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Quo-Shin Chi

2008-01-08
14:00:00 - 15:00:00

Tautness and Algebraicity

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

A compact submanifold in a sphere (or in an Euclidean space) is taut if all the spherical (Euclidean) distance functions are perfect Morse-Bott functions with respect to Z_2 coefficients. Tautness is preserved by stereographic projection, and so the theories in the two ambient spaces are essentially the same. Two important classes of taut submanifolds are isoparametric submanifolds and compact proper Dupin hypersurfaces in spheres, among many others. In a paper published in 1984, Kuiper raised the question as to whether a taut submanifold is real algebraic, i.e., whether a taut submanifold is an irreducible component of a real algebraic subvariety of the ambient space. In the 1980's the answer to this question was widely thought to be true, for a good reason, because intuitively a perfect Morse-Bott function should not be flabby anywhere. In this talk, I shall indicate a proof that the answer to this question of Kuiper's is affirmative if the dimension of the taut submanifold is less than or equal to four.