TY - JOUR
T1 - Ricci Curvature and Fundamental Groups of Effective Regular Sets
AU - Pan , Jiayin
JO - Journal of Mathematical Study
VL - 1
SP - 3
EP - 21
PY - 2025
DA - 2025/03
SN - 58
DO - http://doi.org/10.4208/jms.v58n1.25.01
UR - https://global-sci.org/intro/article_detail/jms/23935.html
KW - Ricci curvature, fundamental groups, Gromov-Hausdorff convergence.
AB - <p style="text-align: justify;">For a Gromov-Hausdorff convergent sequence of closed manifolds $M^n_i\xrightarrow{GH} X$ with ${\rm Ric}\ge -(n-1)$, ${\rm diam}(M_i) ≤ D,$ and ${\rm vol}(M_i) ≥ v &gt; 0,$ we study the relation between $π_1(M_i)$ and $X.$ It was known before that there is a surjective homomorphism $ϕ_i
: π_1(M_i) → π_1(X)$ by the work of Pan-Wei. In this paper, we construct a surjective
homomorphism from the interior of the effective regular set in $X$ back to $M_i,$ that is, $ψ_i
:π_1(\mathcal{R}^◦_{ϵ,δ} )→π_1(M_i).$ These surjective homomorphisms $ϕ_i$ and $ψ_i$ are natural in the
sense that their composition $ϕ_i◦ψ_i$ is exactly the homomorphism induced by the inclusion map $R^◦_{ ϵ,δ}\hookrightarrow X.$</p>