Given a domain $\Omega$ of ${ℝ}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\partial \Omega$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces in $\Omega$ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega$ perpendicularly along their boundaries.

Let $S$ be a Riemann surface. Let ${ℍ}^3$ be the $3$-dimensional hyperbolic space and let ${\partial }_{\infty }{ℍ}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\to {\partial }_{\infty }{ℍ}^3=\widehat{ℂ}$. If $i:S\to {ℍ}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\widehat{ı}$, of $i$ by $\widehat{ı}=N$. Let $\overrightarrow{n}:U{ℍ}^3\to {\partial }_{\infty }{ℍ}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\widehat{ı})$ is complete and $\overrightarrow{n}\circ \widehat{ı}=\varphi$. In this...

In the class of real hypersurfaces $M^{2n-1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_n\left(c\right)$ of constant holomorphic sectional curvature $c$ $(\ne 0)$ which is either a complex projective space $ℂP^n\left(c\right)$ or a complex hyperbolic space $ℂH^n\left(c\right)$ according as $c>0$ or $c<0$, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds....

We study lightlike hypersurfaces $M$ of an indefinite Kaehler manifold $\overline{M}$ of quasi-constant curvature subject to the condition that the characteristic vector field $\zeta$ of $\overline{M}$ is tangent to $M$. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface $M$ of $\overline{M}$ such that (1) the screen distribution $S\left(TM\right)$ is totally umbilical or (2) $M$ is screen conformal.

Let Mⁿ be a hypersurface in $R^{n+1}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $\left(K_{x,y}\circ K_{z,u}\right)\left(u\right)=\left(K_{z,u}\circ K_{x,y}\right)\left(u\right)$, where $K_{x,y}\left(u\right)=R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

We compute the essential dimension of the functors Forms${}_{n,d}$ and Hypersurf${}_{n,d}$ of equivalence classes of homogeneous polynomials in $n$ variables and hypersurfaces in ${\mathbb{P}}^{n-1}$, respectively, over any base field $k$ of characteristic $0$. Here two polynomials (or hypersurfaces) over $K$ are considered equivalent if they are related by a linear change of coordinates with coefficients in $K$. Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application...

We give the definition of $f$-biminimal submanifolds and derive the equation for $f$-biminimal submanifolds. As an application, we give some examples of $f$-biminimal manifolds. Finally, we consider $f$-minimal hypersurfaces in the product space ${ℝ}^n\times {𝕊}^1\left(a\right)$ and derive two rigidity theorems.

For two-dimensional, immersed closed surfaces $f:\Sigma \to {ℝ}^n$, we study the curvature functionals ${ℰ}^p\left(f\right)$ and ${𝒲}^p\left(f\right)$ with integrands ${(1+|A|}^2{)}^{p/2}$ and ${(1+|H|}^2{)}^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p>2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for ${𝒲}^p$-bounded sequences. In the case of ${ℰ}^p$ this is just Langer’s theorem [16], while for ${𝒲}^p$ we have to impose a bound for the Willmore energy strictly below $8\pi$ as an additional...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

We give in ${ℝ}^6$ a real analytic almost complex structure $J$, a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$, such that there is no germ of $J$-holomorphic disc $\gamma$ included in $M$ with $\gamma \left(0\right)=0$ and $\frac{\partial \gamma }{\partial x}\left(0\right)=v$, although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$, we give sufficient conditions under which there exists such a germ of disc.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...