Curve Theorem Research Articles

Rigidity theorems for holomorphic curves in a complex Grassmann manifold G(3,6)

In this paper, we prove some local rigidity theorems of holomorphic curves in a complex Grassmann manifold [Formula: see text] by moving frames. By applying our rigidity theorems, we also give a characterization of all homogeneous holomorphic two-spheres in [Formula: see text] classified by the second author.

Algebraic degeneracy and Uniqueness Theorems for Holomorphic Curves with Infinite Growth Index from a Disc into $${\mathbb {P}}^n({\mathbb {C}})$$ Sharing $$2n+2$$ Hyperplanes

Algebraic degeneracy and Uniqueness Theorems for Holomorphic Curves with Infinite Growth Index from a Disc into $${\mathbb {P}}^n({\mathbb {C}})$$ Sharing $$2n+2$$ Hyperplanes

Uniqueness theorem for holomorphic curves on the disc

Uniqueness theorem for holomorphic curves on the disc

Distribution value of algebraic curves and the Gauss maps on algebraic minimal surfaces

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in [Formula: see text] with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field.

Beck's Theorem for Plane Curves

Let $d\in\mathbb{Z}^+t$, $\mathbb{K}$ be a field of characteristic zero and $A$ be a nonempty finite subset of $\mathbb{K}^2$. Denote by $\mathcal{C}_{d,\mathbb{K}}$ the family of algebraic curves of degree $d$ in $\mathbb{K}^2$ and $\mathcal{C}_{\leq d,\mathbb{K}}:=\bigcup_{e=1}^d\mathcal{C}_{e,\mathbb{K}}$. For any $C_1\in \mathcal{C}_{d,\mathbb{K}}$, we say that $C_1$ is determined by $A$ if for any $C_2\in\mathcal{C}{d,\mathbb{K}}$ such that $C_2\cap A\supseteq C_1\cap A$, we have that $C_1=C_2$; we denote by $\mathcal{D}_{d,\mathbb{K}}(A)$ the family of elements of $\mathcal{C}_{d,\mathbb{K}}$ determined by $A$. Beck's theorem establishes that if $\mathbb{K}=\mathbb{R}$ and $A$ is not collinear, then $$|\mathcal{D}_{1,\mathbb{R}}(A)|=\Theta\left(|A|\min_{C\in \mathcal{C}_{1,\mathbb{R}}}|A\setminus C|\right).$$ In this paper we generalize Beck's theorem showing that for all $d\in\mathbb{Z}^+$, there exists a constant $c=c(d)>0$ such that if $\min_{C\in\mathcal{C}_{\leq d,\mathbb{K}}}|A\setminus C|>c,$ then $$|\mathcal{D}_{d,\mathbb{K}}(A)|=\Theta_d\left(|A|^d\prod_{e=1}^d\left(\min_{C\in \mathcal{C}_{\leq e,\mathbb{K}}}|A\setminus C|\right)^{d-e+1}\right).$$

Degenerated second main theorem for holomorphic curves into algebraic varieties

In this paper, under the refinement of the subgeneral position, we give an improvement for the Second Main Theorem with truncated counting functions of algebraically non-degenerate holomorphic curves into algebraic varieties [Formula: see text] intersecting divisors in subgeneral position with some index.

Slant Curve in Lorentzian BCV Spaces

The aim of this study is to examine the slant curves in Lorentzian manifolds with BCV (Bianchi-Cartan-Vranceanu) metrics. A practical form of the directional derivative in the presence of the BCV metrics is presented. Moreover, some theorems for slant curves in Lorentzian BCV manifolds are proved.

Some uniqueness theorems for holomorphic curves on annulus sharing hypersurfaces

In this article, we prove some uniqueness theorems for algebraically non-degenerate holomorphic curves on annulus sharing sufficiently many hypersurfaces in general position for Veronese embedding.

Applications of difference analogue of Cartan’s second main theorem for holomorphic curves

In this paper, we present refinement of earlier results on sufficient conditions for periodicity of a meromorphic function which are obtained by using the new version of Cartan’s second main theorem for the Casorati determinant.

The Gauss Map of Algebraic Complete Minimal Surfaces Omits Hypersurfaces in Subgeneral Position

The first aim of this paper is to show a second main theorem for algebraic curves into the n-dimensional projective space sharing hypersurfaces in subgeneral position. We then use it to study the value distribution of the generalized Gauss map of the complete (regular) minimal surfaces in \(\mathbb {R}^{m}\) with finite total curvature, as well as the unicity problem, sharing hypersurfaces in subgeneral position. Our results generalize and complete previous results in this area, especially the works of Chern and Osserman (J. Anal. Math. 19, 15–34, 1967), Jin and Ru (Differ. Geom. Appl. 25: 701–712, 2007).

A modification of the Nevanlinna–Cartan theory for holomorphic curve

In this paper, we prove some fundamental theorems for holomorphic curves on intersecting a finite set of fixed hyperplanes in general position in with modified counting and characteristic functions.

A result on the quasi-periodic solutions of forced isochronous oscillators at resonance

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity $$x'' + V'(x) + g(x) = e(t,x,x'),$$ where the assumptions on V, g and e are regular, described precisely in the introduction. Using a variant of Moser’s twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.

Geometric Adeles and the Riemann—Roch Theorem for 1-Cycles on Surfaces

The classical Riemann–Roch theorem for projective irreducible curves over perfect fields can be elegantly proved using adeles and their topological self-duality. This was known already to E. Artin and K. Iwasawa and can be viewed as a relation between adelic geometry and algebraic geometry in dimension one. In this paper we study geo- metric two-dimensional adelic objects, endowed with appropriate higher topology, on algebraic proper smooth irreducible surfaces over perfect fields. We establish several new results about adelic objects and prove topological self-duality of the geometric adeles and the discreteness of the function field. We apply this to give a direct proof of finite dimen- sion of adelic cohomology groups. Using an adelic Euler characteristic we establish an additive adelic form of the intersection pairing on the surfaces. We derive a direct and relatively short proof of the adelic Riemann–Roch theorem. Combining with the relation between adelic and Zariski cohomology groups, this also implies the Riemann–Roch theorem for surfaces.

Uniqueness theorems for holomorphic curves sharing moving hypersurfaces

In this article, we prove some uniqueness theorems for non-constant holomorphic curves of ℂ into ℙ n (ℂ) sharing moving hypersurfaces in general position for the Veronese embedding, ignoring multiplicity.

MANNHEIM PARTNER D-CURVES IN 3-DIMENSIONAL HEISENBERG GROUP IN3

In this study, we give some basic theorems for Mannheim partner curves in three dimensional Heisenberg group. We also found the relations between the curvatures and torsions of these associated curves.

A Uniqueness Theorem for Holomorphic Curves on Annulus Sharing 2n+3 Hyperplanes

In this paper, we will prove a theorem on unicity for linearly nondegenerate holomorphic curves on annulus ignoring multiplicity with 2n+3 hyperplanes.

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.

Effective Siegel's theorem for modular curves

We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.

Uniqueness theorems and uniqueness polynomials for holomorphic curves

We give some uniqueness theorems for nonconstant holomorphic curves and establish the existence of a class of strong uniqueness polynomials for algebraically nondegenerate holomorphic curves. We show that if X is a hypersurface defined by a polynomial in this class, then X is a unique range set for algebraically nondegenerate holomorphic curves.

Multivariate Pólya-Schur classification problems in the Weyl algebra

A multivariate polynomial is {\em stable} if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra $\A_n$ that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of P\'olya-Schur's notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy-Poincar\'e interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in $\A_1$ that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in $\A_n$ preserves stability if and only if its Fischer-Fock adjoint does. These are powerful multivariate extensions of the classical Hermite-Poulain-Jensen theorem, P\'olya's curve theorem and Schur-Mal\'o-Szeg\H{o} composition theorems. Examples and applications to strict stability preservers are also discussed.