Weak finite determinacy of relative map-germs

Abstract

The weak finite determinacy of relative map-germs is studied. The authors first give the concept of weak finite determination, and then give several sufficient conditions for a relative map-germ to be weak finitely determined, which is an important complement to Mather’s work. Moreover, as an application, it is proven that the relative stable map-germs are weak finitely determined.

Keywords

Relative map-germs / Weak finite determinacy / Relative stability

References

[1] Kong L L, Pei D H. Singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space. Sci. China Ser. A, 2008, 51(2): 241-249

[2] Pei D H, Kong L L, Sun J G Singularities of lightlike hypersurface in semi-Euclidean 4-space with index 2. Sci. China Ser. A, 2010, 53(12): 3243-3254

[3] Li Y C, He W. Versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups. Acta Math. Sci., 2008, 4: 915-923

[4] Su D, Zhang D M. The Newton filtration and d-determination of bifurcation problems related to C ⁰-contact equivalence. Sci. China Ser. A, 2006, 5: 660-668

[5] Adachi J, Ishikawa G. Classification of phase singularities for complex scalar waves and their bifurcations. Nonlinearity, 2007, 8: 1907-1925

[6] Mather J N. Stability of C ^∞ mappings III: Finitely determined map-germs. Publ. Math. I. H. E. S., 1969, 35: 127-156

[7] Gaffney T, du Pless A A. More on the determinacy of smooth map-germs. Invent. Math., 1982, 66: 137-163

[8] Wall C T C. Finite determinacy of smooth map-germs. Bull. London Math. Soc., 1981, 13: 481-539

[9] Porto P F S, Loibel G F. Relative finite determinacy and relative stability of function-germs. Bol. Soc. Brasil Math., 1978, 9: 1-18

[10] Radamila B, José A G, León K. Relative versality for map-germ. Bolletino U. M. I., 1987, 7: 305-320

[11] Kushner L, Leme B T. Finite relative determinacy and relative stability. Pacific Journal of Math., 2000, 192(2): 315-327

[12] Sun B H, Wilson L C. Determinacy of smooth germs with real isolated line singularities. Amer. Math. Soc., 2001, 127(9): 2789-2797

[13] Grandjean V. Infinite relative determinacy of smooth function germs with transverse isolated singularity and relative Lojasiewicz conditions. J. London Math. Soc., 2004, 69(2): 518-530

[14] Chen L, Sun W Z, Pei D H. Contact finite determinacy of relative map germs. Comm. Math. Res., 2010, 26(1): 1-6

[15] Mather J N. Stability of C ^∞ mappings IV: Classification of stable germs by ℝ-algebras. Publ. Math. I. H. E. S., 1970, 37: 223-248