PreprintCC BY 4.0arXiv
Stability of Einstein 4-manifolds satisfying a chiral curvature condition
Abstract
Let (M,g) be a compact oriented Einstein four-manifold with Einstein constant E and let R̂+ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if R̂+ < 0, then g is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterion for stability. Our result is stronger than the one by Fine-Krasnov-Singer, who conclude local rigidity from R̂+ < 0 by proving stability for a different action functional. Our proof proceeds by showing that (M,g) admits a natural spinh structure carrying a non-zero parallel spinh-spinor. We then apply a lower bound on the Lichnerowicz Laplacian on traceless symmetric two-tensors in the presence of such a spinor.
Related
- Einstein four-manifolds of positive sectional curvature and ADM mass
- K{ä}hler Einstein manifolds and the Calabi curvature operator
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- Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
- Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation
BibTeX
@misc{artacho2026stability,
title = {{Stability of Einstein 4-manifolds satisfying a chiral curvature condition}},
author = {Diego Artacho},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15159},
eprint = {2609.15159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15159},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stability of Einstein 4-manifolds satisfying a chiral curvature condition
AU - Diego Artacho
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15159
UR - https://arxiv.org/abs/2609.15159
AB - Let (M,g) be a compact oriented Einstein four-manifold with Einstein constant E and let R̂^+ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if R̂^+ < 0, then g is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterion for stability. Our result is stronger than the one by Fine-Krasnov-Singer, who conclude local rigidity from R̂^+ < 0 by proving stability for a different action functional. Our proof proceeds by showing that (M,g) admits a natural spin^h structure carrying a non-zero parallel spin^h-spinor. We then apply a lower bound on the Lichnerowicz Laplacian on traceless symmetric two-tensors in the presence of such a spinor.
ER -