Kähler Sobolev Liouville conjecture

Prove that every positive solution of the equation \(-\square u+\lambda u=u^\alpha\) on a closed Kähler manifold of complex dimension \(m\geq 2\) satisfying \(\operatorname{Ric}\geq g\), for \(1<\alpha\leq (m+1)/(m-1)\) and \(0<\lambda<1/(\alpha-1)\), is constant, namely \(u\equiv \lambda^{1/(\alpha-1)}\).

Background

The paper studies positive solutions of the Euler–Lagrange equation associated with the Sobolev branch of the Beckner–Sobolev inequality on closed Kähler manifolds. The first positive eigenvalue of the ∂ˉ\bar\partial-Laplacian satisfies Λ≥1\Lambda\geq 1 under the curvature condition Ric⁡≥g\operatorname{Ric}\geq g, and linearization around constant solutions yields the parameter range in the conjecture.

The authors prove a weaker result with an upper bound on λ\lambda depending explicitly on α\alpha, mm, and Λ\Lambda. The conjecture asserts the optimal curvature-only threshold 1/(α−1)1/(\alpha-1), independent of the first eigenvalue.

References

The lower bound \Lambda\geqslant 1, together with the necessary condition obtained by linearizing BS around constant functions, motivates the following two conjectures. Suppose $(M,g,J)$ is a closed Kähler manifold of complex dimension $m\geqslant2$, $\operatorname{Ric}\geqslant g$, $1<\alpha\leqslant\frac{m+1}{m-1}$, and $0<\lambda<\frac{1}{\alpha-1}$. Then every positive solution of eq:Sobolev is constant; more precisely, $u\equiv\lambda{\frac{1}{\alpha-1}}$.

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, $

eq:Sobolev:

−□u+λu=uα,1<α⩽m+1m−1,λ>0,-\square u+\lambda u=u^\alpha,\quad1<\alpha\leqslant\frac{m+1}{m-1},\quad\lambda>0,

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

Suppose $(M,g,J)$ is a closed Kähler manifold of complex dimension $m\geqslant2$, $\operatorname{Ric}\geqslant g$, $0<\alpha<1$, and $0<\lambda<\frac{1}{1-\alpha}$. Then every positive solution of eq:Beckner is constant; more precisely, $u\equiv\lambda{\frac{1}{\alpha-1}}$.

eq:Beckner:

□u+λu=uα,0<α<1,λ>0.\square u+\lambda u=u^\alpha,\quad0<\alpha<1,\quad\lambda>0.

— Spectral Improvements of Geometric Inequalities on Closed Kähler Manifolds  (2609.03287 - Chakraborty et al., 3 Sep 2026) in Section 1, Conjecture 2 (labelled \ref{conj:Beckner})