[Paper Review] Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$

Manuel del Pino, Michał Kowalczyk+1 more|ArXiv.org|Feb 12, 2009

Nonlinear Partial Differential Equations19 references13 citations

TL;DR

This paper constructs bounded, entire solutions of the Allen-Cahn equation in $ ^3$ with finite Morse index by gluing together approximations near complete, embedded minimal surfaces of finite total curvature with $m \geq 2$ ends. It proves the existence of $m-1$ non-rigid, parameter-dependent solutions whose level sets approximate the blown-up minimal surface $M_\alpha = \alpha^{-1}M$, and shows these solutions are $L^\infty$-non-degenerate and have Morse index equal to that of the minimal surface, establishing a strong link between minimal surface geometry and phase transition solutions.

ABSTRACT

We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $Δu + f(u) = 0 \hbox{in} \R^3 $. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. 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 $α>0$, 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 $M_α:= α^{-1} M$, with ends possibly diverging logarithmically from $M_\A$. We prove that these solutions are $L^\infty$-{\em 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.

Motivation & Objective

• To establish a correspondence between bounded, entire solutions of the Allen-Cahn equation with finite Morse index and complete, embedded minimal surfaces of finite total curvature in $ ^3$.
• To construct a one-parameter family of solutions to the Allen-Cahn equation that approximate a blown-up minimal surface $M_\alpha = \alpha^{-1}M$ for small $\alpha > 0$, with $m-1$ parameters independent of rigid motions.
• To prove that the constructed solutions are $L^\infty$-non-degenerate up to rigid motions and have Morse index equal to that of the underlying minimal surface $M$.
• To explore the parallels between the De Giorgi conjecture and finite Morse index solutions of the Allen-Cahn equation in higher dimensions.

Proposed method

• Use of a gluing construction to build solutions to the Allen-Cahn equation near a blown-up minimal surface $M_\alpha = \alpha^{-1}M$, leveraging local geometry and asymptotic behavior of the minimal surface.
• Application of a first-order approximation based on the Jacobi operator of the minimal surface, followed by iterative refinement to improve the approximation.
• Reduction of the nonlinear problem to a finite-dimensional gluing system by projecting onto the kernel of the linearized operator and solving the projected problem via contraction mapping.
• Use of weighted Sobolev and $L^\infty$ estimates to control error terms in different regions: near the surface, in the ends, and in the far field.
• Analysis of the linearized operator around the approximate solution, proving a priori estimates and existence of solutions via a contraction argument.
• Study of Jacobi fields of logarithmic growth at the ends of the minimal surface to understand the kernel of the linearized operator and ensure non-degeneracy.

Experimental results

Research questions

• RQ1 Can bounded, entire solutions of the Allen-Cahn equation with finite Morse index be constructed near complete, embedded minimal surfaces of finite total curvature in $ ^3$?
• RQ2 What is the dimension of the solution space of such solutions, and how does it relate to the number of ends of the minimal surface?
• RQ3 Are the constructed solutions $L^\infty$-non-degenerate up to rigid motions, and does their Morse index match that of the minimal surface?
• RQ4 How do the level sets of the solutions behave asymptotically, particularly in relation to the ends of the minimal surface?
• RQ5 What is the role of logarithmic growth in Jacobi fields at the ends in determining the solution space structure?

Key findings

• For any small $\alpha > 0$, the Allen-Cahn equation admits a family of bounded, entire solutions depending on $m-1$ parameters (modulo rigid motions), whose level sets are embedded surfaces close to $M_\alpha = \alpha^{-1}M$, with ends possibly diverging logarithmically.
• The constructed solutions are $L^\infty$-non-degenerate up to rigid motions, meaning no non-trivial bounded Jacobi fields exist except those arising from rigid motions.
• The Morse index of the constructed solutions is exactly equal to the Morse index of the minimal surface $M$, establishing a precise correspondence between the two indices.
• The solutions are asymptotically close to the minimal surface $M_\alpha$ in the $C^2$ topology away from the ends, with controlled error terms in the nonlinear equation.
• The existence of solutions is tied to the geometry of the minimal surface: the construction relies on the non-degeneracy of $M$ (i.e., bounded Jacobi fields come only from rigid motions), which holds for surfaces like the catenoid and Costa-Hoffman-Meeks surfaces.
• The analysis reveals that Jacobi fields of logarithmic growth at the ends play a crucial role in the kernel of the linearized operator, and their presence is essential for the existence of the $m-1$ parameter family of solutions.