Some properties of the tubular minimal surfaces of arbitrary codimension

V. M. Miklyukov   ; V. G. Tkachev

Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 133-150

Résumé

A tubular surface is an immersion $u\colon M\to\mathbf R^n$ for which the section $\Pi\cap u(M)$ by an arbitrary hyperplane $\Pi$ orthogonal to a fixed vector $e\in\mathbf R^n$ is a compact set. For tubular minimal surfaces in $\mathbf R^n$ we prove that (a) if $\dim M=2$ and $u(M)$ lies in a half-space, then $u(M)$ also lies in some hyperplane; and (b) if $\dim M\geqslant3$, then a tubular minimal surface lies in the layer between two hyperplanes orthogonal to $e$. We obtain the corresponding results about the structure of the Gaussian image of two-dimensional tubular minimal surfaces. The case $\operatorname{codim}M=1$ was investigated earlier (RZh.Mat., 1987, 2 B 807). Bibliography: 19 titles.

Bibliographie

[1] Kobayasi Sh., Nomadzu K., Osnovy differentsialnoi geometrii, t. 2, Nauka, M., 1981

[2] Nitsche J. C. C., “A uniqueness theorem of Bernschtein's type for minimal surfaces in cilindrical coordinates”, J. Math. Mech., 6 (1957), 859–864 | MR | Zbl

[3] Vedenyapin A. D., Miklyukov V. M., “Vneshnie razmery trubchatykh minimalnykh giperpoverkhnostei”, Matem. sb., 131(173) (1986), 240–250 | MR | Zbl

[4] Miklyukov V. M., “O nekotorykh svoistvakh trubchatykh minimalnykh poverkhnostei v $\mathbf{R}^n$”, DAN SSSR, 247:3 (1979), 549–552 | MR | Zbl

[5] Miklyukov V. M., “Ob odnom novom podkhode k teoreme Bernshteina i blizkim voprosam uravnenii tipa minimalnoi poverkhnosti”, Matem. sb., 108(150) (1979), 268–289 | MR

[6] Grigoryan A. A., “O suschestvovanii polozhitelnykh fundamentalnykh reshenii uravnenii Laplasa na rimanovykh mnogoobraziyakh”, Matem. sb., 128(170) (1985), 354–363 | MR | Zbl

[7] Simons J., “Minimal varieties in Rimannian manifolds”, Ann. Math., 88:1 (1968), 62–105 | DOI | MR | Zbl

[8] Miklyukov V. M., Tkachev V. G., “O stroenii v tselom vneshne polnykh minimalnykh poverkhnostei v $\mathbf{R}^3$”, Izv. vuzov. Matem., 1987, no. 7, 30–36 | MR | Zbl

[9] Tkachev V. G., “Priznaki parabolichnosti konformnogo tipa dvumernykh minimalnykh poverkhnostei v $\mathbf{R}^n$”, Vsesoyuznaya konferentsiya po geometrii “v tselom”, Novosibirsk, 1987, S. 119

[10] Barbashov B. M., Nesterenko V. V., Model relyativistskoi struny v fizike adronov, Energoatomizdat, M., 1987

[11] Barbashov B. M., Nesterenko V. V., “Superstruny – novyi podkhod k edinoi teorii fundamentalnykh vzaimodeistvii”, UFN, 150:4 (1986), 489–524

[12] Kurant R., Printsip Dirikhle, konformnye otobrazheniya i minimalnye poverkhnosti, IL, M., 1953

[13] Suvorov G. D., Obobschennyi “printsip dliny i ploschadi” v teorii otobrazhenii, Naukova dumka, Kiev, 1985 | MR

[14] Fomenko A. T., “O skorosti rosta i naimenshikh ob'emakh globalno minimalnykh poverkhnostei v kobordizmakh”, Tr. seminara po vekt. i tenz. analizu, vyp. 21, MGU, M., 1983, 3–12 | MR

[15] Fomenko A. T., “Algebraicheskaya struktura nekotorykh klassov vpolne integriruemykh gamiltonovykh sistem na algebrakh Li”, Geom. teoriya funktsii i topol., Naukova dumka, Kiev, 1981, 85–126 | MR

[16] Fomenko A. T., Variatsionnye metody v topologii, Nauka, M., 1982 | MR

[17] Burago Yu. D., Zalgaller V. A., Geometricheskie neravenstva, Nauka, L., 1980 | MR | Zbl

[18] Kasue A., “Gap theorems for minimal submanifolds of Euclidean spase”, J. Math. Soc. Japan, 38:3 (1986), 473–492 | DOI | MR | Zbl

[19] Schoen R., “Uniqueness, symmetry and embeddedness of minimal surfaces”, J. of Diff. Geom., 18 (1983), 791–809 | MR | Zbl