Stephen Lynch

High codimension mean curvature flow with surgery (w. H. T. Nguyen). [arXiv:2004.07163]

Following the Hamilton--Perelman Ricci flow with surgery for 3-manifolds and Huisken--Sinestrari's mean curvature flow with surgery for 2-convex hypersurfaces, we develop a mean curvature flow with surgery for submanifolds of arbitrary codimension. Instead of 2-convexity, we impose a natural quadratic pinching condition introduced by Andrews--Baker. This gives the first mean curvature flow through singularities with topological control in codimension in higher codimension. The estimates that make the construction work may be of independent interest: these include a gradient estimate, cylindrical estimate and planarity estimate. The most subtle point in the construction is to argue that these estimates hold uniformly, independently of the number of surgeries. 

Translators asymptotic to planes (w. G. Tinaglia) [arxiv:2509.17391]

Solutions to the mean curvature flow which move by symmetries (shrinking, translating, expanding) play an important role in studying singularity formation. We show that a translating solution which is asymptotic to a collection of planes must itself be a plane. This is in stark contrast to the situation for minimal surfaces, where there are an abundance of nontrivial examples with planar ends, such as the catenoid. The proof relies on a flux formula for the Willmore energy, which (we later found out) was discovered earlier by Neves--Tian in the Lagrangian setting. 

Canonical foliation of bubblesheets (w. J. Lagacé).  [arXiv:2411.14340]

High-curvature regions in geometric flows often look like generalised cylinders after rescaling, and it is useful to have a canonical parameterisation of such regions. For cylinders with one Euclidean factor, the standard tool (found by Hamilton) is a foliation by constant mean curvature hyperspheres. But for cylinders with more than one Euclidean factor, the spherical cross-sections have higher codimension and the natural analogue condition, parallel mean curvature, is too rigid---such foliations need not exist. We introduce a new curvature condition, quasi-parallel mean curvature (QPMC), which includes all CMC hypersurfaces and all submanifolds with parallel mean curvature, and prove that any approximately cylindrical Riemannian manifold admits a canonical foliation by QPMC spheres.

Rotational symmetry of ancient solutions to fully nonlinear curvature flows (w. A. Cogo and O. Vičánek Martínez). [arXiv:2310.08301]

For mean curvature flow, work of Brendle–Choi shows that the noncompact, noncollapsed, uniformly two-convex ancient solutions are exactly the cylinders and the bowl soliton. We extend this to a broad class of fully nonlinear curvature flows: every convex, noncollapsing, uniformly two-convex, noncompact ancient solution is either a shrinking cylinder or a rotationally symmetric translating soliton. Our motivation was to obtain a complete classification of blow-up limits at singularities of the flow introduced by Brendle--Huisken, which can probably be used to give an alternative construction of the flow with surgery built in that paper. It remains to consider the corresponding problem for compact ancient solutions, where one would expect to see only the shrinking spheres and a unique ancient oval. 

Some new ingredients which could be useful elsewhere include: a proof that blow-down limits are shrinking cylinders, even though Huisken's montonicity formula is not available; an existence theorem for rotationally symmetric shrinking discs, which make useful barriers; and an asymptotic expansion for the profile curve of the bowl soliton solutions to a large class of flows.  

A differential Harnack inequality for noncompact evolving hypersurfaces. [arXiv:2310.07369]

The Li--Yau differential Harnack inequality for positive solutions to the heat equation is arguably one of the most beautiful results in paralobic PDE theory. Hamilton found powerful analogues of this inequality for curvature quantities along the Ricci flow and mean curvature flow. Andrews later generalised the differential Harnack inequality for the mean curvature flow to a large class of fully nonlinear flows, where the speed of motion is replaced by a symmetric function of the principal curvatures. Here I generalise Andrews' inequality, which required the solution to be compact, to the noncompact setting.  An important new ingredient is a notion of uniform inverse-concavity for the speed function. I needed to assume bounds on the curvature of the solution and some of its derivatives, which are satisfied in many situations of interest, but are a bit unsatisfying. I would be very interested to know whether these hypotheses can be removed. 

Plateau's problem via the Allen--Cahn functional (w. M. A. M. Guaraco).  [arXiv:2305.00363]

Critical points of the Allen–Cahn functional are intimately related to closed minimal hypersurfaces, but Plateau's problem — minimal surfaces with prescribed boundary Γ — had not been treated this way, despite a suggestion of Fröhlich and Struwe. We take Γ to be a compact codimension-two submanifold and study the Allen–Cahn functional on sections of a nontrivial real line bundle over the complement of Γ; the twisting of the bundle forces energy concentration around a generalised surface with boundary Γ. 

We develop a boundary version of the Hutchinson–Tonegawa theory, showing that critical sections with uniformly bounded energy converge to an integer rectifiable varifold in the sharp interface limit. We then develop a regularity theory for minimisers, which gives a new proof that the smooth Plateau problem has a solution for every boundary curve in 3-dimensional Euclidean space. It would be very interesting to know whether the nodal set of a minimizer is already a smooth surface with boundary Γ, even before passing to the sharp interface limit. 

One result here which may be of independent interest is an improved version of Hutchinson--Tonegawa's Modica-type gradient estimate in interior balls, which seems to be essentially sharp in the radius of the ball. 

Ancient solutions of Ricci flow with Type I curvature growth (w. A. Royo Abrego). [arXiv:2211.06253]

Collapsing and noncollapsing in convex ancient mean curvature flow (w. T. Bourni & M. Langford).  [arXiv:2106.06339]

Uniqueness of convex ancient solutions to hypersurface flows, [arXiv:2103.02314]

Convexity estimates for hypersurfaces moving by concave curvature functions, [arXiv:2007.07791]

Convexity estimates for high codimension mean curvature flow (w. H. T. Nguyen).  [arXiv:2006.05227]

Pinched ancient solutions to the high codimension mean curvature flow (w. H. T. Nguyen). [arXiv:1709.09697]

Sharp one-sided curvature estimates for fully nonlinear curvature flows with applications to ancient solutions (w. M. Langford). [arXiv:1704.03802]