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

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Abstract

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 \(\sigma _k\)-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 (Xg) for which this problem admits multiple non-homothetic 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.

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Correspondence to Yi Wang.

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JSC was supported by a Grant from the Simons Foundation (Grant No. 524601). YW was partially supported by NSF Grant No. DMS-1612015.

Communicated by A. Chang.

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Case, J.S., Moreira, A.C. & Wang, Y. Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory. Calc. Var. 58, 106 (2019). https://doi.org/10.1007/s00526-019-1566-4

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Mathematics Subject Classification

  • Primary 58J32
  • Secondary 53A30
  • 58J40