Related papers: How curved is a random complex curve?
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric.…
We prove a uniform Fourier extension-restriction estimate for a certain class of curves in d-dimensional Euclidean space.
We consider the curves whose all normal planes are at the same distance from a fixed point and obtain some characterizations of them in the 3-dimensional Euclidean space.
Contours may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold and can be studied using statistical shape analysis.…
This work addresses a modification of the random geometric graph (RGG) model by considering a set of points uniformly and independently distributed on the surface of a $(d-1)$-sphere with radius $r$ in a $d-$dimensional Euclidean space,…
We consider a quantum particle constrained to a curved layer of a constant width built over an infinite smooth surface. We suppose that the latter is a locally deformed plane and that the layer has the hard-wall boundary. Under this…
In this paper we obtain density estimates for compact surfaces immersed in R^n with total boundary curvature less than 4pi and with sufficiently small L^p norm of the mean curvature, p>2. Our results generalize the main results in [2]. We…
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…
We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More…
We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally…
In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is…
This is a continuation of "Rational curves on hypersurfaces of low degree", math.AG/0203088. We prove that if d^2+d+1 < n and d > 2, then for a general hypersurface X_d in P^n of degree d, for each degree e the space of rational curves of…
A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…
We classify ACM curves contained in a surface of degree d in $\mathbb{P}^{3}$ in terms of weak admissible pairs. In the case of a very general smooth determinantal quartic surface, we provide a geometric description of these curves and…
Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected…
A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…
We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…
We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in…
Non-relativistic particles that are effectively confined to two dimensions can in general move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we…