Comparison geometry of manifolds with boundary under lower 𝑁-weighted Ricci curvature bounds with 𝜀-range

Abstract

We study comparison geometry of manifolds with boundary under a lower 𝑁-weighted Ricci curvature bound for 𝑁 ∈ ] −∞, 1] ∪ [𝑛, +∞] with 𝜀-range introduced by Lu–Minguzzi–Ohta in 2022. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted 𝑝-Laplacian. Our results interpolate those for 𝑁 ∈ [𝑛, +∞[ and 𝜀 = 1, and for 𝑁 ∈ ] −∞, 1] and 𝜀 = 0 by the second named author.