A non-existence result for minimal catenoids in asymptotically flat spaces

TL;DR

This paper proves that in asymptotically Schwarzschildean 3-manifolds, minimal catenoids cannot be deformed from Euclidean ones, regardless of the ADM mass or catenoid size, highlighting a sharp three-dimensional obstruction.

Contribution

It establishes a non-existence result for minimal catenoids in asymptotically flat spaces, revealing a specific geometric obstruction.

Findings

Minimal catenoids cannot be perturbed in asymptotically Schwarzschildean spaces.

The non-existence holds regardless of the ADM mass or catenoid size.

The obstruction is specific to three-dimensional asymptotically flat geometries.

Abstract

We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold for more general classes of asymptotically flat data.