Integration of Monge-Ampère equations and surfaces with negative gaussian curvature
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 27 (1998) no. 2, p. 309-330
@article{ASNSP_1998_4_27_2_309_0,
     author = {Ha Tien Ngoan and Kong, Dexing and Tsuji, Mikio},
     title = {Integration of Monge-Amp\`ere equations and surfaces with negative gaussian curvature},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 27},
     number = {2},
     year = {1998},
     pages = {309-330},
     zbl = {0978.53006},
     mrnumber = {1664691},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_1998_4_27_2_309_0}
}
Ha Tien Ngoan; Kong, Dexing; Tsuji, Mikio. Integration of Monge-Ampère equations and surfaces with negative gaussian curvature. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 27 (1998) no. 2, pp. 309-330. http://www.numdam.org/item/ASNSP_1998_4_27_2_309_0/

[ 1 ] M.H. Amsler, Des surfaces à courbure constante négative dans l'espace à trois dimensions et de leurs singularités, Math. Ann. 130 (1955), 234-256. | MR 73225 | Zbl 0068.35102

[2] R. Courant - D. Hilbert, "Methods of Mathematical Physics", vol. 2, Interscience, New York, 1962. | Zbl 0099.29504

[3] G. Darboux, "Leçons sur la théorie générale des surfaces", tome3, Gauthier-Villars, Paris, 1894.

[4] N.V. Efimov, Generation of singularities on surfaces of negative curvature, Maht. USSR-Sb. 64 (1964), 286-320. | MR 167938 | Zbl 0126.37402

[5] E. Goursat, "Leçons sur l'intégration des équations aux dérivées partielles du second ordre", tome 1, Hermann, Paris, 1896.

[6] E. Goursat, "Cours d'analyse mathématique", tome 3, Gauthier-Villars, Paris, 1927. | JFM 53.0180.05

[7] J. Hadamard, "Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques", Hermann, Paris, 1932. | JFM 58.0519.16 | Zbl 0006.20501

[8] D. Hilbert, Über Flächen von constanter Gausscher Krümmung, Trans. Amer. Math. Soc. 2 (1901), 87-99. | JFM 32.0608.01 | MR 1500557

[9] F. Holmgen, Sur les surfaces à courbure constante négative, C. R. Acad. Sci. Paris 134 (1902), 740-743. | JFM 33.0643.01

[10] S. Izumiya, Geometric singularities for Hamilton-Jacobi equation, Adv. Stud. Pure Math. 22 (1993), 89-100. | MR 1274941 | Zbl 0837.35090

[11] S. Izumiya, Characteristic vector fields for first order partial differential equations, preprint. | MR 1611166

[12] S. Izumiya - G.T. Kossioris, Semi-local classification of geometric singularities for Hamilton-Jacobi equations, J. Differential Equations 118 (1995), 166-193. | MR 1329407 | Zbl 0837.35091

[13] M. Kossowski, Local existence of multivalued solutions to analytic symplectic Monge-Ampère equations, Indiana Univ. Math. J. 40 (1991), 123-148. | MR 1101224 | Zbl 0718.53043

[14] H. Levy, Über das Anfangswertproblem einer hyperbolischen nichtlinearen partiellen Differentialgleichung zveiler Ordnung mit zwei unabhänggigen Veränderlichen, Math. Ann. 97 (1927), 179-191. | JFM 53.0473.15

[15] H. Levy, A priori limitations for solutions of Monge-Ampère equations I, II, Trans. Amer. Math. Soc. 37 (1934), 417-434; 41 (1937), 365-374. | JFM 63.0442.01 | MR 1501906

[16] V.V. Lychagin, Contact geometry and non-linear second order differential equations, Russian Math. Surveys 34 (1979), 149-180. | MR 525652 | Zbl 0427.58002

[17] T.K. Milnor, Efimov's theorem about complete immersed surfaces of negative curvature, Adv. Math. 8 (1972), 454-543. | MR 301679 | Zbl 0236.53055

[18] S. Nakane, Formation of singularities for Hamilton-Jacobi equations in several space variables, J. Math. Soc. Japan 43 (1991), 89-100. | MR 1082424 | Zbl 0743.35043

[19] S. Nakane, Formation of shocks for a single conservation law, SIAM J. Math. Anal. 19 (1988), 1391-1408. | MR 965259 | Zbl 0681.35057

[20] A Pliś, Characteristics of nonlinear partial differential equations, Bull. Polish Acad. Sci. Math, Cl.III2 (1954), 419-422. | MR 67296 | Zbl 0056.31902

[21] Tran Dinh Son, On surfaces with negative analytic Gaussian curvature, Diploma at Hanoi Institute of Mathematics (1996).

[22] M. Tsuji, Formation of singularities for Hamilton-Jacobi equations II, J. Math. Kyoto Univ. 26 (1986), 299-308. | MR 849221 | Zbl 0655.35009

[23] M. Tsuji, Prolongation of classical solutions and singularities of generelized solutions, Ann. Inst. H. Poincaré - Anal. Non Linéaire 7 (1990), 505-525. | Numdam | MR 1079570 | Zbl 0722.35025

[24] M. Tsuji, Formation of singularities for Monge-Ampère equations, Bull. Sci. Math. 119 (1995), 433-457. | MR 1354246 | Zbl 0845.35005

[25] M. Tsuji, Monge-Ampère equations and surfaces with negative Gaussian curvature, Banach Center Publ. 39 (1997), 161-170. | MR 1458658 | Zbl 0890.35093

[26] M. Tsuji, Geometric approach to blow-up phenomena in nonlinear problems, In "Real Analytic and Algebraic Singularities" edited by T. Fukuda, et al. (Pitman Research Notes in Math. 381. Longman, 1998), 164-180. | MR 1607635 | Zbl 0894.35072

[27] H. Whitney, On singularities of mappings of Euclidean spaces I, Ann. Math. 62 (1955), 374-410. | MR 73980 | Zbl 0068.37101