Kähler diameter conjecture

Prove that every closed Kähler manifold of complex dimension \(m\geq 2\) satisfying \(\operatorname{Ric}\geq g\) has diameter at most \(\pi\sqrt m\), with equality if and only if it is biholomorphically isometric to the product \((\mathbb{CP}^1)^m\) equipped with the normalization \(\operatorname{Ric}=g\) on each factor.

Background

The conjecture is motivated by the expected sharp Beckner–Sobolev constant A=1A=1 and by comparison with the model spaces CPm\mathbb{CP}^m and (CP1)m(\mathbb{CP}^1)^m. The latter has diameter πm\pi\sqrt m under the stated normalization.

The authors derive improved diameter estimates from existing Beckner–Sobolev inequalities and from their spectral refinement, but these estimates do not establish the proposed sharp bound or its rigidity statement.

References

Taking $A=1$ and letting $q$ approach the critical exponent $\frac{2m}{m-1}$, one is led to the following diameter conjecture. Suppose $(M,g,J)$ is a closed Kähler manifold of complex dimension $m\geqslant2$, and $\operatorname{Ric}\geqslant g$. Then $\operatorname{diam}(M,g)\leqslant\pi\sqrt m$, with equality if and only if $(M,g,J)$ is biholomorphically isometric to $(\mathbb{CP}1)m$, where $\mathbb{CP}1$ is normalized so that $\operatorname{Ric}=g$.

— Spectral Improvements of Geometric Inequalities on Closed Kähler Manifolds  (2609.03287 - Chakraborty et al., 3 Sep 2026) in Section 1, immediately following the discussion of the Bakry–Ledoux diameter estimate