Journal of the Mathematical Society of Japan

Height functions on surfaces with three critical values

Fumiya MORISHITA and Osamu SAEKI

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For a given closed surface, we study height functions with three critical values associated with immersions of the surface into 3-space, where the critical points may not be non-degenerate. We completely characterize the numbers of critical points corresponding to the three critical values that can be realized by such height functions. We also study the cases where the immersion can be replaced by an embedding or the critical points are all non-degenerate. Similar problems are studied for distance functions as well.

Article information

J. Math. Soc. Japan, Volume 63, Number 1 (2011), 153-162.

First available in Project Euclid: 27 January 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58K05: Critical points of functions and mappings
Secondary: 57R45: Singularities of differentiable mappings 57R42: Immersions

height function critical point critical value immersion embedding


MORISHITA, Fumiya; SAEKI, Osamu. Height functions on surfaces with three critical values. J. Math. Soc. Japan 63 (2011), no. 1, 153--162. doi:10.2969/jmsj/06310153.

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