Geometry of Surfaces and Four-Dimensional Manifolds

The goal of this project is to explore critical points of natural geometric functionals, particularly in the fields of minimal surfaces with boundaries and four-dimensional manifolds. These concepts arise naturally such as when one dips a wireframe into a soap solution, forming a soap film, which is an example of a minimal surface. Similarly, the world we are living in can be modeled as a four-dimensional manifold with three spatial and one time directions.

Thus, advancements in this area will have applications in physics, biology, and applied sciences. As a consequence, the project's objective is to advance knowledge by exploring the following research directions. The project also includes mentoring of students (at both the undergraduate and graduate levels) and the organization of a mini-school aimed at broadening participation of under-represented minorities in STEM fields.

The first research direction in this project aims to study stationary surfaces with boundaries, examining the first and second variations to draw out classification and uniqueness results. In particular, motivated by Lawson and Willmore's conjectures for minimal surfaces in a sphere, the PI will conduct a parallel study for free boundary minimal surfaces in a ball.

By introducing creative ideas (Jacobi-Steklov eigenvalue, interpreting constraints as linear functionals), the PI will develop an approach that gives a framework for further investigations. The second thrust studies the connection between the geometry and topology of a four-dimensional manifold. The PI aims to classify these manifolds under natural conditions, particularly resolving a differentiable sphere conjecture of the second kind and providing insights on one of Hopf's conjectures and a folklore conjecture about Einstein metrics.

The guiding principle is that the Hodge star operator gives rise to several elliptic identities in dimension four. The project also includes several mentoring activities as well as impactful activities aimed at broadening participation of under-represented minorities in STEM fields.

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.