Related papers: Dupin Submanifolds in Lie Sphere Geometry (updated…

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmu\c{s}-Montaldo-Oniciuc, we…

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our…

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…

The Gauss map of a hypersurface of a unit sphere $S^{n+1}(1)$ is a Lagrangian immersion into the complex quadric $Q^n$ and, conversely, every Lagrangian submanifold of $Q^n$ is locally the image under the Gauss map of several hypersurfaces…

In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally…

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of…

In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinct constant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$…

We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of $\mathbb R^n$. More precisely, if $K$ is a smooth convex body in…

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…

We address the study of some curvature equations for distinguished submanifolds in para-K\"ahler geometry. We first observe that a para-complex submanifold of a para-K\"ahler manifold is minimal. Next we describe the extrinsic geometry of…

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a…

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean…

We introduce a class of surfaces in euclidean space motivated by a problem posed by \'{E}lie Cartan. This class furnishes what seems to be the first examples of pairs of non-congruent surfaces in euclidean space such that, under a…

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume…

We proof existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two…

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function…

This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $\mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR…

Let M be a closed embedded minimal hypersurface in a Euclidean sphere of dimension n+1, we prove that it is strongly rigid. As applications we confirm the conjecture proposed by Choi and Schoen in [3] and the Chern conjecture for n less…

This is an updated version of a paper which appeared in the proceedings of the 1979 Berlin Colloquium on Global Differential Geometry. This paper contains the original exposition together with some notes by the authors made in 2025 (as…