Is star-shapedness preserved under $W^{2,2}$ perturbations?

by user128470   Last Updated April 25, 2018 21:20 PM

Let $S_R(0)\subset\mathbb{R}^3$ be a centered sphere and $\nu$ be its outward normal. Of course, we have that $\partial_r\cdot\nu=1$. I am wondering if the condition of $S_R$ being star-shaped, that is, $\partial_r\cdot\nu\geq 0$, is stable under $W^{2,2}$ (or $C^{0,\alpha}$ if you will) perturbations. If this is not the case, can we show that $|\overset{\circ}A|_{L^2}\geq \epsilon$ for the traceless part of the second fundamental form for some constant $\epsilon>0$ if the star shaped condition is violated? My intuition is that for $\partial_r\cdot\nu$ to drop to zero, the surface would need to look locally like a cylinder in a small environment, where consequently $|\overset{\circ}A|_{L^2}$ should be large. I don't know if this idea can be made rigorous tho.

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Updated April 29, 2017 04:20 AM