Delaunay-type singular solutions for the fractional Yamabe problem

DelaTorre, Azahara; Pino, Manuel del; González, María del Mar; Wei, Juncheng

doi:10.1007/s00208-016-1483-1

Cited by 59 publications

(54 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, While in Proposition 3.6 we calculated the Fourier symbol forP m γ , now we will write it as an integrodifferential operator for a well behaved convolution kernel. The advantage of this formulation is that immediately yields regularity for v m as in [30]. Now we look at the m = 0 projection, which corresponds to finding radially symmetric singular solutions to (4.1).…”

Section: New Ode Methods For Non-local Equationsmentioning

confidence: 99%

“…, for a different constant c. As in [30], one can calculate its asymptotic behavior, and we refer to this paper for details:…”

Section: New Ode Methods For Non-local Equationsmentioning

confidence: 99%

“…Remark 4.2. Note that (4.4) is a special case of the Funk-Hecke formula for the m = 0 spherical harmonic (see [69], pages [29][30].…”

Section: New Ode Methods For Non-local Equationsmentioning

confidence: 99%

See 2 more Smart Citations

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Ao¹,

Chan²,

DelaTorre³

et al. 2019

Duke Math. J.

Self Cite

View full text Add to dashboard Cite

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard ([63]) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

show abstract

Section: New Ode Methods For Non-local Equationsmentioning

confidence: 99%

“…, for a different constant c. As in [30], one can calculate its asymptotic behavior, and we refer to this paper for details:…”

Section: New Ode Methods For Non-local Equationsmentioning

confidence: 99%

See 1 more Smart Citation

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Ao¹,

Chan²,

DelaTorre³

et al. 2019

Duke Math. J.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The calculation of the scattering operator and the conformal fractional Laplacian in this setting can be found in [10,11] and, in particular:…”

Section: The Model Cylindermentioning

confidence: 99%

Boundary Connected Sum of Escobar Manifolds

González

Sire

2019

J Geom Anal

Self Cite

View full text Add to dashboard Cite

Let (X1,ḡ1) and (X2,ḡ2) be two compact Riemannian manifolds with boundary (M1, g1) and (M2, g2) respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum X = X1#X2 by excising half ball near points on the boundary. The resulting metric on X has zero scalar curvature and a CMC boundary. We fully exploit the nonlocal aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very well-known problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures.

show abstract

“…al. [15], attempting to prove nonuniqueness for the boundary value problem (1.4) using Schoen's symmetry argument runs into the problem that one can only reduce the problem to a local PDE of two variables. Moreover, since σ k is fully nonlinear, it is unclear if one could reduce the problem to a nonlocal ODE on the boundary.…”

Section: Introductionmentioning

confidence: 99%

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

2019

View full text Add to dashboard Cite

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σ k -curvature in the interior and constant H k -curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds (X, g) for which this problem admits multiple nonhomothetic solutions in the case when 2k < dim X. Our examples are all such that the boundary with its induced metric is a Riemannian product of a round sphere with an Einstein manifold.2010 Mathematics Subject Classification. Primary 58J32; Secondary 53A30, 58J40.Thus depending on the different volume constraints, there are two types of boundary Yamabe problem. If one considers,n−1 n , then the Euler-Lagrangian equation (with suitable normalization) is given byThis boundary Yamabe problem is called the scalar flat type. This problem, as remarked by Escobar in [19], is a higher dimensional generalization of the Riemann mapping theorem. It was studied by Escobar [19,21] and Marques [31,32]. If one considers instead inf g∈[g0] X J g dvol g + M H g dvol ι * g Vol g (X) n−1 n+1, then the Euler-Lagrangian equation is given by R g = const, in X, H g = 0, on M .

show abstract

Delaunay-type singular solutions for the fractional Yamabe problem

Abstract: We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularityWe follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.

Cited by 59 publications

References 36 publications

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Boundary Connected Sum of Escobar Manifolds

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Contact Info

Product

Resources

About