Comparison geometry of manifolds with boundary under lower $N$-weighted Ricci curvature bounds with $\varepsilon$-range

Abstract

We study comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N \in ] -\infty, 1] \cup [n, +\infty]$ with $\varepsilon$-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 $N \in [n, +\infty[$ and $\varepsilon = 1$, and for $N \in ] -\infty, 1]$ and $\varepsilon = 0$ by the second named author.