By the results in Morse theory, a smooth function on $T^2$ has at least three critical points, and at least one of them is degenerate. I'm asked to construct a smooth function that has exactly three critical points.

I have tried to construct a smooth function on $\mathbb R^2$ that is periodical with respect to both components, but I cannot find such function with three critical points.

An explicit expression is preferred, but not necessary. Thanks for your help.


1 Answer 1


The idea is to modify the monkey saddle to something with three critical points. I got this (a long time ago) by messing around with bump functions and the monkey saddle if I remember correctly. On the right the gradient flow is depicted. It seems that there are more critical points with the gradient flow, but this is really just an artefact of my programming skills.



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