A deformation of the cubical Fermat-Pham-Brieskorn function on ${\fam\msbfam\tenmsb R}^3$.

The cubical Fermat-Pham-Brieskorn function $f:{\fam\msbfam\tenmsb R}^3 \to {\fam\msbfam\tenmsb R}$ is given by the expression $f(x,y,z)=x^3+y^3+z^3$. The mapping $\phi:{\fam\msbfam\tenmsb R}^3 \to {\fam\msbfam\tenmsb R}^3$, given by $\phi(x,y,z)=(x^3,y^3,z^3)$, is a continuous bijection. The reciprocal map $\psi:{\fam\msbfam\tenmsb R}^3 \to {\fam\msbfam\tenmsb R}^3$ is also continuous and is given by $\psi(u,v,w)= (\sqrt[3]{u},\sqrt[3]{v},\sqrt[3]{w})$. Please notice that the reciprocal map $\psi$ is not differentiable. It follows that $\phi$ maps the level surface $L_f(t):=\{(x,y,z) \in {\fam\msbfam\tenmsb R}^3 \mid f(x,y,z)=x^3+y^3+z^3=t \}, t \in {\fam\msbfam\tenmsb R},$ to the plane $P_t:=\{ (u,v,w) \in {\fam\msbfam\tenmsb R}^3 \mid u+v+w=t \}$. Hence the level surfaces $L_f(t), t \in {\fam\msbfam\tenmsb R}$, are topological planes in ${\fam\msbfam\tenmsb R}^3$. Since the reciprocal map $\psi$ is not differentiable, we can not conclude, that all the level surfaces $L_f(t), t \in {\fam\msbfam\tenmsb R}$, are differentiable submanifolds in ${\fam\msbfam\tenmsb R}^3$. The level surfaces of the function $f$ produce a non differentiable foliation on ${\fam\msbfam\tenmsb R}^3$, which is topologically the standard foliation of ${\fam\msbfam\tenmsb R}^3$ by planes,

The function $f_{\epsilon}(x,y,z):=x^3+y^3+z^3-\epsilon(x+y+z), \epsilon > 0,$ is a deformation of the function $f$. The functions $f_{\epsilon},\epsilon > 0,$ have $8$ critical points, which are the vertices $[\pm\sqrt{{\epsilon \over 3}}, \pm\sqrt{{\epsilon \over 3}}, \pm\sqrt{{\epsilon \over 3}}]$ of a cube in ${\fam\msbfam\tenmsb R}^3$. The maximal number of critical points, which can have a function that is a ``small'' perturbution of the function $f$, is the sum of the Milnor numbers of its singularities. The function $f$ has only a singularity at $(0,0,0)$ with Milnor number $8$. The deformation $f_{\epsilon},\epsilon > 0,$ is called a maximal small deformation of the singularity of the function $f$ at $(0,0,0) \in {\fam\msbfam\tenmsb R}^3$, since this deformation realizes the maximal possible number of critical points. Please notice that for negative values of $\epsilon$ the function $f_{\epsilon}$ has no critical points at all.

The movie SURFACE shows level surfaces of the function $f_1$. One can see a ballet of appearing and disappearing cycles. From the choreography and geometry of this ballet one can determine the toplogical behavior of the corresponding complex map $f:{\fam\msbfam\tenmsb C}^3 \to {\fam\msbfam\tenmsb C}$.

The frames of the movie are made with the software SURF, which is freely available at http://surf.sourceforge.net/. The playoff is done with XANIM, with is part of any Linux distribution.