For every integer n≥ 3, every compact n-dimensional Riemannian manifold (M^n,g) with nonnegative Ricci curvature, operatornameRic_g≥ 0, and every principal curvature of partial M bounded below by a constant kappa>0 satisfies sigma_1(M,g)≥ kappa, where sigma_1(M,g) is the first nonzero Steklov eigenvalue. Moreover, equality is conjectured to characterize the Euclidean ball of radius 1/kappa.

Escobar's conjecture

Escobar conjectured in 1999 that nonnegative Ricci curvature and boundary curvature bounded below by $c>0$ force the first nonzero Steklov eigenvalue to satisfy $\sigma_1\ge c$ , with equality only for the corresponding Euclidean ball.

Known results

Payne (1970): the sharp planar result.
Escobar: the two-dimensional nonnegative-Gaussian-curvature case and the higher-dimensional estimate $\sigma_1>c/2$ under nonnegative Ricci curvature.
Montaño (2013): the conjecture for rotationally symmetric metrics and Euclidean ellipsoids.
Xia and Xiong (2024): the conjecture under the stronger assumption of nonnegative sectional curvature.

August 24, 2026 counterexamples

On August 24, 2026, the preprint Counterexamples to Escobar's conjecture claimed a perturbative criterion and explicit polynomial conformal factors producing counterexamples in every relevant dimension. This would disprove the original conjecture, but the manuscript is unrefereed and the claim is not independently corroborated.

Current status (as of August 2026): the conjecture is claimed to be disproved in every relevant dimension by an unrefereed preprint, while the earlier curvature-restricted cases are established and the counterexamples remain unverified.

Escobar's conjecture

Primary source

Additional references

Progress summary

Solutions 0