Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2024; 61(1): 183-205

Online first article October 23, 2023      Printed January 1, 2024

Copyright © The Korean Mathematical Society.

Complete noncompact submanifolds of manifolds with negative curvature

Ya Gao , Yanling Gao, Jing Mao , Zhiqi Xie

Hubei University; Hubei University; Hubei University; Yulin University


In this paper, for an $m$-dimensional ($m\geq5$) complete noncompact submanifold $M$ immersed in an $n$-dimensional ($n\geq6$) simply connected Riemannian manifold $N$ with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of $M$, the norm of its weighted mean curvature vector $|\textbf{\emph{H}}_{f}|$ and the weighted real-valued function $f$, we can obtain:

$\bullet$ several one-end theorems for $M$; 
$\bullet$ two Liouville theorems for harmonic maps from $M$ to complete Riemannian manifolds with nonpositive sectional curvature. 

Keywords: $L^{p}$ harmonic 1-forms, submanifolds, ends, sectional curvature, $k$-th Ricci curvature

MSC numbers: 53C20, 53C42

Supported by: This work is partially supported by the NSF of China (Grant Nos. 11801496, 11926352 and 12261095), the Fok Ying-Tung Education Foundation (China) and Hubei Key Laboratory of Applied Mathematics (Hubei University).