The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set Ω with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that Ω is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer problem to an arbitrary Riemannian manifold and also...

The objective of this paper is to study singularities of n-ruled (n + 1)-manifolds in Euclidean space. They are one-parameter families of n-dimensional affine subspaces in Euclidean space. After defining a non-degenerate n-ruled (n + 1)-manifold we will give a necessary and sufficient condition for such a map germ to be right-left equivalent to the cross cap × interval. The behavior of a generic n-ruled (n + 1)-manifold is also discussed.

Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

We introduce a skeletal structure $(M,U)$ in ${ℝ}^{n+1}$, which is an $n$- dimensional Whitney stratified set $M$ on which is defined a multivalued “radial vector field” $U$. This is an extension of notion of the Blum medial axis of a region in ${ℝ}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $ℬ$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $ℬ$. Together these allow us to provide sufficient numerical conditions for...

In this paper some properties of an immersion of two-dimensional surface with boundary into $Esp4$ are studied. The main tool is the maximal principle property of a solution of the elliptic system of partial differential equations. Some conditions for a surface to be a part of a 2-dimensional spheren in $Esp4$ are presented.

We consider graphs of positive scalar or Gauss-Kronecker curvature over a punctured disk in Euclidean and hyperbolic $n$-dimensional space and we obtain removable singularities theorems.

Nachdem der Begriff des sphärischen Bildes der Menge $𝐌$ und der Begriff von sphärisch äquivalenten Mengen eingeführt wurde, werden verschiedene Zusammenhänge zwischen der Menge $𝐌$ und ihrem sphärischen Bild untersucht und zwar unter verschiedenen Voraussetzung über $𝐌$ (z. B. ihre Beschränkheit, Unbeschränkheit, strenge Konvexität). Die bewiesene Tatsache, dass die Menge $𝐌$ und ihre $ϵ$-Umgebung sphärisch äquivalent sind, kann - sowie andere Ergebnisse der Arbeit - in der Theorie der konvexen parametrischen...

In this paper we study the pairs of orthogonal foliations on oriented surfaces immersed in R3 whose singularities and leaves are, respectively, the umbilic points and the lines of normal mean curvature of the immersion. Along these lines the immersions bend in R3 according to their normal mean curvature. By analogy with the closely related Principal Curvature Configurations studied in [S-G], [GS2], whose lines produce the extremal normal curvature for the immersion, the pair of foliations by lines...