The Aubin-type inequality for the conformally invariant Dirac equation with convolution nonlinearity is strict on compact spin manifolds unless conformal to the sphere, giving existence for the 4D conformal Dirac-Einstein problem.
Ammann.The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Comm
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A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometry
The Aubin-type inequality for the conformally invariant Dirac equation with convolution nonlinearity is strict on compact spin manifolds unless conformal to the sphere, giving existence for the 4D conformal Dirac-Einstein problem.