On the Parallels Between Minimal Surfaces and Einstein Four-Manifolds

TL;DR

This paper explores deep analogies between minimal surfaces and Einstein four-manifolds, proposing that under certain conditions, Einstein manifolds can be viewed as minimal immersions, revealing potential underlying geometric connections.

Contribution

It establishes a set of parallels between minimal surfaces and Einstein four-manifolds and demonstrates that some Einstein manifolds can be realized as minimal submanifolds in higher-dimensional spheres.

Findings

Einstein four-manifolds can admit minimal immersions into spheres.

The Veronese embedding of P^2 into S^7 exemplifies a minimal Einstein submanifold.

Analogies suggest potential unified geometric frameworks.

Abstract

Minimal surfaces and Einstein manifolds are among the most natural structures in differential geometry. Whilst minimal surfaces are well understood, Einstein manifolds remain far less so. This exposition synthesises together a set of parallels between minimal surfaces embedded in an ambient three-manifold, and Einstein four-manifolds. These parallels include variational formulations, topological constraints, monotonicity formulae, compactness and epsilon-regularity theorems, and decompositions such as thick/thin and sheeted/non-sheeted structures. Though distinct in nature, the striking analogies between them raises a profound question: might there exist circumstances in which these objects are, in essence, manifestations of the same underlying geometry? Drawing on foundational results such as Jensen's theorem, Takahashi's theorem, and a conjecture of Song, this work suggests a bridge between the two structures. In particular, it shows that certain Einstein four-manifolds admit a minimal immersion into a higher-dimensional sphere. A key example of this is the embedding of $\mathbb{CP}^{2}$ into $S^{7}$ via the Veronese map, where it arises as a minimal submanifold.