Uniqueness of solutions of the Yamabe problem on manifolds with boundary

Cárdenas, Elkin; Sierra, Willy

doi:10.1016/j.na.2019.04.010

Cited by 3 publications

(1 citation statement)

References 16 publications

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“…Compactness results for other boundary Yamabe-type problems can be found in Han and Lin [29], Djadli et al [16,17], Disconzi and Khuri [15], and so on. By using the compactness property, Cádenas and Sierra [10] also yielded uniqueness of solutions to (1.1) for some manifolds whose metrics are non-degenerate.…”

Section: Introductionmentioning

confidence: 98%

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Kim

Musso

Wei

2019

Preprint

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We concern C 2 -compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Together with the positive mass theorem, we conclude the C 2 -compactness holds for all 4-manifolds (which may be non-umbilic). For 5-and 6-manifolds, we also prove that the C 2 -compactness is true if the trace-free second fundamental form on the boundary never vanishes.

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Section: Introductionmentioning

confidence: 98%

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Kim

Musso

Wei

2019

Preprint

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Bifurcation of metrics with null scalar curvature and constant mean curvature on the boundary of compact manifolds

Cárdenas

Sierra

2021

manuscripta math.

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Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Kim

Musso

Wei

2021

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We concern C^{2} -compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the C^{2} -compactness for all 4-manifolds (which may be non-umbilic). For the 5-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the C^{2} -compactness for all 5-manifolds. Finally, we show that the C^{2} -compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

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Uniqueness of solutions of the Yamabe problem on manifolds with boundary

Cited by 3 publications

References 16 publications

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Bifurcation of metrics with null scalar curvature and constant mean curvature on the boundary of compact manifolds

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

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