Kondrashov A.N. Liouville-Type Theorems for Tubular A-Surfaces

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https://doi.org/10.15688/mpcm.jvolsu.2026.1.2

Alexander N. Kondrashov
Candidate of Sciences (Physics and Mathematics), Associate Professor, Department of Computer Sciences and Experimental Mathematics, Volgograd State University
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https://orcid.org/0000-0003-1614-0393
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. This paper is devoted to establishing analogs of the geometric versions of the Liouville and Phragmen — Lindel of theorems, known from the works of  V.M. Miklyukov and his students (V.G. Tkachev, V.A. Klyachin, A.N. Kondrashov), for entire minimal tubes in Euclidean space and maximal ones in Minkowski spacetime. Specifically, we obtain two theorems that serve as such analogs for certain systems of divergence-type differential equations. More precisely, consider solutions f(x) = (f₁(x), . . . , f_N (x)) : D ⊂ R² → R^N , x = (x₁, x₂), N ≥ 3 of the above system, which define an proper immersion of D in R^N . To each such solution we associate a two-dimensional surface M in the Euclidean space R^N . The properties of some types of such surfaces are the same as those of surfaces of zero mean curvature with a positive-definite metric, in general, in pseudo-Euclidean space. Such surfaces M are called A-surfaces. By analogy with tubular minimal surfaces in Euclidean space, we introduce the concept of tubular A-surfaces (A-tubes) with respect to a given direction and study their global geometric properties. The two theorems have been established for A-tubes. The first theorem (a Liouville-type result) states that if an entire A-tube lies entirely in one of the two closed half-spaces into which a hyperplane divides the space R^N , then the tube must in fact be a subset of a hyperplane parallel to the bounding one. The second theorem concerns A-tubes with a semi-infinite projection and provides an estimate for the growth of the surface width. It represents a geometric version of the Phragmen – Lindel ´ of theorem. The proofs are based on ¨ capacity methods developed in the aforementioned works of V.M. Miklyukov

Key words: zero mean curvature type systems, Liouville-type theorems, A- surface, A-tubular surfaces, δ-harmonic functions, weighted capacity, Phragmen – ´ Lindelof principle

Kondrashov A.N. Liouville-Type Theorems for Tubular A-Surfaces is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 29 No. 1 2026, pp. 13-25

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