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In this talk, I will discuss the geometry of smooth complete manifolds where the first eigenvalue of the operator −γΔ + Ric, for γ > 0, is bounded from below. Here, Ric denotes the (pointwise) lowest eigenvalue of the Ricci tensor. This condition is weaker than a pointwise lower bound on Ricci curvature. In particular, I will focus on the following result. Let (Mⁿ,g) be a complete non-compact Riemannian manifold with at least two ends, and with n≥2. Assume u is a positive function on M satisfying −γΔu+Ric*u≥0 and γ<4/(n-1). Then, Mⁿ is isometric to the product of IR x Nⁿ⁻¹. This extends a result of Cheeger–Gromoll from 1971, and the bound required on γ is sharp. Joint work with M. Pozzetta and K. Xu.

https://yale.zoom.us/j/91979343134