April  2014, 34(4): 1269-1284. doi: 10.3934/dcds.2014.34.1269

Prescription of Gauss curvature on compact hyperbolic orbifolds

1. 

Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse cedex 9, France

Received  October 2012 Revised  March 2013 Published  October 2013

In this paper, we generalize a result by Alexandrov on the Gauss curvature prescription for Euclidean convex bodies. We prove an analogous result for hyperbolic orbifolds. In addition to the duality theory for convex sets, our main tool comes from optimal mass transport.
Citation: Jérôme Bertrand. Prescription of Gauss curvature on compact hyperbolic orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1269-1284. doi: 10.3934/dcds.2014.34.1269
References:
[1]

A. D. Aleksandrov, Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Dokl.) Acad. Sci. URSS (N. S.), 35 (1942), 131-134.

[2]

A. D. Alexandrov, "Convex Polyhedra," Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.

[3]

I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations," With an obituary for the author by W. Rundell, Edited by S. D. Taliaferro, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-69881-1.

[4]

T. Barbot, F. Béguin and A. Zeghib, Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: Application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier (Grenoble), 61 (2011), 511-591. doi: 10.5802/aif.2622.

[5]

J. Bertrand, Existence and uniqueness of optimal maps on Alexandrov spaces, Adv. Math., 219 (2008), 838-851. doi: 10.1016/j.aim.2008.06.008.

[6]

D. Burago, Y. Burago and S. Ivanov, "A Course in Metric Geometry," Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.

[7]

G. Carlier, On a theorem of Alexandrov, J. Nonlinear Convex Anal., 5 (2004), 49-58.

[8]

F. Fillastre, Fuchsian convex bodies: Basics of Brunn-Minkowski theory, Geom. Funct. Anal., 23 (2013), 295-333. doi: 10.1007/s00039-012-0205-4.

[9]

W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.

[10]

P. Guan, C. Lin and X.-N. Ma, The existence of convex body with prescribed curvature measures,, Int. Math. Res. Not. IMRN, 2009 (): 1947.  doi: 10.1093/imrn/rnp007.

[11]

I. Iskhakov, "On Hyperbolic Surfaces Tesselations and Equivariant Spacelike Convex Polyhedra," Ph.D thesis, Ohio State University, 2000.

[12]

L. Kantorovitch, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N. S.), 37 (1942), 199-201.

[13]

F. Labourie and J.-M. Schlenker, Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante, Math. Ann., 316 (2000), 465-483. doi: 10.1007/s002080050339.

[14]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.

[15]

_______, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.

[16]

V. I. Oliker, Embedding $ S^n$ into $ R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $ S^n$, Adv. Math., 213 (2007), 600-620. doi: 10.1016/j.aim.2007.01.005.

[17]

V. I. Oliker, The Gauss curvature and Minkowski problems in space forms, in "Recent Developments in Geometry" (Los Angeles, CA, 1987), Contemp. Math., 101, Amer. Math. Soc., Providence, RI, (1989), 107-123. doi: 10.1090/conm/101/1034975.

[18]

B. O'Neill, "Semi-Riemannian Geometry. With Applications to Relativity," Pure and Applied Mathematics, 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

[19]

A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces," Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 35, American Mathematical Society, Providence, R.I., 1973.

[20]

J. G. Ratcliffe, Foundations of hyperbolic manifolds, Second edition, Graduate Texts in Mathematics, 149, Springer, New York, 2006.

[21]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.

[22]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[23]

L. Rüschendorf, On $c$-optimal random variables, Statist. Probab. Lett., 27 (1996), 267-270. doi: 10.1016/0167-7152(95)00078-X.

[24]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.

[25]

C. S. Smith and M. Knott, Note on the optimal transportation of distributions, J. Optim. Theory Appl., 52 (1987), 323-329. doi: 10.1007/BF00941290.

[26]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[27]

_______, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

A. D. Aleksandrov, Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Dokl.) Acad. Sci. URSS (N. S.), 35 (1942), 131-134.

[2]

A. D. Alexandrov, "Convex Polyhedra," Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.

[3]

I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations," With an obituary for the author by W. Rundell, Edited by S. D. Taliaferro, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-69881-1.

[4]

T. Barbot, F. Béguin and A. Zeghib, Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: Application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier (Grenoble), 61 (2011), 511-591. doi: 10.5802/aif.2622.

[5]

J. Bertrand, Existence and uniqueness of optimal maps on Alexandrov spaces, Adv. Math., 219 (2008), 838-851. doi: 10.1016/j.aim.2008.06.008.

[6]

D. Burago, Y. Burago and S. Ivanov, "A Course in Metric Geometry," Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.

[7]

G. Carlier, On a theorem of Alexandrov, J. Nonlinear Convex Anal., 5 (2004), 49-58.

[8]

F. Fillastre, Fuchsian convex bodies: Basics of Brunn-Minkowski theory, Geom. Funct. Anal., 23 (2013), 295-333. doi: 10.1007/s00039-012-0205-4.

[9]

W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.

[10]

P. Guan, C. Lin and X.-N. Ma, The existence of convex body with prescribed curvature measures,, Int. Math. Res. Not. IMRN, 2009 (): 1947.  doi: 10.1093/imrn/rnp007.

[11]

I. Iskhakov, "On Hyperbolic Surfaces Tesselations and Equivariant Spacelike Convex Polyhedra," Ph.D thesis, Ohio State University, 2000.

[12]

L. Kantorovitch, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N. S.), 37 (1942), 199-201.

[13]

F. Labourie and J.-M. Schlenker, Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante, Math. Ann., 316 (2000), 465-483. doi: 10.1007/s002080050339.

[14]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.

[15]

_______, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.

[16]

V. I. Oliker, Embedding $ S^n$ into $ R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $ S^n$, Adv. Math., 213 (2007), 600-620. doi: 10.1016/j.aim.2007.01.005.

[17]

V. I. Oliker, The Gauss curvature and Minkowski problems in space forms, in "Recent Developments in Geometry" (Los Angeles, CA, 1987), Contemp. Math., 101, Amer. Math. Soc., Providence, RI, (1989), 107-123. doi: 10.1090/conm/101/1034975.

[18]

B. O'Neill, "Semi-Riemannian Geometry. With Applications to Relativity," Pure and Applied Mathematics, 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

[19]

A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces," Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 35, American Mathematical Society, Providence, R.I., 1973.

[20]

J. G. Ratcliffe, Foundations of hyperbolic manifolds, Second edition, Graduate Texts in Mathematics, 149, Springer, New York, 2006.

[21]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.

[22]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[23]

L. Rüschendorf, On $c$-optimal random variables, Statist. Probab. Lett., 27 (1996), 267-270. doi: 10.1016/0167-7152(95)00078-X.

[24]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.

[25]

C. S. Smith and M. Knott, Note on the optimal transportation of distributions, J. Optim. Theory Appl., 52 (1987), 323-329. doi: 10.1007/BF00941290.

[26]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[27]

_______, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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