Related papers: Exponential rarefaction of maximal real algebraic …

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4,…

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal…

The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using…

We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary.…

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on…

We deal with a generalization of a Theorem of P. Gordan and M. Noether on hypersurfaces with vanishing (first) Hessian. We prove that for any given $N\geq 3$, $d \geq 3$ and $2\leq k < \frac{d}{2}$ there are infinitely many irreducible…

We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of…

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane…

Unexpected hypersurfaces arise when vanishing in points of a set $Z$ and higher-order vanishing along a general linear subspace fails to impose the expected number of independent conditions on forms of a fixed degree. The phenomenon was…

In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher…

The purpose of this article is to investigate the holomorphic vector fields tangent to a real hypersurface in $\mathbb C^2$ vanishing at an infinite type point.

In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set…

In this paper, we study rare events in spherical and Gaussian random geometric graphs in high dimensions. In these models, the vertices correspond to points sampled uniformly at random on the $d$ dimensional unit sphere or correspond to $d$…

We study Laguerre isotropic hypersurfaces in the Euclidean space, which are hypersurfaces whose Laguerre form is zero and the eigenvalues of the Laguerre tensor are constant and equal to $\lambda\geq 0$. We prove a rigidity theorem for the…

We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.

In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or $\lambda_1$ have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities.…

Let $X=\Lambda\backslash\mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $\delta$ denote the Hausdorff dimension of the limit set of $\Lambda$. We prove that for any compact subset…

We will generalize a Maximum Principle at Infinity in the parabolic case given by De Lima [Ann. Global Anal. Geom. ${\bf 20}$, 325-343 2001] and De Lima and Meeks [Indiana Univ. Math. Journal ${\bf 53}$ 5, 1211-1223 2004], for disjoints…

Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f…

We observe that the notions of a topological space being extremally disconnected, and of a continuous map of compact Hausdorff spaces being proper, and being surjective proper, can each be defined in terms of the Quillen lifting property…