Picture below is from 294 page of Huisken, Gerhard, Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom. 31, No.1, 285-299 (1990). ZBL0694.53005.

I don't understand the notation $y$. If it is a graph, it should be ordered pairs liking $(x,y(x))$. But if so, I can't understand $y^{-1}$. If $y$ just be $y(x)$, then , we can't represent this rotating surface, just have a curve. how to understand it ?

Besides, what is the mean of $p,q$ ? How to calculate 1 ?

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  • $\begingroup$ Once you have the surface $M$, for any $\bf x\in M$ you have $y({\bf x})=Distance({\bf x},x_1-axis)$. $\endgroup$ – san May 26 '17 at 20:30
  • $\begingroup$ @san Thanks, do you know how to calculate $\nabla_1 y =-qy$ ? $\endgroup$ – lanse7pty May 28 '17 at 1:45
  • $\begingroup$ I'm not sure if I understand well, (I'm not sure about muy first comment, either), according to my understanding, I have $\nabla_1y=-q/p$. $\endgroup$ – san May 29 '17 at 3:50
  • $\begingroup$ @san From the below of this paper, it should be $\nabla_1 y =-qy$. Could you talk about how to get $\nabla_1y=-q/p$ ? $\endgroup$ – lanse7pty May 30 '17 at 2:37
  • $\begingroup$ Maybe I misunderstood $\nabla_1 y$, I thought it was $\nabla_{\iota_1}y$, but it seems to be $\nabla_{e_1}y$. In that case the formula seems to be ok. You should work with $qy$ and $py$ instead of $q$, then the formulas are geometrically clear. $\endgroup$ – san May 30 '17 at 15:27

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