Solution surfaces of Monge-Ampère equations

Differential Geometry and its Application

Volumn 14, Issue 2, 2001, Pages 113-124

  Ishikawa, Go O ^a   Morimoto, Tohru ^b  

^a HOKKAIDO UNIVERSITY   (Japan)

^b NARA WOMEN'S UNIVERSITY   (Japan)

Author keywords

53A05; 53A20; 53D10; 58K05; Contact geometry; Hartman Nirenberg's theorem; Legendre submanifold; Monge Amp re equation; Open umbrella; Projective duality; Tangent developable

Indexed keywords

EID: 0041334010     PISSN: 09262245     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0926-2245(01)00033-X     Document Type: Article

References (33)

1. Arnol'd V.I. Singularities of Caustics and Wave Fronts. Mathematics and its Applications. 62:1990;Kluwer, Dordrecht.
2. Arnol'd V.I., Gusein-Zade S.M., Varchenko A.N. Singularities of Differentiable Maps I, The Classification of Critical Points, Caustics and Wave. Monographs in Mathematics. 82:1985;Birkhäuser, Boston.
3. Ballesteros J.J.N. On the number of the triple points of the tangent developable. Geom. Dedicata. 47:1993;241-254.
4. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A. Exterior Differential Systems. 1991;Springer, New-York.
5. Cleave J.P. The form of the tangent developable at points of zero torsion on space curves. Math. Proc. Cambridge Philos. Soc. 88:1980;403-407.
6. Givental' A.B. Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funktsional. Anal. i Prilozhen. 20:1986;35-41.
7. E. Goursat, Leçon sur l'Intégration des Équations aux Derivées Partielles du Second Ordre à deux Variables Indépendentes, I, II (Hermann, Paris, 1896, 1898).
8. Griffiths P., Harris J. Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. 12:1979;355-432.
9. Hartman P., Nirenberg L. On spherical image whose Jacobians do not change sign. Amer. J. Math. 81:1959;901-920.
10. Ishikawa G. Determinacy of the envelope of the osculating hyperplanes to a curve. Bull. London Math. Soc. 25:1993;603-610.
11. Ishikawa G. Developable of a curve and determinacy relative to osculation-type. Quart. J. Math. Oxford. 46:1995;437-451.
12. Ishikawa G. Developable hypersurfaces and algebraic homogeneous spaces in a real projective space. Lobachevskii J. Math. 3:1999;113-125.
13. Izumiya S. Geometric singularities for Hamilton-Jacobi equation. Adv. Stud. Pure Math. 22:1993;89-100.
14. 2=1 . Math. Ann. 127:1954;130-134.
15. Kossowski M. Local existence of Multivalued solutions to analytic symplectic Monge-Ampère equations. Indiana Univ. Math. J. 40:1991;123-148.
16. Kossowski M. Homotopy invariants for solutions to symplectic Monge-Ampère equations. J. Differential Equations. 106:1993;294-311.
17. Landsberg J.M. On the second fundamental forms of projective varieties. Invent. Math. 117:1994;303-315.
18. Landsberg J.M. On degenerate secant and tangential varieties and local differential geometry. Duke Math. J. 85:1996;605-634.
19. Lychagin V.V., Rubtsov V.N., Chekalov I.V. A classification of Monge-Ampère equations. Ann. Sci. École Norm. Sup. 26:1993;281-308.
20. Milnor T.K. Efimov's theorem about complete immersed surfaces of negative curvature. Advances in Math. 8:1972;474-543.
21. Mond D. On the tangent developable of a space curve. Math. Proc. Cambridge Philos. Soc. 91:1982;351-355.
22. Mond D. Singularities of the tangent developable surface of a space curve. Quart. J. Math. Oxford. 40:1989;79-91.
23. Morimoto T. La géométrie des équations de Monge-Ampère. C.R. Acad. Sci. Paris. 289:1979;A25-A29.
24. Morimoto T. Geometric structures on filtered manifolds. Hokkaido Math. J. 22:1993;263-347.
25. Morimoto T. Monge-Ampère equations viewed from contact geometry. Symplectic Singularities and Geometry of Gauge Fields. Banach Center Publ. 39:1995;105-120 Polish Acad. Sci. Warsaw.
26. Nitsche J.C.C. Elementary proof of Bernstein's theorem on minimal surfaces. Ann. of Math. 66:1957;543-544.
27. Scherbak O.P. Projectively dual space curves and Legendre singularities. Trudy Tbiliss. Univ. 232-233:1982;280-336.
28. Sternberg S. Lectures on Differential Geometry. 1983;Printice-Hall, New York.
29. Stoker J.J. Differential Geometry. 1969;John Wiley & Sons, New York.
30. Tsuji M. Formation of singularities for Monge-Ampère equations. Bull. Sci. Math. 119:1995;433-456.
31. Tsuji M. Monge-Ampère equations and surface with negative Gaussian curvature. Symplectic Singularities and Geometry of Gauge Fields. Banach Center Publ. 39:1997;161-170 Polish Acad. Sci. Warsaw.
32. Tunitskii D.V. On the contact linearization of Monge-Ampère equations. Izv. Math. 60:1996;425-451.
33. Wu H. Complete developable submanifolds in real and complex Euclidean spaces. Internat. J. Math. 6:1995;461-489.