Related papers: Entire holomorphic curves into $\mathbb{P}^n(\math…

There is a belief that the Ehrenfest theorem holds true universally. We demonstrate that for a classically nonrelativistic particle constrained on an $N-1$ ($N\geq 2$) curved hypersurface embedded in $N$ flat space, the theorem breaks down.

Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is…

Let $D$ be a disk in $\mathbb{R}^n$ and $f\in C^{r+2}(D, \mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of…

We consider the algebraic degeneracy of holomorphic curves from a point of view of meromorphic vector fields. Employing the notion of Jocabian sections introduced by W. Stoll, we establish a Second Main Theorem type inequality. As…

Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space $(\mathbb{C}P^3, g_a)$, including nearly K\"ahler $\mathbb{C}P^3$. Notably, all extrinsically homogeneous hypersurfaces are…

In a recent paper by Cook, et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here we expand the definition to hypersurfaces of any dimension and, 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.…

Let $D\subset\subset\mathbb{C}^n$ be a complex manifold of dimension $p\geq 2$ with $\C^2$ boundary in $\mathbb{C}^n$. Let $f$ be a $\C^1$ function on $bD$ and $V$ a generic and large enough family of complex $(n-p+1)$-planes. Let suppose…

The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness…

A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures (Dorfmeister-Neher's theorem) is given in a method applicable to the multiplicity two case.

In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : M^k --> C^N with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. The principal…

The paper develops an equidistribution theory of meromorphic mappings from a complete K\"ahler manifold with non-negative Ricci curvature into a complex projective manifold intersecting normal crossing divisors. When the domain manifolds…

Based on a geometric interpretation of Brotbek's symmetric differential forms, for the intersection family $\mathcal{X}$ of generalized Fermat-type hypersurfaces in $\mathbb{P}_{\mathbb{K}}^N$ defined over any field $\mathbb K$, we…

The purpose of this article is to prove that there exists a real smooth pseudoconvex hypersurface germ $(M,p)$ of D'Angelo infinite type in $\mathbb C^{n+1}$ such that it does not admit any (singular) holomorphic curve in $\mathbb C^{n+1}$…

The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and…

Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $\Gamma\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph…

The Milnor number, \mu(X,0), and the singularity genus, p_g(X,0), are fundamental invariants of isolated hypersurface singularities (more generally, of local complete intersections). The long standing Durfee conjecture (and its…

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…

We show that for any degree $d$ hypersurface $Y \subset X$ in a possibly singular projective variety $X \subset \mathbf{P}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$ for some explicit constant…

In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinct constant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$…