arXiv subjects
Publications and source records attributed to Jinchuan Bai.
We prove that in the anti-Gaussian space $\bigl(\R^{m+1},\bar g_{\Euc},e^{|x|^2/4}\,dx\bigr)$, $m\ge2$, round spheres centered at the origin are the only closed, connected, two-sided immersed hypersurfaces with constant weighted mean curvature that are stable under weighted-volume-preserving variations.
In this paper, we establish a Heintze--Karcher inequality for closed embedded hypersurfaces in a class of warped product manifolds endowed with radial density. As a consequence, we prove Alexandrov-type theorem for constant weighted mean curvature hypersurfaces in such spaces. In particular, we prove that a closed embedded $\lambda$-self-expander in the Euclidean space must be a round sphere centered at the origin.
Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v \right) ^ { \frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.