Related papers: Extremal general affine surface areas

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionals parametric surfaces in R3 as the sum of the area integral with an anisotropic term of suitable form. In the class of parametric surfaces with…

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than $\pi$) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with…

Let $1\leq i \leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\subset\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\subset \mathbb{R}^n$: \begin{align*} &V_i \big(…

For fixed g and T we show that finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech groups contain a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we show that any…

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety,…

In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By…

This paper establishes strong profinite rigidity results for K\"ahler groups, showing that certain groups are determined within the class of residually finite K\"ahler groups by their profinite completion. Examples include products of…

We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in…

Fractal geometry and analysis constitute a growing field, with numerous applications, based on the principles of fractional calculus. Fractals sets are highly effective in improving convex inequalities and their generalisations. In this…

For a multigraph $F$, the $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$. Conlon and Lee conjectured that if $k$ is even, then the…

This work studies the Fourier transform of the characteristic function of planar convex bodies averaged over affine transformations. We establish lower and upper bounds on the latter quantities in terms of the geometric properties of the…

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating…

We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.

We study aspects of entanglement and extremal surfaces in various families of spacetimes exhibiting cosmological, Big-Crunch, singularities, in particular isotropic $AdS$ Kasner. The classical extremal surface dips into the bulk radial and…

We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many…

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given…

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups…

It is shown that each continuous even Minkowski valuation on convex bodies of degree $1 \leq i \leq n - 1$ intertwining rigid motions is obtained from convolution of the $i$th projection function with a unique spherical Crofton…

Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f…

In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces…