A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary

TL;DR

This work generalizes the Cheeger–Gromoll splitting phenomenon to noncompact manifolds with boundary under a spectral Ricci condition. By introducing the α-weighted length functional L_u^α and analyzing its second variation, the authors show that minimizing weighted curves carry rigidity information that forces Ricci-type constraints along these curves. The curve-capture construction produces L_u^α-minimizing rays and lines through any point, enabling a global splitting conclusion once an interior end is present. The main theorem thus yields a product structure M ≅ Σ×ℝ_{≥0} with Σ compact and Ric_Σ≥0 or asserts the absence of interior ends, with a corollary interpreting biRic-type curvature conditions in a free-boundary setting.

Abstract

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $λ_1(-αΔ+\operatorname{Ric})\geq 0$ for some $α<\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $Σ\times \mathbb{R}_{\geq 0}$ for a closed manifold $Σ$ with nonnegative Ricci curvature or it has no interior ends.

Figures (1)

• Figure 1: The case where $\gamma_r$ is a line

Theorems & Definitions (31)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6