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- W2007761567 abstract "We study stable, two-sided, smooth, properly immersed solutions to the Gaussian Isoperimetric Problem. That is, we study hyper-surfaces $$Sigma ^n subset {{mathbb {R}}}^{n+1}$$ that are second order stable critical points of minimizing $${{mathcal {A}}}_mu (Sigma ) = int _Sigma e^{-|x|^2/4} , d {{mathcal {A}}}$$ for compact variations that preserve weighted volume. Such variations are represented by $$u in C^infty _0(Sigma )$$ such that $$int _Sigma e^{-|x|^2/4} u , d {{mathcal {A}}}= 0$$ . We show that such $$Sigma $$ satisfy the curvature condition $$H = langle x, N rangle /2 + C$$ where $$C$$ is a constant. We also derive the Jacobi operator $$L$$ for the second variation of such $$Sigma $$ . Our first main result is that for non-planar $$Sigma $$ , bounds on the index of $$L$$ , acting on volume preserving variations, gives us that $$Sigma $$ splits off a linear space. A corollary of this result is that hyper-planes are the only stable smooth, complete, properly immersed solutions to the Gaussian Isoperimetric Problem, and that there are no hypersurfaces of index one. Finally, we show that for the case of $$Sigma ^2 subset {{mathbb {R}}}^3$$ , there is a gradient decay estimate for fixed bound $$|C| le M$$ ( $$C$$ is from the curvature condition) and $$Sigma $$ obeying an appropriate $${{mathcal {A}}}_mu $$ condition. This shows that for fixed $$C$$ , in the limit as $$R rightarrow infty $$ , stable $$(Sigma , partial Sigma ) subset (B_{2R}(0), partial B_{2R}(0))$$ with good volume growth bounds approach hyper-planes." @default.
- W2007761567 created "2016-06-24" @default.
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- W2007761567 date "2015-02-20" @default.
- W2007761567 modified "2023-09-26" @default.
- W2007761567 title "The hyperplane is the only stable, smooth solution to the Isoperimetric Problem in Gaussian space" @default.
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- W2007761567 doi "https://doi.org/10.1007/s10711-015-0057-9" @default.
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