Related papers: Explicit Minimal Surface Models in $\mathbb{R}^5$ …

We prove that the hierarchical hull of any finite set of interior points, hierarchy rays, and boundary points in a hierarchically hyperbolic space (HHS) is quasi-median quasi-isometric to a CAT(0) cube complex of bounded dimension. Our…

Here we prove that for each Hamiltonian function $H\in \mathcal{C}^\infty(\mathbb{R}^4, \mathbb{R})$ defined on the standard symplectic $(\mathbb{R}^4, \omega_0)$, for which $M:=H^{-1}(0)$ is a non-empty compact regular energy level, the…

We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…

We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature…

This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…

As a generalization of semi-invariant submersions, we introduce conformal semi-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from…

We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete…

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are…

Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of these methods, however, learn only closed surfaces and are…

Nondegenerate covariance, correlation and spectral density matrices are necessarily symmetric or Hermitian and positive definite. The main contribution of this paper is the development of statistical data depths for collections of Hermitian…

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be…

The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean…

Explicit neural surface representations allow for exact and efficient extraction of the encoded surface at arbitrary precision, as well as analytic derivation of differential geometric properties such as surface normal and curvature. Such…

Conformal symmetry underlies the mathematical description of various two-dimensional integrable models (e.g. for their Lax representation, Poisson algebra, zero curvature representation,...) or of conformal models (for the anomalous Ward…

Let H:(M,p)->(M',p') be a formal mapping between two germs of real-analytic generic submanifolds in C^N with nonvanishing Jacobian. Assuming M to be minimal at p and M' holomorphically nondegenerate at p', we prove the convergence of the…

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the…

We consider general integrable systems on graphs as discrete flat connections with the values in loop groups. We argue that a certain class of graphs is of a special importance in this respect, namely quad-graphs, the cellular…

We present an intuitive diagrammatic representation of a new class of integrable $\s$-models. It is shown that to any given diagram corresponds an integrable theory that couples $N$ WZW models with a certain number of each of the following…

In this paper, we prove that a closed minimally immersed hypersurface $M^4\subset\mathbb S^5$ with constant $S:=\sum\limits_{i=1}^4\lambda_i^2$ and $A_3:=\sum\limits_{i=1}^4\lambda_i^3$ whose scalar curvature $R_M$ is nonnegative must be…