On the parallels between minimal surfaces and Einstein four-manifolds

Preprint

Minimal surfaces and Einstein manifolds are among the most natural structures in differential geometry: the former being well understood, the latter far less so. In this exposition, we survey the striking parallels between minimal surfaces in three-manifolds and Einstein metrics in four dimensions. 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 objects, 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 and modern results, 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. As a key example, we realise $\mathbb{CP}^2$ as a minimal submanifold of $S^7$ via the Veronese map, demonstrating a deep unity between the two seemingly distinct worlds.

Authors: Beard, M

• Publication status: Published
• Peer review status: Not peer reviewed
• Preprint server: SSRN
• Publication date: 2025-08-06
• DOI: 10.2139/ssrn.5381156
• EISSN: 1556-5068
• Server owner: Elsevier
• Language: English
• Keywords: geometric analysis; minimal surfaces; Einstein manifolds; differential geometry