Some hypersurface has a positive second fundamental form potentially

Notation : $$r^2=x^2+y^2$$.

Exercise : Define $$F_\sigma (x,y)= (f_\sigma (x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2 r^2)(x-\sigma x^3,y-\sigma y^3)$$

Define $$G_\sigma: \mathbb{R}^4\rightarrow \mathbb{R}^4$$ by $$G_\sigma(x,y,a,b)=(dF_\sigma )_{(x,y)} (a,b) =( (df_\sigma)_{(x,y)}(a,b) ,R(x,y))$$ where $$R$$ is $$\angle ((a,b),(1,0))$$-rotation

Then there is a sequence of balls $$U_\sigma$$ and unit circle $$S$$ s.t. $$\lim_{\sigma\rightarrow 0}\ {\rm diam}\ U_\sigma =0$$ and $$G_\sigma (U_\sigma \times S)$$ has positive second fundamental form.

And $$G_0 (U\times S)$$, homeomorphic to solid torus, has no positive second fundamental form for any ball $$U$$ (These are obatined from direct computation)

Question : Intuitively, given saddle $$F_0$$, how can we find saddle $$F_\sigma$$ s.t. $$G_\sigma (U_\sigma\times S)$$ has a positive second fundamental form ?

Reference : On intrinsic geometry of surface in normed spaces - Burago and Ivanv

Exercise (cf. Reference) : Hypersurface has a positive second fundamental form at $$p$$ if there is a linear function $$L$$ and $$c>0$$ s.t. $$L(q)\leq L(p)- c|q-p|^2$$ for all $$q$$ in a neighborhood of $$p$$
If $$F_0(x,y)=(x,y,\frac{x^2-y^2}{2},xy)$$, then $$dF_0\ U\times S$$ has out normal $$W$$ at a point $$(a,b,0,0),\ (a,b)\in S$$.
$$\langle W,(a,b,0,0) -(a_0,b_0,0,0)\rangle \leq -c \{ |(a,b)-(a_0,b_0)|^2+|(x,y)|^2\}$$ does not hold since there is a term $$|(x,y)|^2$$.
Hence we have a strategy to find $$f_\sigma$$ s.t. $$| (df_\sigma)_{(x,y)}(a,b) -(a,b)| \leq O(x^2,xy,y^2)$$