1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

This is an application of our previous quantitative rigidity result via pinching Heintze-Reilly's inequality. Based on work by Hu and Xu [Recognizing shape via 1st eigenvalue, mean curvature, and upper curvature bound, arXiv:1905.01664v2], we prove that for any closed convex hypersurface Mⁿ lying in a convex ball B(p, R) of the ambient (n + 1)-manifold N^{n +1}, whose sectional curvature μ ≤ K_N ≤ δ, if λ₁(M) approaches n(δ + ∥H∥² _∞), then M (resp., its enclosed domain) is Hausdorff (resp., C^1,α) close to a sphere (resp., a geodesic ball) of constant curvature, where λ₁(M) is the 1st eigenvalue of M and ∥H∥_∞ is the maximum of M's mean curvature in N.