Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics

Abstract

Let $g$ be a complete Ricci-flat Böhm metric on $\R^{p+1}\times S^q$, where $p, q\ge2$ and $p+q\le8$. We classify the geometric Dirichlet kernel on every compact truncation and prove that the reduced Morse index counts the preceding umbilic orbits. Every negative $L^2$ Lichnerowicz eigentensor is $O(p+1)\times O(q+1)$-invariant, and the quadratic form is coercive in homogeneous energy on the complementary subspace. These results imply symmetry of sufficiently small ancient Ricci--DeTurck solutions whose non-invariant component is bounded in $L^2$, without an assumption of backward convergence. They also give stationary rigidity in a fixed background gauge. Each degeneracy of the smooth induced-metric map is a fold. Arbitrarily large finite collections of pairwise nonisometric Ricci-flat fillings, with arbitrarily large reduced index, persist under nonsymmetric boundary perturbations. The reduced negative spectrum converges to the complete negative spectrum; a transverse Dirichlet-to-Neumann form has a positive-measure representation. All scalar coefficient identities are given explicitly, with their exact integer data.