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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

A. D. Alexandrov, "Convex Polyhedra,", Translated from the 1950 Russian edition by N. S. Dairbekov, (1950).   Google Scholar

[3]

I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations,", With an obituary for the author by W. Rundell, (1994).  doi: 10.1007/978-3-642-69881-1.  Google Scholar

[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.  doi: 10.5802/aif.2622.  Google Scholar

[5]

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

[6]

D. Burago, Y. Burago and S. Ivanov, "A Course in Metric Geometry,", Graduate Studies in Mathematics, 33 (2001).   Google Scholar

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[11]

I. Iskhakov, "On Hyperbolic Surfaces Tesselations and Equivariant Spacelike Convex Polyhedra,", Ph.D thesis, (2000).   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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.  doi: 10.1016/j.aim.2007.01.005.  Google Scholar

[17]

V. I. Oliker, The Gauss curvature and Minkowski problems in space forms,, in, 101 (1989), 107.  doi: 10.1090/conm/101/1034975.  Google Scholar

[18]

B. O'Neill, "Semi-Riemannian Geometry. With Applications to Relativity,", Pure and Applied Mathematics, 103 (1983).   Google Scholar

[19]

A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces,", Translated from the Russian by Israel Program for Scientific Translations, (1973).   Google Scholar

[20]

J. G. Ratcliffe, Foundations of hyperbolic manifolds,, Second edition, 149 (2006).   Google Scholar

[21]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[22]

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

[23]

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

[24]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (1993).  doi: 10.1017/CBO9780511526282.  Google Scholar

[25]

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

[26]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[27]

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

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.   Google Scholar

[2]

A. D. Alexandrov, "Convex Polyhedra,", Translated from the 1950 Russian edition by N. S. Dairbekov, (1950).   Google Scholar

[3]

I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations,", With an obituary for the author by W. Rundell, (1994).  doi: 10.1007/978-3-642-69881-1.  Google Scholar

[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.  doi: 10.5802/aif.2622.  Google Scholar

[5]

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

[6]

D. Burago, Y. Burago and S. Ivanov, "A Course in Metric Geometry,", Graduate Studies in Mathematics, 33 (2001).   Google Scholar

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[11]

I. Iskhakov, "On Hyperbolic Surfaces Tesselations and Equivariant Spacelike Convex Polyhedra,", Ph.D thesis, (2000).   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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.  doi: 10.1016/j.aim.2007.01.005.  Google Scholar

[17]

V. I. Oliker, The Gauss curvature and Minkowski problems in space forms,, in, 101 (1989), 107.  doi: 10.1090/conm/101/1034975.  Google Scholar

[18]

B. O'Neill, "Semi-Riemannian Geometry. With Applications to Relativity,", Pure and Applied Mathematics, 103 (1983).   Google Scholar

[19]

A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces,", Translated from the Russian by Israel Program for Scientific Translations, (1973).   Google Scholar

[20]

J. G. Ratcliffe, Foundations of hyperbolic manifolds,, Second edition, 149 (2006).   Google Scholar

[21]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[22]

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

[23]

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

[24]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (1993).  doi: 10.1017/CBO9780511526282.  Google Scholar

[25]

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

[26]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[27]

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

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