arXiv subjects

Dmitrii Korshunov

Publications and source records attributed to Dmitrii Korshunov.

5 recordsLinked to original sources

arXiv2026

Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces

We prove that if $M$ is a closed Riemannian surface of diameter $d$ and area $v$ with sectional curvature in the $[-1,1]$ interval, then a closed geodesic net of length $l$ has at most $f(l,d,v)$ branch points, where $f(l,d,v)=(400\bar{l})^{(180\bar{l})^4}$ for $\bar{l}= \max\{l, \frac{\exp(d)}{\min\{1, \frac{v}{4}\}}\}$ This answers a question posed by S. Becker-Kahn. We also prove that for each geodesic net in the Euclidean plane with at most n unbalanced (boundary) vertices such that all its unbalanced vertices have degree 1, the number of balanced vertices of degree $\ge 3$ (=branch points) does not exceed $(25n)^{2n^2}$. This answers a question posed in [GM] and [NP].

math.DG↗

arXiv2026

Polynomial contractions of $\mathbb C^d$ and degree growth

We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincaré-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which is holomorphically but not algebraically linearizable.

math.CV↗

arXiv2024

Sub-twistor metrics

We consider a natural distance function on the period space of a hyperkähler manifold associated to non-holonomic constraints imposed by twistor lines. These metrics were introduced by Verbitsky in the context of the global Torelli theorem for hyperkäler manifolds. We show that they are Finsler and explicitly describe their Finsler norms. To achieve this goal we consider a class of distance functions (called here sub-conic metrics) that slightly generalize sub-Riemann metrics. The main technical result is the statement that every sub-conic metric is sub-Finsler. The content and methods of the paper lie within basic metric geometry. The complex geometrical background and motivation are isolated in the appendix.

math.DG↗

arXiv2024

A note on lattice knots

The aim of this note is to share the observation that the set of elementary operations of Turing on lattice knots can be reduced to just one type of simple local switches.

math.GT↗

arXiv2022

Moduli spaces of polygons and deformations of polyhedra with boundary

We prove a conjecture of Ian Agol: all isometric realizations of a polyhedral surface with boundary sweep out an isotropic subset in the Kapovich-Millson moduli space of polygons isomorphic to the boundary. For a generic polyhedral disk we show that boundaries of its isometric realizations make up a Lagrangian subset. As an application of this result, we obtain a new solution to the problem of Richard Kenyon about spanning domes of piecewise linear curves comprised of unit intervals in R^3.

math.SG↗