arXiv subjects

Egor Morozov

Publications and source records attributed to Egor Morozov.

6 recordsLinked to original sources

arXiv2026

Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class

We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the $k$-th normalized Laplace-Beltrami eigenvalue functional $\bar\lambda_k$ in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain $\mathbb S^1$-equivariant harmonic maps $\mathbb T^2\to\mathbb S^3$. For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for $\bar\lambda_1$ in this conformal class with the value of $\bar\lambda_1$ being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for $\bar\lambda_1$ in its conformal class, then it is $\mathbb S^1$-equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in $\mathbb S^3$ called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals.

math.DG↗

arXiv2022

Surfaces containing two isotropic circles through each point

We prove (under some technical assumptions) that each surface in $\mathbb R^3$ containing two arcs of parabolas with axes parallel to $Oz$ through each point has a parametrization $\left(\frac{P(u,v)}{R(u,v)},\frac{Q(u,v)}{R(u,v)},\frac{Z(u,v)}{R^2(u,v)}\right)$ for some $P,Q,R,Z\in\mathbb R[u,v]$ such that $P,Q,R$ have degree at most 1 in $u$ and $v$, and $Z$ has degree at most 2 in $u$ and $v$. The proof is based on the observation that one can consider a parabola with vertical axis as an isotropic circle; this allows us to use methods of the recent work by M. Skopenkov and R. Krasauskas in which all surfaces containing two Euclidean circles through each point are classified. Such approach also allows us to find a similar parametrization for surfaces in $\mathbb R^3$ containing two arbitrary isotropic circles through each point (under the same technical assumptions). Finally, we get some results concerning the top view (the projection along the $Oz$ axis) of the surfaces in question.

math.DG↗

arXiv2022

Index of bipolar surfaces to Otsuki tori

For each rational number $p/q\in (1/2,\sqrt 2/2)$ one can construct an $\mathbb S^1$-equivariant minimal torus in $\mathbb S^3$ called Otsuki torus and denoted by $O_{p/q}$. The Lawson's bipolar surface construction applied to $O_{p/q}$ gives a minimal torus $\widetilde O_{p/q}$ in $\mathbb S^4$. In this paper we give upper and lower bounds on the Morse index and the nullity of these tori for $p/q$ close to $\sqrt 2/2$. We also state a numerically assisted conjecture concerning the general case.

math.DG↗

arXiv2022

On the Index of Fraser-Sargent-type minimal surfaces

Fraser-Sargent surfaces are free boundary minimal surfaces in the four-dimensional unit Euclidean ball. Extended infinitely they define immersed minimal surfaces in the Euclidean space. In the present paper we compute the Morse index and the nullity of these extended minimal surfaces. The parts of these surfaces outside the ball are exterior free boundary minimal surfaces. We provide a numerical evidence that they are stable. As a corollary of these results we obtain a lower bound on the index of Fraser-Sargent surfaces inside the ball. The obtained lower bound is not sharp. We provide computational experiments and state a conjecture about an improved index lower bound. Independently of it we also find an upper bound on the index of Fraser-Sargent surfaces inside the ball.

math.DG↗

arXiv2020

Symmetries of 3-polytopes with fixed edge lengths

We consider an interesting class of combinatorial symmetries of polytopes which we call \emph{edge-length preserving combinatorial symmetries}. These symmetries not only preserve the combinatorial structure of a polytope but also map each edge of the polytope to an edge of the same length. We prove a simple sufficient condition for a polytope to realize all edge-length preserving combinatorial symmetries by isometries of ambient space. The proof of this condition uses Cauchy's rigidity theorem in an unusual way.

math.MG↗

arXiv2017

Generalized problem of Apollonius

The aim of this paper is to generalize Apollonius' problem. The problem is to construct a circle that is tangent to three given circles in a plane. We find the maximum possible number of solution circles in the case of more than the three given circles. We show that if all the given circles are not tangent at the same point, then there exist at most six solutions in the case of the four given generalized circles and there exist at most four solutions in the case of the five given generalized circles. We also describe all quadruples of generalized circles with exactly six solutions.

math.HO↗