Comparison geometry of manifolds with boundary under lower $N$-weighted Ricci curvature bounds with $\epsilon$-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 $\epsilon$-range introduced by Lu-Minguzzi-Ohta. 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 $\epsilon =1$, and for $N\in ]-\infty ,1]$ and $\epsilon =0$ by the second named author.

24 pages