arXiv subjects
Publications and source records attributed to Huixin Tan.
Let $M^n\subset\mathbb{S}^{n+1}(1)$ be a closed minimal hypersurface with constant $S=|h|^2$ and constant $f_3=\operatorname{tr} h^3$, where $h$ is the shape operator. We prove that $S>n$ implies $S\geq2n$, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal curvatures, which occur only in dimensions $n=3, 6, 12, 24$.
We prove that the values set of \(S=|A|^2\) attained by closed embedded minimal hypersurfaces in \(\mathbb S^{n+1}(1)\) with constant \(S\) and constant \(f_3=\operatorname{tr}(A^3)\) is locally finite, where \(A\) denotes the shape operator. Neither the topology of the hypersurface nor the value of \(f_3\) is fixed.
We prove that the angle function associated with the canonical product structure is constant for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{S}^{m}$, $\mathbb{S}^{n}\times \mathbb{H}^{m}$, or $\mathbb{H}^{n}\times \mathbb{H}^{m}$. This rigidity result enables us to provide a complete classification of isoparametric and homogeneous hypersurfaces in $\mathbb{S}^{n}\times \mathbb{S}^{m}$ and $\mathbb{S}^{n}\times \mathbb{H}^{m}$. Furthermore, we establish a geometric characterization in these two spaces: a hypersurface is isoparametric if and only if it has constant principal curvatures and a constant angle function.
We first show that every isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ possesses a constant angle function with respect to the canonical product structure. Exploiting this rigidity, we achieve a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces. Furthermore, we prove that an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ also has constant principal curvatures.
In this paper, we prove that for an $n$-dimensional closed minimal Willmore hypersurface $M^n$ with constant scalar curvature in the unit sphere $\mathbb{S}^{n+1}$, the squared norm $S$ of the second fundamental form of $M^n$ satisfies $S\geqslant n+\frac{4n+9-\sqrt{4 n^{2}+60 n+81}}{2}$ if $S>n$. This proves, in the approximate sense, the Chern conjecture about the second gap ($S\geqslant 2n$ if $S>n$), which will be fully verified under a further inequality condition about the 4-th mean curvature.