Related papers: A sharp necessary condition for rectifiable curves…

In this article, we study a calibrated version of Reifenberg theorem "with holes". In particular we study sets that are suitably approximable at all points and scales by calibrated planes and show that, without any additional hypotheses on…

In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by…

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture in 1977 that an affine maximal graph of a smooth, locally uniformly convex function on two-dimensional Euclidean space…

The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the…

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first…

In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version…

In this paper, we discuss some necessary and sufficient condition for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in…

Given a Riemannian metric on the 2-sphere, sweep the 2-sphere out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as…

In 1962 P\'osa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of P\'osa's Conjecture. They also proved that it…

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean…

We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

An isoperimetric inequality on the Hamming cube for exponents $\beta\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $\beta\ge \log_2(3/2)\approx 0.585$. Improved bounds are also obtained at the…

The study of curvature properties of homogeneous Finsler spaces with $(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler…

For a compact connected set $X\subseteq \ell^{\infty}$, we define a quantity $\beta'(x,r)$ that measures how close $X$ may be approximated in a ball $B(x,r)$ by a geodesic curve. We then show there is $c>0$ so that if $\beta'(x,r)>\beta>0$…

We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of…

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power…

Let $\mathbb{G}$ be any Carnot group. We prove that, if a subset of $\mathbb{G}$ is contained in a rectifiable curve, then it satisfies Peter Jones' geometric lemma with some natural modifications. We thus prove one direction of the…

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. {\color{blue}For a metric space, we define its boundary to be the completion of the space minus…

Let $\mathbb{K}$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $\mathbb{K}$, containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$, $nP_2+P_3$, and…