Some rigidity theorems for spectral curvature bounds

We investigate the geometric implications of spectral curvature bounds, extending classical rigidity results in scalar curvature geometry to the spectral setting. By systematically employing the warped $\mu$-bubble method, we show classification theorems for stable weighted minimal hypersurfaces in 3-manifolds with nonnegative spectral scalar curvature, and we establish band width estimates for both spectral Ricci and spectral scalar curvatures. Furthermore, we prove some splitting theorems under spectral curvature conditions, including a spectral version of the Geroch conjecture for manifolds with arbitrary ends and a result related to the Milnor conjecture.