# MathSciDoc: An Archive for Mathematician ∫

#### Differential Geometrymathscidoc:2206.10006

Mathematische Annalen, 381, 1723–1743, 2020.9
Let M be a compact Riemannian manifold, π : \tilde{M} → M be the universal covering and ω be a smooth 2-form on M with π^∗ω cohomologous to zero. Suppose the fundamental group π_1(M) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth 1-form η on \tilde{M} of linear (resp. bounded) growth such that π^∗ω = dη. As applications, we prove that on a compact Kahler manifold (M,ω) with π^∗ω cohomologous to zero, if π_1(M) is CAT(0) or automatic (resp. hyperbolic), then M is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic (−1)^{dim_RM/2χ(M) ≥ 0 (resp. >0).
@inproceedings{bing-long2020on,
title={On Euler characteristic and fundamental groups of compact manifolds},
author={Bing-Long Chen, and Xiaokui Yang},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622154552520091447},
booktitle={Mathematische Annalen},
volume={381},
pages={1723–1743},
year={2020},
}

Bing-Long Chen, and Xiaokui Yang. On Euler characteristic and fundamental groups of compact manifolds. 2020. Vol. 381. In Mathematische Annalen. pp.1723–1743. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622154552520091447.