Singularities in minimal surfaces

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Minimal surfaces are surfaces which (locally) minimize area for their boundary, and are mathematical models of soap films, or more generally any physical interface whose area and energy are proportional. Minimal surfaces are important as both models and tools, and have found applications ranging from knot theory to general relativity. Like the soap films they model, minimal surfaces may not always be smooth -- for example bubble clusters in a bathtub will exhibit singular junctions where multiple bubbles meet. The goal of this project is to better understand the singular nature of minimal surfaces, when singularities can or cannot exist, and what the surface looks like near singularities. In addition to these research goals, the PI will mentor graduate and undergraduate students, and will continue supporting the local math outreach programs for elementary and middle school students. A significant goal of this project is to understand boundary singularities of capillary minimal surfaces, which are surfaces meeting a container at a prescribed angle, like liquid in a cup. The PI aims to construct examples of singular capillary surfaces, to investigate notions of generic regularity in the capillary setting, and push further the relationship between small-angle capillary minimal surfaces and the one-phase Bernoulli problem. A second goal of the project is to better understand entire minimal surfaces asymptotic to cylindrical cones, which effectively model degeneration towards cylindrical singularities. A third goal is to investigate the nature of minimal surfaces near tetrahedral singularities (which model the junction of 4 bubbles meeting at a point), and in particular whether tetrahedral singularities must persist or can be perturbed away into Y-type junctions. 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.