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.

We study the notion of strong $r$-stability for the context of closed hypersurfaces ${\Sigma }^n$ ($n\ge 3$) with constant $(r+1)$-th mean curvature $H_{r+1}$ immersed into the Euclidean sphere ${𝕊}^{n+1}$, where $r\in \{1,...,n-2\}$. In this setting, under a suitable restriction on the $r$-th mean curvature $H_r$, we establish that there are no $r$-strongly stable closed hypersurfaces immersed in a certain region of ${𝕊}^{n+1}$, a region that is determined by a totally umbilical sphere of ${𝕊}^{n+1}$. We also provide a rigidity result for such hypersurfaces.

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.

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....

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.

In the present paper we give some properties of $f$-biharmonic hypersurfaces in real space forms. By using the $f$-biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$-biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$-biharmonic vertical cylinders in $S^2\times ℝ$.

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 $(M,\theta )$ be a compact CR manifold of dimension $2n+1$ with a contact form $\theta$, and $L=(2+2/n){\Delta }_b+R$ its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form $\tilde{\theta }$ on $M$ conformal to $\theta$ which has a constant Webster curvature. This problem is equivalent to the existence of a function $u$ such that $Lu=u^{1+2/n}$, $u>0$ on $M$. D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where $n\ge 2$ and $(M,\theta )$ is not locally CR equivalent to the sphere $S^{2n+1}$ of ${𝐂}^n$. In a join work with R. Yacoub, the CR Yamabe...

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...

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...

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 ${\Phi }_1,...,{\Phi }_{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U{\subset }^{N+m}$, let $𝕄$ be a $N$-dimensional $C^k$ submanifold of $U$ and define $$𝕋:=\{z\in 𝕄:{\Phi }_1\left(z\right),...,{\Phi }_{k+1}\left(z\right)\in T_z𝕄\}$$ where $T_z𝕄$ is the tangent space to $𝕄$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $𝕄$) of $𝕋$: If $z_0\in 𝕄$ is a $(N+k)$-density point (relative to $𝕄$) of $𝕋$ then all the iterated Lie brackets of order less or equal to $k$ $${\Phi }_{i_1}\left(z_0\right),[{\Phi }_{i_1},{\Phi }_{i_2}]\left(z_0\right),[[{\Phi }_{i_1},{\Phi }_{i_2}],{\Phi }_{i_3}]\left(z_0\right),...(h,i_h\le k+1)$$ belong to $T_{z_0}𝕄$. Such a property has been proved in [9] for $k=1$ and its proof in the...

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. $k=n-1$.

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some...

Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k\left(t\right)$ with respect to some point $p\in M$, where $t=d(·,p)$ is the Riemannian distance on $M$ to $p$, $k\left(t\right)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty$, satisfy $$\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}+\cdots +\frac{1}{{\mu }_n}\ge \frac{l^{n+2}}{(n+2){\int }_0^l{f}^{n-1}\left(t\right)\mathrm{d}t},$$ where $f\left(t\right)$ is the solution to $$\left\{\begin{array}{c}{f}^{\text{'}\text{'}}\left(t\right)+k\left(t\right)f\left(t\right)=0\text{on}(0,\infty ),\hfill \\ f\left(0\right)=0,f^{\text{'}}\left(0\right)=1.\hfill \end{array}\right.$$

Let $X$ be a connected closed manifold and $f$ a self-map on $X$. We say that $f$ is almost quasi-unipotent if every eigenvalue $\lambda$ of the map $f_{*k}$ (the induced map on the $k$-th homology group of $X$) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ odd is equal to the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ even. We prove that if $f$ is $C^1$ having finitely many periodic points all of them...