Minimal surfaces and harmonic maps into singular targets

Many central objects in geometry arise as optimal configurations: they minimize, or are stationary for, quantities such as energy or area. Familiar physical analogues include soap films, elastic membranes, and equilibrium states of materials, but the same ideas also play a powerful role in modern pure mathematics. This project studies two important classes of such objects. The first is harmonic maps, which minimize energy subject to boundary, symmetry, or topological constraints; these maps have become important tools in areas including geometric group theory. The second is minimal surfaces, which are surfaces that minimize area under prescribed constraints. The project seeks to understand when these objects exist, how regular they are, what their singularities look like, and when they are uniquely determined by the data. Progress on these questions will deepen connections between geometric analysis, topology, and geometric group theory. The project investigates several regularity and rigidity questions for harmonic maps and minimal surfaces. One direction concerns harmonic maps from two-dimensional domains into singular metric spaces, including Euclidean buildings and smooth surfaces with cone points. A central problem is whether such maps have unique tangent maps at singular points. Results for Q-valued functions provide a useful model, but the techniques are not immediately tractable: Fourier-analytic methods available in the Q-valued setting do not directly extend to maps into Euclidean buildings. A second direction concerns pluriharmonic maps from Kähler manifolds into Euclidean buildings. Building on the investigator’s previous proof of discreteness of order for harmonic maps from surfaces, the project studies the corresponding discreteness problem in higher-dimensional Kähler settings. This requires a detailed analysis of the spectral cover associated to a multi-valued holomorphic section of a vector bundle over a Kähler manifold, and the resulting ideas may also inform the tangent-map question. Finally, the project studies uniqueness of equivariant minimal surfaces in infinite-dimensional homogeneous spaces. A key first step is to determine whether immersed minimal two-spheres in such spaces must lie in proper subspaces. For a broad class of homogeneous spaces, this containment property is expected to reduce the uniqueness question to classical geometric-flow methods This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.