Abstract:
We investigate the properties of stable (and unstable) hypersurfaces
with prescribed mean curvature in Euclidean space and establish
some necessary and sufficient tests for stability stated in terms of
the external geometric structure of the surface. We prove an
analogue of a well-known theorem of A. D. Aleksandrov that
generalizes the variational property of the sphere and find an exact
estimate for the extent of a stable tubular surface of constant mean
curvature. Our method is based on an analysis of the first and
second variations of area-type functionals for the surfaces under
consideration.
\Bibitem{Kly06}
\by V.~A.~Klyachin
\paper On some properties of stable and unstable surfaces with prescribed mean curvature
\jour Izv. Math.
\yr 2006
\vol 70
\issue 4
\pages 717--730
\mathnet{http://mi.mathnet.ru/eng/im587}
\crossref{https://doi.org/10.1070/IM2006v070n04ABEH002325}
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Linking options:
https://www.mathnet.ru/eng/im587
https://doi.org/10.1070/IM2006v070n04ABEH002325
https://www.mathnet.ru/eng/im/v70/i4/p77
This publication is cited in the following 13 articles:
Vladimir Klyachin, “THE ESTIMATES OF PRINCIPLE FREQUENCY OF DOMAINS ON RIEMANNIAN MANIFOLDS AND MINIMAL SURFACES STABILITY”, Mathematical Physics and Computer Simulation, 27:3 (2024), 15
M. B. Karmanova, “O minimalnykh poverkhnostyakh nad mnogoobraziyami Karno proizvolnoi glubiny”, Matem. tr., 25:1 (2022), 74–101
M. B. Karmanova, “Minimal Surfaces Over Carnot Manifolds”, Sib. Adv. Math., 32:3 (2022), 211
N. M. Poluboyarova, “Relations between length and instability of tubular extremal surfaces”, Ufa Math. J., 13:1 (2021), 77–84
N. M. Poluboyarova, “On stable extremals of the potential energy functional”, Siberian Math. J., 62:3 (2021), 482–488
A. A. Klyachin, V. A. Klyachin, “Research in the field of geometric analysis at Volgograd state university”, Mathematical Physics and Computer Simulation, 23:2 (2020), 5–21
N. M. Poluboyarova, “On instability of extremals of potential energy functional”, Ufa Math. J., 10:3 (2018), 77–85
N. M. Poluboyarova, “Some Properties of Extremals of the Functional of Potential Energy”, J. Math. Sci. (N. Y.), 252:2 (2021), 225–231
N. M. Poluboyarova, “Uravneniya ekstremalei funktsionala potentsialnoi energii”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 5(36), 60–72
Kubis S., Wojcik W., “Geometric approach to nuclear pasta phases”, Phys. Rev. C, 94:6 (2016), 065805
Z. S. Akhtemov, N. N. Stepanyan, V. G. Fainshtein, G. V. Rudenko, “Structure of the magnetic field at altitudes of 1–1.15 solar radii”, Astron. Rep., 60:9 (2016), 839
V. A. Klyachin, E. G. Grigoreva, “Chislennoe issledovanie ustoichivosti ravnovesnykh poverkhnostei s ispolzovaniem paketa NumPy”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2015, no. 2(27), 17–30
M. M. Molodenskiǐ, L. I. Starkova, “Solar structures related to coronal holes”, Astron. Rep., 51:12 (2007), 1036
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