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Abstract

For every integer n≥2, we construct a bounded connected nonconvex domain with smooth boundary on which the affine 2-Laplacian has two nonproportional strictly positive first eigenfunctions and a strictly positive eigenfunction above the first eigenvalue. These examples give negative answers at exponent two to both the simplicity and positive-eigenvalue uniqueness questions of Haddad, Jiménez and Montenegro. The construction starts with a planar pentagon pinched into two triangular lobes. A closing odd spectral gap creates negative shear curvature at an exactly isotropic position. A global minimum comparison on the symmetry-preserving linear slice then forces every positive first affine eigenfunction to break a domain symmetry. Parameter-controlled smooth exhaustion and a balanced product lift preserve both properties in every dimension. We also obtain sharp inverse-gap curvature limits and nonparallel-hull isotropic saddles. In contrast, a normalization correspondence and Faifman and Polterovich's convex-position theorem show that on bounded convex domains the positive affine 2-eigenfunction is unique up to scale and belongs to the first affine eigenvalue.