## Duke Mathematical Journal

### Hypersurfaces of prescribed curvature measure

#### Abstract

We consider the corresponding Christoffel–Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$th curvature measures ($k\textgreater 0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial a priori $C^{2}$-estimate for the corresponding curvature equation on $\mathbb{S}^{n}$.

#### Article information

Source
Duke Math. J., Volume 161, Number 10 (2012), 1927-1942.

Dates
First available in Project Euclid: 27 June 2012

https://projecteuclid.org/euclid.dmj/1340801627

Digital Object Identifier
doi:10.1215/00127094-1645550

Mathematical Reviews number (MathSciNet)
MR2954620

Zentralblatt MATH identifier
1254.53073

#### Citation

Guan, Pengfei; Li, Junfang; Li, YanYan. Hypersurfaces of prescribed curvature measure. Duke Math. J. 161 (2012), no. 10, 1927--1942. doi:10.1215/00127094-1645550. https://projecteuclid.org/euclid.dmj/1340801627

#### References

• [1] A. D. Alexandrov, Zur theorie der gemishchten volumina von knovexen korpern, 2, Math. USSR-Sb. 2 (1937), 1205–1238.
• [2] A. D. Alexandrov, Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35 (1942), 131–134.
• [3] C. Berg, Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk. 37 (1969), no. 6, 64 pp.
• [4] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261–301.
• [5] L. Caffarelli, L. Nirenberg, and J. Spruck, “Nonlinear second order elliptic equations IV: Starshaped compact Weingarten hypersurfaces” in Current Topics in Partial Differential Equations, Kinokuniya, Tokyo, 1986, 1–26.
• [6] S. Y. Cheng and S. T. Yau, On the regularity of the solution of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), 495–516.
• [7] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.
• [8] W. J. Firey, Christoffel’s problem for general convex bodies, Mathematika 15 (1968), 7–21.
• [9] P. Guan and Y. Li, $C^{1,1}$ estimates for solutions of a problem of Alexandrov, Comm. Pure Appl. Math. 50 (1997), 789–811.
• [10] P. Guan and Y. Li, Unpublished research notes, 1995.
• [11] P. Guan, C. Lin, and X.-N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not. IMRN 2009, no. 11, 1947–1975.
• [12] P. Guan and X.-N. Ma, The Christoffel-Minkowski problem, I: Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), 553–577.
• [13] P. Guan and X.-N. Ma, “Convex solutions of fully nonlinear elliptic equations in classical differential geometry” in Geometric Evolution Equations, Contemp. Math. 367, Amer. Math. Soc., Providence, 2005, 115–127.
• [14] H. Lewy, On differential geometry in the large, I: Minkowski’s problem, Trans. Amer. Math. Soc. 43 (1938), 258–270.
• [15] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394.
• [16] V. I. Oliker, Existence and uniqueness of convex hypersurfaces with prescribed Gaussian curvature in spaces of constant curvature, Sem. Inst. Mate. Appl. “Giovanni Sansone”, Univ. Studi Firenze, 1983, 1–39.
• [17] A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sbornik N.S. 31(73) (1952), 88–103.
• [18] A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, Transl. Math. Monogr. 35, Amer. Math. Soc., Providence, 1973.
• [19] A. V. Pogorelov, The Minkowski Multidimensional Problem, V. H. Winston, Washington, D.C., 1978.
• [20] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1993.