## Journal of Differential Geometry

- J. Differential Geom.
- Volume 93, Number 1 (2013), 67-131.

### Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$

Manuel del Pino, Michal Kowalczyk, and Juncheng Wei

#### Abstract

We consider minimal surfaces $M$ which are complete, embedded,
and have finite total curvature in $\mathbb{R}^3$, and bounded, entire
solutions with finite Morse index of the Allen-Cahn equation $\Delta u+
f(u) = 0$ in $\mathbb{R}^3$. Here $f = −W'$ with $W$ bi-stable and balanced, for
instance $W(u) = \frac{1}{4} (1 − u^2)^2$. We assume that $M$ has $m \ge 2$ ends,
and additionally that $M$ is non-degenerate, in the sense that its
bounded Jacobi fields are all originated from rigid motions (this
is known for instance for a Catenoid and for the Costa-Hoffman-Meeks surface of any genus). We prove that for any small $\alpha \gt 0$,
the Allen-Cahn equation has a family of bounded solutions depending
on $m − 1$ parameters distinct from rigid motions, whose
level sets are embedded surfaces lying close to the blown-up surface
$M_\alpha := \alpha^{−1}M$, with ends possibly diverging logarithmically
from $M_\alpha$. We prove that these solutions are $L^\infty$-*non-degenerate*
up to rigid motions, and find that their Morse index coincides
with the index of the minimal surface. Our construction suggests
parallels of De Giorgi conjecture for general bounded solutions of
finite Morse index.

#### Article information

**Source**

J. Differential Geom., Volume 93, Number 1 (2013), 67-131.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1357141507

**Digital Object Identifier**

doi:10.4310/jdg/1357141507

**Mathematical Reviews number (MathSciNet)**

MR3019512

**Zentralblatt MATH identifier**

1275.53015

#### Citation

del Pino, Manuel; Kowalczyk, Michal; Wei, Juncheng. Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$. J. Differential Geom. 93 (2013), no. 1, 67--131. doi:10.4310/jdg/1357141507. https://projecteuclid.org/euclid.jdg/1357141507