Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

Hyder, Ali; Martinazzi, Luca

doi:10.2422/2036-2145.201905_001

Cited by 2 publications

(1 citation statement)

References 28 publications

(86 reference statements)

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“…In fact, in case ( ii ) of Theorem 1.2, one can prescribe the blow‐up set

S_{φ}

, in the sense that given any

φ \in K (Ω, \emptyset)

, one can construct a sequence

(u_{k})

solving () and () with

u_{k} \to + \infty

S_{φ}

, as shown in [28]. Moreover in the radial case of dimension 6, it was also shown in [29] that the blow‐up set

S_{1} = false{0 false}

and

S_{φ} = false{x : false| x false| = 1 false}

can coexist; see also [23, 31, 34] for the case of a closed manifold of even dimension 4 and higher.…”

Section: Introductionmentioning

confidence: 99%

Concentration phenomena for the fractional Q‐curvature equation in dimension 3 and fractional Poisson formulas

DelaTorre

González

Hyder

et al. 2021

J. London Math. Soc.

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We study the compactness properties of metrics of prescribed fractional Q-curvature of order 3 in R 3 . We will use an approach inspired from conformal geometry, seeing a metric on a subset of R 3 as the restriction of a metric on R 4 + with vanishing fourth-order Q-curvature. We will show that a sequence of such metrics with uniformly bounded fractional Q-curvature can blow up on a large set (roughly, the zero set of the trace of a non-positive bi-harmonic function Φ in R 4 + ), in analogy with a four-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

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“…In fact, in case ( ii ) of Theorem 1.2, one can prescribe the blow‐up set