Related papers: Affine surface area
Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential…
An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution…
Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact…
We consider the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature),…
In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in…
We identify the moduli space of complex affine surfaces with the moduli space of regular meromorphic connections on Riemann surfaces and show that it satisfies a corresponding universal property. As a consequence, we identify the tangent…
We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such…
We prove a hyperplane inequality for the surface area of projection bodies.
We prove that an affine cone $X$ admits a surjective morphism from an affine space if and only if $X$ is unirational.
The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.
In this revised version, we add some expository material and references and make some minor corrections.
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…
The article is devoted to affine and wrap algebras over quaternions and octonions. Residues of functions of quaternion and octonion variables are studied. They are used for construction of such algebras. Their structure is investigated.
We give a combinatorial/geometric argument of the classical result that an affine connection, which is both torsion free and curvature free, is locally an affine space.
We introduce a simple property, affine invariance, which characterizes within the class of fuzzy topological spaces those which are induced from an underlying topology on the space. We illustrate it by considering the simple notions of…
Chapters : Old and new inequalities; Surfaces with $\chi=1$ and the bicanonical map; Surfaces with $p_g=4$; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF…
We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a…
By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…
In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct interesting actions…
The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both…