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 $S\sp n$ into ${\bf R}\sp {n+1}$ with prescribed Gauss curvature and its solution by variational methods

Author: V. I. Oliker
Journal: Trans. Amer. Math. Soc. 295 (1986), 291-303
MSC: Primary 53C45; Secondary 49F99, 58G30
DOI: https://doi.org/10.1090/S0002-9947-1986-0831200-1
MathSciNet review: 831200
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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.

[A] A. D. Aleksandrov, Convex polyhedrons, GITTL, Moscow and Leningrad, 1950 (Russian); German transl., Akademie-Verlag, Berlin, 1958.
[A1] -, On the mixed volumes of convex bodies, Mat. Sb. 3(45) (1938).
[A2] -, Intrinsic geometry of convex surfaces, "Nauka", 1948 (Russian); German transl., Akademie-Verlag, Berlin, 1955.
[A3] -, Uniqueness theorems for surfaces in the large. III, Vestnik Leningrad Univ. 13 (1958), no. 7, 14-26. MR 0102112 (21:907)
[B] I. Bakelman, A variational problem associated with the Monge-Ampère equation, Soviet Math. 2 (1961), no. 6, 1585-1589. MR 0166501 (29:3776)
[B1] -, Variational problems and elliptic Monge-Ampère equations, J. Differential Geometry 18 (1983), 669-699. MR 730922 (85h:58045)
[BH] W. Blaschke and G. Herglotz, Über die Realisierung einer geschlossenen Flächen mit vorgeschriebenem Bogenelement..., S.-B. Bayer Akad. Wiss. 2 (1937), 229-230.
[Bu] H. Buseman, Convex surfaces, Wiley, New York, 1958. MR 0105155 (21:3900)
[CH] R. Courant and D. Hilbert, Methods of mathematical physics, vol. II, Wiley, New York, 1962. MR 1013360 (90k:35001)
[D] Ph. Delanoë, Plongements radiaux ${S^n} \to {R^{n + 1}}$ a courbure de Gauss positive prescrite, preprint.
[FJ] W. Fenchel and B. Jessen, Mengen funktionen und konvexe Körper, Dansk. Vid. Selsk. Math.-Fys. Medd. 16 (1938).
[H] D. Hilbert, Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen, Leipzig and Berlin, 1915.
[M] H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), 447-495. MR 1511220
[O] V. I. Oliker, Hypersurfaces in ${R^{n + 1}}$ with prescribed Gaussian curvature and related equations of Monge-Ampère type, Comm. Partial Differential Equations 9 (1984), 807-838. MR 748368 (85h:53047)
[P] A. V. Pogorelov, Extrinsic geometry of convex surfaces, "Nauka", Moscow, 1969 (Russian); English transl., Transl. Math. Monographs, vol. 35, Amer. Math. Soc., Providence, R.I., 1973. MR 0244909 (39:6222)
[P1] -, The Minkowski multidimensional problem, "Nauka", Moscow, 1975 (Russian); English transl., Wiley, New York, 1978. MR 0478079 (57:17572)
[V] Yu. 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. MR 0224037 (36:7084)
[Y] S. T. Yau, Problem Section, Seminar on Differential Geometry, Ed. by S. T. Yau, Ann. of Math. Studies, no. 102, Princeton Univ. Press, 1982, pp. 669-706. MR 645762 (83e:53029)