2025/06/17 by Beard, Mia
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2506.14972
openalex publication_date 2025/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbCP2 into S7 via the Veronese map, where it arises as a minimal submanifold.