”

Section: Introductionmentioning

confidence: 99%

Two Flow Approaches to the Loewner–Nirenberg Problem on Manifolds

2021

J Geom Anal

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We introduce two flow approaches to the Loewner-Nirenberg problem on compact Riemannian manifolds (M n , g) with boundary and establish the convergence of the corresponding Cauchy-Dirichlet problems to the solution of the Loewner-Nirenberg problem. In particular, when the initial data u 0 is a subsolution to (1.1), the convergence holds for both the direct flow (1.3)-(1.5) and the Yamabe flow (1.6). Moreover, when the background metric satisfies R g ≥ 0, the convergence holds for any positive initial data u 0 ∈ C 2,α (M) for the direct flow; while for the case the first eigenvalue λ 1 < 0 for the Dirichlet problem of the conformal Laplacian L g , the convergence holds for u 0 > v 0 where v 0 is the largest solution to the homogeneous Dirichlet boundary value problem of (1.1) and v 0 > 0 in M • . We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of (M, g) and inf u∈C 1 (M)−{0} Q(u) > −∞ when (M, g) is smooth, provided that the positive mass theorem holds, where Q is the energy functional (see (3.2)) of the second type Escobar-Yamabe problem.

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“…The content of this paper is part of the author's doctoral thesis [Cavalcante 2006]. The author would like to express his gratitude to Professor Manfredo do Carmo for the encouragement and to Professor Fernando Coda Marques for many useful discussions during this work.…”

Section: Acknowledgementsmentioning

confidence: 99%

Convexity in locally conformally flat manifolds with boundary

Cavalcante¹

2008

Pacific J. Math.

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Given a closed subset of the open unit ball B 1 ⊂ ‫ޒ‬ n for n ≥ 3, we consider a complete Riemannian metric g on B 1 \ of constant scalar curvature equal to n(n − 1) and conformally related to the Euclidean metric. We prove that every closed Euclidean ball B ⊂ B 1 \ is convex with respect to the metric g, assuming the mean curvature of the boundary ∂ B 1 is nonnegative with respect to the inward normal.

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Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds

Cited by 85 publications

References 3 publications

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

Two Flow Approaches to the Loewner–Nirenberg Problem on Manifolds

Convexity in locally conformally flat manifolds with boundary

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