On gravitational instantons and several special-holonomy manifolds, nonzero-eigenvalue L^2 eigenforms exist if and only if L^2 eigenfunctions exist, and on ALE, A_n ALF, and asymptotically conical manifolds they never exist.
Exhaustion functions and the spectrum of Riemannian manifolds.Indiana Univ
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Eigenforms and special holonomy
On gravitational instantons and several special-holonomy manifolds, nonzero-eigenvalue L^2 eigenforms exist if and only if L^2 eigenfunctions exist, and on ALE, A_n ALF, and asymptotically conical manifolds they never exist.