Diameter Based Lower Bounds for the First p-Form Eigenvalue on Positively Curved Warped Products

A main topic of spectral geometry is to determine the geometry of a manifold given its vibrational tones in the form of eigenvalues. For manifolds of curvature greater than the unit sphere, the first eigenvalue is greater than or equal to the dimension of the manifold, with equality precisely for the sphere (Lichnerowicz-Obata, 1962). This is a fundamental result in Geometric Analysis and has been developed over the past 60 years. Kroger (1992) has shown a larger bound when the diameter of the manifold is small. We show for warped products of positive curvature, the first eigenvalue on $p$-forms has a bound depending on radial diameter, extending p-form eigenvalue bounds of Tachibana in the warped product case.