  In this article, we investigate the connection between scalar curvature and
first eigenfunctions via positive mass theorem for Brown-York mass. For compact
manifolds with nice boundary, we show that a sharp inequality holds for first
eigenfunctions when posing appropriate assumptions on scalar curvature and
first eigenvalue. This inequality implies that for a compact n-dimensional
manifold with boundary, its first eigenvalue is no less that n, if its scalar
curvature is at least n(n-1) with appropriate boundary conditions posed, where
equality holds if and only the manifold is isometric to the canonical upper
hemisphere. As an application, we derive an estimate for the area of event
horizon in a vacuum static space with positive cosmological constant, which
reveals an interesting connection between the area of event horizon and
Brown-York mass. This estimate generalizes a similar result of Shen for three
dimensional vacuum static spaces and also improves the uniqueness result of de
Sitter space-time due to Hizagi-Montiel-Raulot.
