Volume 33, Issue 1
$L^2$-Harmonic 1-Forms on Complete Manifolds

Commun. Math. Res., 33 (2017), pp. 1-7.

Published online: 2019-12

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• Abstract

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 $\bigg(\delta>\dfrac{(n-1)^2}{n^2}\bigg)$ 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's theorems on hypersurfaces and quaternionic Kähler manifolds.

• Keywords

minimal hypersurface, end, quaternionic manifold, weighted Poincaré inequality

53C21, 54C42

Zhupeng2004@126.com (Peng Zhu)

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@Article{CMR-33-1, author = {Zhu , Peng and Zhou , Jiuru}, title = {$L^2$-Harmonic 1-Forms on Complete Manifolds}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {1}, pages = {1--7}, abstract = {

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 $\bigg(\delta>\dfrac{(n-1)^2}{n^2}\bigg)$ 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's theorems on hypersurfaces and quaternionic Kähler manifolds.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.01.01}, url = {http://global-sci.org/intro/article_detail/cmr/13440.html} }
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 -

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 $\bigg(\delta>\dfrac{(n-1)^2}{n^2}\bigg)$ 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's theorems on hypersurfaces and quaternionic Kähler manifolds.

Peng Zhu & Jiuru Zhou. (2019). $L^2$-Harmonic 1-Forms on Complete Manifolds. Communications in Mathematical Research . 33 (1). 1-7. doi:10.13447/j.1674-5647.2017.01.01
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