Related papers: Rigidity of CR-immersions into spheres

We establish the holomorphic wedge extendability of CR functions, defined on an everywhere locally minimal generic submanifold M of C^n and having singularities contained in a submanifold N of codimension 1, 2 or 3, assuming some…

In this paper we classify Euclidean hypersurfaces $f\colon M^n \rightarrow \mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$.…

Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic…

Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M.$ Let $% \Phi $ denote a $CR$ developing map {equation*} \Phi :\widetilde{M}\rightarrow S^{2n+1}…

We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and…

Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies…

We show that every formal embedding sending a real-analytic strongly pseudoconvex hypersurface in $M\subset \C^N$ into another such hypersurface in $M'\subset \C^{N+1}$ is convergent. More generally, if $M$ and $M'$ are merely…

We study the (global) Bishop problem for small perturbations of $\mathbf{S}^n$ --- the unit sphere of $\mathbb{C}\times\mathbb{R}^{n-1}$ --- in $\mathbb{C}^n$. We show that if $S\subset\mathbb{C}^n$ is a sufficiently-small perturbation of…

Let $M$ be an $n(\geq3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature of…

We establish an asymptotic rigidity result for isometric immersions of codimension-1. Specifically, we consider a sequence of immersions from a compact $d$-dimensional manifold into a complete $(d+1)$-dimensional manifold whose elastic…

Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[…

In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the…

We study the non-embddability property for a class of real hypersurfaces, called real hypersurfaces of involution type, into the sphere in the low codimensional case, by making use of property of a naturally related Gauss curvature. We also…

We prove that every complete, minimally immersed submanifold $f\: M^n \to \mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface…

We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of…

I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi}{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere can be rigid under smooth deformations that increase…

Let $\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=\phi(x),v$ and…

Totally real immersions $f$ of a closed real surface $\Sigma$ in an almost complex surface $M$ are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes $\frak{M}(f)$ of mappings from…

An isometric immersion $f: M^{n} \rightarrow \tilde M^{m}$ from an $n$-dimensional Riemannian manifold $M^{n}$ into an almost Hermitian manifold $\tilde M^{m}$ of complex dimension $m$ is called pointwise slant if its Wirtinger angles…

Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$ $(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain $D$ in $Z$ such that $K\subset D$ and there is a complete proper…