Relations between length and instability of tubular extremal surfaces

Abstract

In the paper we study surfaces being extremals of the potential energy functional. In our case, the potential energy is the sum of two functionals, one being a functional of the area type, and the other being a functional of the volume density of forces. Extremal surfaces are stable if the second variation of the functional is sign-definite, otherwise they are unstable. In order to obtain the instability, we impose additional conditions on the surface and integrands, then we apply the properties of positive definite symmetric matrices, employ the Kronrod-Federer formula, the Cauchy-Bunyakovsky inequality, and the Weingarten homomorphism estimate. This allows us to estimate the second variation of the functional. Such technique, being a developing of an approach proposed by V.A. Klyachin, allows us to obtain conditions ensuring the instability. We establish that the length of the tubular extremal surface can be estimated in terms of the minimal and maximal $(n-1)$-dimensional measure of the cross-sections of the surface by hyperplanes. The obtained statement means that too long tubes with a non-zero mean curvature are unstable. The physical aspects of this phenomenon were considered in a work by V.A. Saranin.