Kählerian almost rigidity conjecture for the first eigenvalues

Let $(M,\omega)$ be a compact Kähler manifold of complex dimension $n$, with Ricci curvature satisfying

$$\operatorname{Ric}(\omega)\geqslant\omega.$$

Write $\lambda_j$ for the eigenvalues of the relevant Laplacian on $(M,\omega)$, counted with multiplicity, and let $(\mathbb{CP}^n,\omega_{\mathbb{CP}^n})$ denote complex projective space with its normalized Fubini–Study Kähler metric. Kählerian almost rigidity conjecture. For every $\varepsilon>0$, there exists $\delta=\delta(n,\varepsilon)>0$ such that if

$$\lambda_{n^2+3}<1+\delta,$$

then $M$ is biholomorphic to $\mathbb{CP}^n$ and

$$d_{\mathrm{GH}}\big((M,\omega),(\mathbb{CP}^n,\omega_{\mathbb{CP}^n})\big)<\varepsilon.$$

This is the proposed Kähler analogue of the Petersen–Aubry almost-rigidity theorem for Riemannian manifolds. The preceding discussion indicates that the threshold $n^2+3$ lies immediately above the gap after the maximal first-eigenvalue multiplicity of $\mathbb{CP}^n$; the conjecture asserts that near-attainment of this eigenvalue bound forces both the complex structure and the Gromov–Hausdorff geometry to be close to the projective-space model.