Complete Conformal Metrics of Negative Ricci Curvature on Euclidean Spaces

Sui, Zhenan

doi:10.1007/s12220-016-9703-1

Cited by 19 publications

(14 citation statements)

References 31 publications

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“…Recall that when dimension n ≥ 3, the existence of solutions of the σ k -Yamabe problem has been proved for k ≥ n/2, k = 2 or when (M, g) is locally conformally flat, the compactness of the set of solutions has been proved for k ≥ n/2 when the manifold is not conformally equivalent to the standard sphere − they were established in [11,22,27,31,45,54,65]. For more recent works on σ k -Yamabe type problems, see for example [1,3,4,7,8,9,19,20,21,23,30,32,33,38,39,40,41,42,55,56,60,66,67,68] and references therein. However, there are still many challenging open problems on general compact Riemannian manifolds -the compactness remains open for 2 ≤ k ≤ n/2 and the existence remains open for 2 < k < n/2.…”

Section: Letmentioning

confidence: 99%

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

Li,

Lu,

2021

Preprint

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We establish theorems on the existence and compactness of solutions to the σ2-Nirenberg problem on the standard sphere S 2 . A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ2-Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ2-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L ∞ a priori estimates for solutions of the σ2-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ2-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.

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Section: Letmentioning

confidence: 99%

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

Li,

Lu,

2021

Preprint

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“…Some key results for this problem and its analogues on manifolds have been obtained by Chang, Han and Yang [6], González, Li and Nguyen [12], Guan [15], Gursky, Streets and Warren [16], Gurksy and Viaclovsky [18], and Li and Nguyen [25]. For other related works, see also Li and Sheng [22], Sui [32], Wang [36] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem

Li¹,

Nguyen²,

Xiong³

2022

Preprint

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We study the regularity of the viscosity solution u of the σ k -Loewner-Nirenberg problem on a bounded smooth domain Ω ⊂ R n for k ≥ 2. It was known that u is locally Lipschitz in Ω. We prove that, with d being the distance function to ∂Ω and δ > 0 sufficiently small, u is smooth in {0 < d(x) < δ} and the first (n−1) derivatives of d n−2 2 u are Hölder continuous in {0 ≤ d(x) < δ}. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of ∂Ω and its covariant derivatives and vanishes if and only ifUsing a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when ∂Ω contains more than one connected components, u is not differentiable in Ω.

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“…(1.2) and their counterparts on Riemannian manifolds were first studied by Viaclovsky in [63]. Since then, these equations have been addressed by various authors -for a partial list of references, see [1][2][3][8][9][10][11][12]14,[16][17][18][19]24,27,28,[31][32][33]35,[39][40][41]43,44,46,47,53,54,56,64,65] in the positive case and [13,23,25,29,30,42,45,55] in the negative case. When k = 1, these equations reduce to the original Yamabe equation.…”

Section: Introductionmentioning

confidence: 99%

Local pointwise second derivative estimates for strong solutions to the $$\sigma _k$$-Yamabe equation on Euclidean domains

Nguyen

2021

Calc. Var.

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We prove local pointwise second derivative estimates for positive $$W^{2,p}$$ W 2 , p solutions to the $$\sigma _k$$ σ k -Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.

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Complete Conformal Metrics of Negative Ricci Curvature on Euclidean Spaces

Cited by 19 publications

References 31 publications

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem

Local pointwise second derivative estimates for strong solutions to the $$\sigma _k$$-Yamabe equation on Euclidean domains

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