Abstract : Let $M$ be a real hypersurface in the complex hyperbolic quadric ${Q^{m}}^{*}$, $m \geq 3$. The Riemannian curvature tensor field $R$ of $M$ allows us to define a symmetric Jacobi operator with respect to the Reeb vector field $\xi$, which is called the structure Jacobi operator $R_{\xi} = R(\, \cdot \, , \xi) \xi \in \mathrm{End}(TM)$. On the other hand, in [Math. Z. 245 (2003), 503-527], Semmelmann showed that the cyclic parallelism is equivalent to the Killing property regarding any symmetric tensor. Motivated by his result above, in this paper we consider the cyclic parallelism of the structure Jacobi operator $R_{\xi}$ for a real hypersurface $M$ in the complex hyperbolic quadric ${Q^{m}}^{*}$. Furthermore, we give a complete classification of Hopf real hypersurfaces in ${Q^{m}}^{*}$ with such a property.
Abstract : Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}}~, \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vetex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critial value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.
Abstract : In this paper, we study the unicorn’s Landsberg problem from an intrinsic point of view. Precisely, we investigate a coordinate-free proof of Numata’s theorem on Landsberg spaces of scalar curvature. In other words, following the pullback approach to Finsler geometry, we prove that all Landsberg spaces of dimension n ≥ 3 of non-zero scalar curvature are Riemannian spaces of constant curvature
Abstract : In this paper, for a $m$-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 $|\textbf{H}_{f}|$ and the weighted real-valued function $f$, we can obtain: several one-end theorems for $M$; two Liouville theorems for harmonic maps from $M$ to complete Riemannian manifolds with nonpositive sectional curvature.
Nguyen Thi Anh Hang, Michael Hoff , Hoang Le Truong
J. Korean Math. Soc. 2022; 59(1): 87-103
https://doi.org/10.4134/JKMS.j210140
Manouchehr Shahamat
J. Korean Math. Soc. 2022; 59(1): 1-12
https://doi.org/10.4134/JKMS.j200112
Pengtao Li, Zhiyong Wang, Kai Zhao
J. Korean Math. Soc. 2022; 59(1): 129-150
https://doi.org/10.4134/JKMS.j210224
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Pengtao Li, Zhiyong Wang, Kai Zhao
J. Korean Math. Soc. 2022; 59(1): 129-150
https://doi.org/10.4134/JKMS.j210224
Manouchehr Shahamat
J. Korean Math. Soc. 2022; 59(1): 1-12
https://doi.org/10.4134/JKMS.j200112
Jung Hee Cheon, Yongha Son, Donggeon Yhee
J. Korean Math. Soc. 2022; 59(1): 35-51
https://doi.org/10.4134/JKMS.j200650
Nguyen Thi Anh Hang, Michael Hoff , Hoang Le Truong
J. Korean Math. Soc. 2022; 59(1): 87-103
https://doi.org/10.4134/JKMS.j210140
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