Related papers: Minimal surface systems, maximal surface systems, …
We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat $H$ minor is quasi-isometric to a graph with no $H$ minor, for an arbitrary…
Motivated by Calabi's calculation of the second variation sign for locally strongly convex affine maximal surfaces in equiaffine geometry, we first prove that every Calabi extremal surface is also maximal in the Calabi affine geometry. By…
In this article we give a construction of the resolution graphs of hypersurface surface singularities (X_k,0) given by generalized Iomdin series. All these resolution graphs are coordinated by an ``universal bi-colored graph'' which is…
Discrete Lagrangian Systems on graphs are considered. Vector-valued closed differential 2-form on the space of solutions is constructed. This form takes values in the first homology group of the graph. This construction generalizes the…
We review the construction of Lagrangians for higher spin fields of mixed symmetry in the framework of graded geometry. The main advantage of the graded formalism in this context is that it provides universal expressions, in the sense that…
For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $\gamma$ projects this into the…
A covariant pseudodifferential calculus on Riemann surfaces, based on the Krichever-Novikov global picture, is presented. It allows defining scalar and matrix KP operators, together with their reductions, in higher genus. Globally defined…
This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. The Lagrange map from the Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite dimensional…
For a Lagrangian system with nonholonomic constraints, we construct extensions of the equations of motion to sets of second-order ordinary differential equations. In the case of a purely kinetic Lagrangian, we investigate the conditions…
We explicitly compute certain Douglas algebras that are invariant under both the Bourgain map and the minimal envelope map. We also compute the Bourgain algebra and the minimal envelope of the maximal subalgebras of a certain singly…
We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way…
The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with…
In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved…
We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in $\mathbb{R}^{n+1}$ $(n \geq 7$) to arbitrary codimension and each $n \geq 3$. We then apply the procedure…
The so called Jenkins-Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either…
We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a klt pair $(X,\Delta)$ can be detected on the base of the $(K_{X}+\Delta)$-trivial reduction map. Thus we show that…
This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portoro\v{z} on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of…
We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat…
Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line $K^2=2\chi-6$ or are somewhat scattered. A…
We present globally supersymmetric models of gauged scale covariance in ten, six, and four-dimensions. This is an application of a recent similar gauging in three-dimensions for a massive self-dual vector multiplet. In ten-dimensions, we…