Related papers: Tight Spherical Embeddings (Updated Version)
The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space $\mathbb{H}^{2n+p}$ with low codimension $2\leq p\leq n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for…
Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $\Phi\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of…
In this paper we find necessary and sufficient conditions for a nondegenerate arbitrary signature manifold $M^n$ to be realized as a submanifold in the large class of warped product manifolds $\varepsilon…
This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces…
We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be…
For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a…
An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of ${\rm Aut}(T_p M)$ for all $p\in M$, which preserves (pointwise) the affine metric $h$, the difference tensor $K$ and the affine shape…
M. Freedman showed that every homology 3-sphere embeds as a locally flat submanifold of $S^4$. This is in striking contrast to the state of our knowledge of smooth embeddings of homology spheres. This book surveys what is presently known…
We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface…
We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched…
Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean…
We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism…
We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially…
Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition…
We propose methods towards a systematic determination of d dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity…
We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…
By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of…
In this paper, we study degenerate CR embeddings $f$ of a strictly pseudoconvex hypersurface $M\subset \bC^{n+1}$ into a sphere $\bS$ in a higher dimensional complex space $\bC^{N+1}$. The degeneracy of the mapping $f$ will be characterized…
A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at…
This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and…