Mark Haskins

• Riemannian geometry
• Systems of elliptic partial differential equations
• Einstein manifolds, especially Ricci-flat manifolds
• Riemannian metrics with special or exceptional holonomy.
• Minimal submanifolds, particularly calibrated submanifolds
• Calibrated currents and their singularities and regularity, especially special Lagrangian and associative currents
• Singular spaces with special or exceptional holonomy
• Gauge theory on spaces with special and exceptional holonomy
• Riemannian collapse under Ricci-curvature bounds
• Solitons and singularity formation in in geometric flows, especially Bryant's Laplacian flow
• Heat and other parabolic equation methods for PDEs on manifolds

Some of my most recent research relates to a geometric flow of closed G₂-structures, called the Laplacian flow. Laplacian flow was introduced by Bryant as a way to try to flow a closed G₂-structure to a torsion-free G₂-structure. As with many geometric flows, when curvature concentrates too fast the nonlinearity of the equation can overwhelm its diffusive tendencies and finite-time singularities can form. In certain geometric flows, like the Ricci flow and Mean Curvature Flow, singularity formation has long been a topic of intensive research and in some cases, like Ricci flow on 3-manifolds, it is largely understood.

The study of singularity formation in Laplacian flow is, however, still in its infancy. The first step to understanding singularity formation in geometric flows is often to study special types of self-similar solutions to the flow, called solitons. With various collaborators, I've been trying to understand solitons in Laplacian flow, as a stepping stone toward understanding finite-time singularity formation in Laplacian flow.

In this preprint,  Johannes Nordstrom and I studied highly symmetric Laplacian solitons. We found a pair of shrinking Laplacian solitons and a 1-parameter family of steady Laplacian solitons, all with asymptotically conical geometry. The asymptotically conical steady solitons are a distinctive feature of Laplacian flow, since such solutions cannot arise in Ricci flow.

In this preprint, my Duke postdoc Alec Payne and our Oxford collaborator Ilyas Khan proved that any asymptotically conical Laplacian shrinker is uniquely determined by its conical end. This is proven by establishing a suitable backwards uniqueness existence result for asymptotically conical solutions to Laplacian flow.