Related papers: Landau's theorem for holomorphic curves in project…
The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex…
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…
Using Green's hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an…
It is shown that the canonical formulation of the abelian BF theory in D = 3 allows to obtain topological invariants associated to curves and points in the plane. The method consists on finding the Hamiltonian on-shell of the theory coupled…
Let $1\leq p\leq n$ be two positive integers. For a linearly nondegenerate holomorphic mapping $f\colon\mathbb{C}^p\rightarrow\mathbb{P}^n(\mathbb{C})$ of maximal rank intersecting a family of hyperplanes in general position, we obtain a…
In this paper we extend the notion of the Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. The main theorem on coincidence of the…
In this paper, we give some extension of fundamental theorems in Nevanlinna - Cartan theory for holomorphic curve on M punctured complex planes. As an application, we establish a result for uniqueness problem of holomorphic curve by inverse…
The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in…
On any complex smooth projective curve with positive genus, we construct Hilbert bundles that admit Hermitian--Einstein metrics. Our main constructive step is by investigating the arithmetic property of the upper half plane in Bridgeland's…
The main goal of the paper is to provide a quantitative lower bound greater than $1$ for the relative projection constant $\lambda(Y, X)$, where $X$ is a subspace of $\ell_{2p}^m$ space and $Y \subset X$ is an arbitrary hyperplane. As a…
Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show…
In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For…
We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for homogeneous ideals in polynomial rings. Our theorem allows us to give…
In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in $\mathbb{R}^n$, such that all the vertices of $\{0, 1\}^n \setminus \{\vec{0}\}$ are covered at least $k$ times, and $\vec{0}$ is…
We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension $2n+1$). We prove a codimension reduction theorem. We…
We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal…
Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation.
We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space…