The hyperplane is the only stable, smooth solution to the Isoperimetric Problem in Gaussian space

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.

Appendix: mean value inequality

Here we give a proof of the Mean Value Inequality, Lemma 7.2. The techniques are well-known (see Colding-Minicozzi [8]), but we include a proof for completeness.

Proof of Lemma 7.2

Assume ${|}H{|} \le M$. Lemma 7.2 is stated in terms of Euclidean quantities, so we are free to translate so that we are considering $B_s(0)$. Recall that $\Delta {|}x{|}^2 = 2n - 2\langle x,N \rangle H$. Then

$$\begin{aligned} 2n\int \limits _{B_s \cap \Sigma } f \, d {{\mathcal {A}}}&= \int \limits _{B_s \cap \Sigma } f\Delta {|}x{|}^2 \, d {{\mathcal {A}}}+ 2\int \limits _{B_s \cap \Sigma }f\langle x, N\rangle H \, d {{\mathcal {A}}}, \nonumber \\&= \int \limits _{B_s \cap \Sigma }{|}x{|}^2 \Delta f \, d {{\mathcal {A}}}+ 2\int \limits _{\partial B_s \cap \Sigma } f {|}x^T{|}\, d {{\mathcal {A}}}\nonumber \\&- s^2 \int \limits _{B_s \cap \Sigma } \Delta f \, d {{\mathcal {A}}}+ 2 \int \limits _{B_s \cap \Sigma } \langle x,N \rangle Hf \, d {{\mathcal {A}}}. \end{aligned}$$

(8.1)

Let $g(s) = s^{-n}\int \limits _{B_s \cap \Sigma } f\, d {{\mathcal {A}}}$. Using the coarea formula and (8.1), we get

$$\begin{aligned} g'(s) \ge \frac{1}{2}s^{-n+1}\int \limits _{B_s \cap \Sigma } \Delta f \, d {{\mathcal {A}}}- s^{-n-1}\int \limits _{B_s \cap \Sigma }\langle x, N\rangle f H \, d {{\mathcal {A}}}. \end{aligned}$$

(8.2)

Here we have used the positivity of $f$. Additionally, if we assume $\Delta f \ge -\lambda t^{-2} f$ on $B_t$, our bound on ${|}H{|}$ gives us

$$\begin{aligned} g'(s)&\ge \frac{-\lambda }{2} s^{1-n}\int \limits _{B_s \cap \Sigma }f t^{-2}\, d {{\mathcal {A}}}- Ms^{-1-n}\int \limits _{B_s \cap \Sigma }s f \, d {{\mathcal {A}}},\nonumber \\&\ge -\left( \frac{\lambda }{2t} + M\right) g(s) \end{aligned}$$

(8.3)

for all $s\le t$. Therefore,

$$\begin{aligned} \frac{d}{ds}\left( g(s)e^{\left( \frac{\lambda }{2t} + M\right) s}\right) \ge 0. \end{aligned}$$

(8.4)

Integrating (8.4) from $s_0$ to $s_1$ (both assumed to be less than $t$) and letting $s_0 \searrow 0$, we get

$$\begin{aligned} e^{\left( \frac{\lambda }{2t} + M\right) s_1} s_1^{-n}\int \limits _{B_{s_1}\cap \Sigma } f \, d {{\mathcal {A}}}\ge \omega _nf(p). \end{aligned}$$

(8.5)

$\square$

Note, that we get the following corollary (monotonicity):

Corollary 8.1

Let $p \in \Sigma$, and let ${|}H{|} \le M$ in $B_{t}(p) \cap \Sigma$. Then for $s\le t$, we have ${|}B_s \cap \Sigma {|} \ge \omega _n e^{-Ms}s^n$, where $\omega _n$ is the volume of the standard unit ball in ${{\mathbb {R}}}^n$.

Proof

Use the Mean Value Inequality, Lemma 7.2 with $f\equiv 1$ and $\lambda = 0$.$\square$