Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics
Abstract
Let g be a complete Ricci-flat Böhm metric on ℝp+1× Sq, where p, q≥2 and p+q≤8. We classify the geometric Dirichlet kernel on every compact truncation and prove that the reduced Morse index counts the preceding umbilic orbits. Every negative L2 Lichnerowicz eigentensor is O(p+1)× 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 L2, 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.
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BibTeX
@misc{nifa2026dirichlet,
title = {{Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = jun,
doi = {10.5281/zenodo.22810800},
url = {https://hal.science/hal-05770000v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics
AU - Anass Nifa
PY - 2026
DA - 2026/06/07
PB - HAL
DO - 10.5281/zenodo.22810800
UR - https://hal.science/hal-05770000v1
AB - Let g be a complete Ricci-flat Böhm metric on ℝ^p+1× S^q, where p, q≥2 and p+q≤8. 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)× 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.
ER -