Related papers: Isoparametric and Dupin Hypersurfaces

We study Lorentz hypersurfaces $M_{1}^{n}$ in $E_{1}^{n+1}$ satisfying $\triangle \vec {H}= \alpha \vec {H}$ with non diagonal shape operator, having complex eigenvalues. We prove that every such Lorentz hypersurface in $E_{1}^{n+1}$ having…

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

Explicit representations of complex structures on closed manifolds are valuable, but relatively rare in the literature. Using isoparametric theory, we construct complex structures on isoparametric hypersurfaces with $g=4, m=1$ in the unit…

We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…

In this paper we study biconservative hypersurfaces $M$ in space forms $\overline M^{n+1}(c)$ with four distinct principal curvatures whose second fundamental form has constant norm. We prove that every such hypersurface has constant mean…

We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic…

We classify complete orientable hypersurfaces of constant isotropic curvature in space forms. We show that such a hypersurface has constant mean curvature only if it is an isoparametric hypersurface, and that it is minimal if and only if it…

We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the…

In this paper, we study biconservative hypersurfaces in $\mathbb S^{n}$ and $\mathbb H^{n}$. Further, we obtain complete explicit classification of biconservative hypersurfaces in $4$-dimensional Riemannian space form with exactly three…

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions.…

We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an orientable, complete immersion with finite strong total curvature, then $X$ is proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite number…

We showed the existence of non-radial solutions of the equation $\Delta u -\lambda u + \lambda u^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $\lambda$. We show that for any…

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature…

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric…

We investigate the existence of minimal hypersurfaces in $\mathbb{S}^{n+1}$ that are generated by the isoparametric foliation of a subsphere $\mathbb{S}^n$. By considering a generalized rotational ansatz formed by the union of homothetic…

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold.…

In this paper, we give a full classification of the separable hypersurfaces of constant sectional curvature in the Euclidean $n$-space $\mathbb{R}^n$. In dimension $n=3$, this classification was solved by Hasanis and L\'opez [Manuscripta…

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower…

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable…

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that in a conic Minkowski space, besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski…