Sharp Kähler Beckner–Sobolev inequality

Establish that the Beckner–Sobolev inequality on every closed Kähler manifold of complex dimension \(m\geq 2\) with \(\operatorname{Ric}\geq g\), for every \(q\in[1,2)\cup(2,2m/(m-1)]\), holds with the sharp constant \(A=1\).

Background

The conjectured inequality is presented as a consequence of the two Liouville conjectures because the associated Euler–Lagrange equations would then admit only constant positive solutions in the relevant parameter ranges.

The paper proves improved constants depending on the first positive eigenvalue of the ∂ˉ\bar\partial-Laplacian, but does not reach the conjectured universal value A=1A=1.

References

In view of the Euler--Lagrange equations associated with BS, Conjectures \ref{conj:Sobolev} and \ref{conj:Beckner} would imply the following sharp Beckner--Sobolev inequality. Suppose $(M,g,J)$ is a closed Kähler manifold of complex dimension $m\geqslant2$, $\operatorname{Ric}\geqslant g$, and $q\in[1,2)\cup(2,\frac{2m}{m-1}]$. Then the Beckner--Sobolev inequality BS holds with $A=1$.

BS:

$\frac{1}{q-2}\Big[\Big(\fint_M|f|^q\Big)^{\frac 2 q}-\fint_M|f|^2\Big]\leqslant A\fint_M|\partial f|^2, $

— Spectral Improvements of Geometric Inequalities on Closed Kähler Manifolds  (2609.03287 - Chakraborty et al., 3 Sep 2026) in Section 1, Conjecture 3 (labelled \ref{conj:BS})

Besides, in the Kahler case, the current estimates are far from optimal. What other structures can one exploit to get better estimates?

— Spectral Improvements of Geometric Inequalities on Closed Kähler Manifolds  (2609.03287 - Chakraborty et al., 3 Sep 2026) in Section 3, final paragraph of the paper