The problem of embedding 𝑆ⁿ into 𝑅ⁿ⁺¹ with prescribed Gauss curvature and its solution by variational methods

Oliker, V. I.

doi:10.1090/S0002-9947-1986-0831200-1

The problem of embedding $S^ n$ into $\textbf {R}^ {n+1}$ with prescribed Gauss curvature and its solution by variational methods
HTML articles powered by AMS MathViewer

by V. I. Oliker PDF

Trans. Amer. Math. Soc. 295 (1986), 291-303 Request permission

Abstract:

A way to recover a closed convex hypersurface from its Gauss curvature is to find a positive function over ${S^n}$ whose graph would represent the hypersurface in question. Then one is led to a nonlinear elliptic problem of Monge-Ampère type on ${S^n}$. Usually, geometric problems involving operators of this type are too complicated to be suggestive for a natural functional whose critical points are candidates for solutions of such problems. It turns out that for the problem indicated in the title, such a functional exists and has interesting geometric properties. With the use of this functional, we obtain new existence results for hypersurfaces with prescribed curvature as well as strengthen some that are already known.

References

Convex polyhedrons

On the mixed volumes of convex bodies

3(45)

Intrinsic geometry of convex surfaces

A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. III, Vestnik Leningrad. Univ. 13 (1958), no. 7, 14–26 (Russian, with English summary). MR 0102112
I. Ja. Bakel′man, A variational problem associated with the Monge-Ampère equation, Leningrad. Gos. Ped. Inst. Učen. Zap. 238 (1962), 119–131 (Russian). MR 0166501
Ilya J. Bakelman, Variational problems and elliptic Monge-Ampère equations, J. Differential Geom. 18 (1983), no. 4, 669–699 (1984). MR 730922

Über die Realisierung einer geschlossenen Flächen mit vorgeschriebenem Bogenelement...

2

Herbert Busemann, Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, no. 6, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. MR 0105155
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360

Plongements radiaux

a courbure de Gauss positive prescrite

Mengen funktionen und konvexe Körper

16

Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen

Hermann Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), no. 4, 447–495 (German). MR 1511220, DOI 10.1007/BF01445180
V. I. Oliker, Hypersurfaces in $\textbf {R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampère type, Comm. Partial Differential Equations 9 (1984), no. 8, 807–838. MR 748368, DOI 10.1080/03605308408820348
A. V. Pogorelov, Vneshnyaya geometriya vypuklykh poverkhnosteĭ, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0244909
Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg. MR 0478079
Ju. A. Volkov, Estimate of the deformation of a convex surface as a function of the change in its intrinsic metric, Dokl. Akad. Nauk SSSR 178 (1968), 1238–1240 (Russian). MR 0224037
Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR 645762