Evolving Surfaces and Evolving Implicit Differential Equations Via Contact Geometry and Singularities

Abstract

We present the list of unavoidable local phenomena (transitions) occurring on the configuration of the parabolic and flecnodal curves of evolving smooth surfaces in R^3 (or RP^3). We also present the list of transitions occurring on the curve of inflections of the solutions of evolving implicit differential equations (IDE). Our results are based on the properties of the contours of surfaces (in a contact 3-space) for projections all whose fibres are Legendrian. Keywords: Surface, flecnodal curve, contact geometry, implicit differential equations.

35 pages + 3 short appendices, 25 figures. Many results of this paper were exposed in the Singularity Theory Semester at Newton Institute (2000) and in a minicourse (Int. Conf. of Real and Complex Singularities Sao Carlos, 2002), where a preprint was distributed. This is a completely revamped version: shorter, clearer and includes several new results

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