On the Parallels Between Minimal Surfaces and Einstein Four-Manifolds

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.