Journal of Differential Geometry

Nonlocal $s$-minimal surfaces and Lawson cones

Juan Dávila, Manuel del Pino, and Juncheng Wei

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The nonlocal $s$-fractional minimal surface equation for $\Sigma = \partial E$ where $E$ is an open set in $\mathbb{R}^N$ is given by

\[ H_\Sigma^ s (p) := \int_{\mathbb{R}^N} \frac{\chi_E(x) - \chi_{E^c}(x)} {{\lvert x-p \rvert}^{N+s}}\, dx \ =\ 0 \; \textrm{for all} \; p\in\Sigma \textrm{ .} \]

Here $0 \lt s \lt 1 , \chi$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s \to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s$-minimal surfaces. When $s$ is close to $1$, we first construct a connected embedded $s$-minimal surface of revolution in $\mathbb{R}^3$, the nonlocal catenoid, an analog of the standard catenoid $\lvert x_3 \rvert = \log(r+ \sqrt{r^2 - 1})$. Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone $\lvert x_3 \rvert = r \sqrt{1 - s}$. We also find a two-sheet embedded s-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes.

On the other hand, for any $0 \lt s \lt 1 , n , m \geq 1$, $s$-minimal Lawson cones $\lvert v \rvert = \alpha \lvert u \rvert, (u, v) \in \mathbb{R}^n \times \mathbb{R}^m$, are found to exist. In sharp contrast with the classical case, we prove their stability for small $s$ and $n + m = 7$, which suggests that unlike the classical theory (or the case $s$ close to $1$), the regularity of $s$-area minimizing surfaces may not hold true in dimension $7$.

Article information

Source
J. Differential Geom., Volume 109, Number 1 (2018), 111-175.

Dates
Received: 9 December 2014
First available in Project Euclid: 4 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1525399218

Digital Object Identifier
doi:10.4310/jdg/1525399218

Mathematical Reviews number (MathSciNet)
MR3798717

Zentralblatt MATH identifier
06868032

Citation

Dávila, Juan; del Pino, Manuel; Wei, Juncheng. Nonlocal $s$-minimal surfaces and Lawson cones. J. Differential Geom. 109 (2018), no. 1, 111--175. doi:10.4310/jdg/1525399218. https://projecteuclid.org/euclid.jdg/1525399218


Export citation