Determine the optimal boundary first-eigenvalue constant under Ricci and convexity bounds

Determine the optimal universal constant c_n for compact connected (n+1)-dimensional Riemannian manifolds with nonempty boundary satisfying Ric_g ≥ ng and II_g ≥ 0, where c_n is the infimum of the first nonzero Laplace eigenvalue of the boundary; in particular, determine how close c_n can be to the Reilly lower bound n/2.

Background

The paper defines c_n as the infimum of the first nonzero boundary Laplace eigenvalue over compact connected (n+1)-dimensional manifolds satisfying Ric_g ≥ ng and II_g ≥ 0. Reilly’s argument yields the lower bound c_n ≥ n/2, while the paper’s perturbative construction proves only that c_n < n, thereby disproving the proposed sharp lower bound n.

The authors explicitly state that their construction does not determine the infimum or establish how close the boundary eigenvalue can be driven to n/2 under strict convexity. Thus the exact optimal constant remains unresolved.