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As their smooth analogue the irreducible symplectic varieties appear as elementary bricks in the generalizations of the Bogomolov decomposition theorem (arXiv:math/0402243, arXiv:2012.00441). Let $S$ be a K3 surface; generalizing the Fujiki…

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly on compacts in $M$ by proper conformal…

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic…

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial…

We study the infinitesimal variation of Hodge structure associated with families of reduced algebraic curves with singularities. The analysis applies to curves beyond the nodal case and is not restricted to plane curves, encompassing curves…

We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first author. In this paper we further extend it to the singular…

Hyperbolic polynomials elegantly encode a rich class of convex cones that includes polyhedral and spectrahedral cones. Hyperbolic polynomials are closed under taking polars and the corresponding cones, the derivative cones, yield…

We study the cone of transverse measures to a fixed geodesic lamination on an infinite type hyperbolic surface. Under simple hypotheses on the metric, we give an explicit description of this cone as an inverse limit of finite-dimensional…

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…

Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have non-trivial Gaussian curvature localised at tips, curved interfaces, and…

We define branched bending deformations as deformations supported on a piecewise totally geodesic complex of $(n-1)$-dimensional faces meeting along $(n-2)$-dimensional branching loci. These are a generalization of bending deformations, as…

This paper considers the cohomology and bounded interpolation of nonstandard finite element complexes, e.g. Stokes, Hessian, Elasticity, divdiv. Compared to the standard finite element exterior calculus, the main challenge is the existence…

We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a…

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine…

The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension…

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of…

The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…

In this note we explicitly give all the equivalent classes of deformations of the 5-dimensional Heisenberg Lie algebra $\mathfrak{h}_2$ over complex or real number fields. We show that there are altogether 20 infinitesimal deformations…

We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3-sphere. We use Baker-Akhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral…