TY - JOUR
T1 - $L^2$-Harmonic 1-Forms on Complete Manifolds
AU - Zhu , Peng
AU - Zhou , Jiuru
JO - Communications in Mathematical Research 
VL - 1
SP - 1
EP - 7
PY - 2019
DA - 2019/12
SN - 33
DO - http://doi.org/10.13447/j.1674-5647.2017.01.01
UR - https://global-sci.org/intro/article_detail/cmr/13440.html
KW - minimal hypersurface, end, quaternionic manifold, weighted Poincaré inequality
AB - <p style="text-align: justify;">We study the global behavior of complete minimal $\delta$-stable hypersurfaces in $\mathbf{R}^{n+1}$ by using $L^2$-harmonic 1-forms. We show that a complete minimal $\delta$-stable $\bigg(\delta&gt;\dfrac{(n-1)^2}{n^2}\bigg)$ hypersurface in $\mathbf{R}^{n+1}$ has only one end. We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincaré inequality. These results are improvements of the first author&#39;s theorems on hypersurfaces and quaternionic Kähler manifolds.&nbsp;</p>