In this article I partially classify the space of eternal mean convex flows in R3 of finite total curvature type, a condition implied by finite total curvature. In particular I show that topologically nonplanar ones must flow out of a catenoid in a natural sense. In a nutshell I observe that the (time) asymptotic behavior of the flow can be well understood under the assumptions, and I combine this with a strong rigidity result for genus zero minimal surfaces of finite total curvature due to Lopez and Ros (namely that they must be planes or catenoids).

    This paper was started in late 2020 but I’ve recently revisited and almost completely rewritten it. The result is now that asymptotically conical self shrinkers which bound a handlebody, defined in a natural sense, are unknotted/topologically standard, which with a bit of extra arguing gives that asymptotically conical self shrinkers with two ends are unknotted. In the conjectural picture, this gives that all self shrinkers with two ends are unknotted.

    In this article, Theo and I construct an ancient curve shortening flow in R^3 from the ansatz that piecing together grim reapers along their ends should nearly be a curve shortening flow in its own right (one may do this with any number of reapers, but we concentrated on the case of just two reapers for simplicity). Since they are nonplanar, they give more examples of ancient flows which take up as much room as possble in the natural sense, in contrast to a number of results which restrict minimal embedding dimension in terms of entropy or curvature conditions.

    In this article, following along similar lines as the previous two papers below, I show that asymptotically conical self shrinkers with Colding-Minicozzi entropy less than the round circle are diffeomorphic to the 3 sphere with some cylindrical ends S^2 x R_+ attached. This generalizes a number of works, including my joint work with Shengwen listed below. I call this entropy level ”medium” because typically a low entropy condition/upper bound for surfaces in R^4 means bounding the entropy by Lambda_2, the entropy of the round 2-sphere. I was a little worried that calling this bound medium was a little bombastic considering the obvious higher dimensional analogue, because in higher dimensions there are more, higher, natural entropy bounds to consider that are interesting so saying ”medium” is a little vague. In the case of R^4 though medium is the largest natural bound to consider, aside from perhaps 2 for multiplicity reasons, that is probably useful at least in the type of argument I use.

    In this article I extend an unknottedness result Shengwen and I showed for compact self shrinkers to the mean curvature flow to shrinkers with finite topology and one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end.

    • Compactness and finiteness theorems for rotationally symmetric self shrinkers. To appear in the Journal of Geometric Analysis.

    In this note I show some compactness and finiteness theorems for rotationally symmetric self shrinkers: first for general such self shrinkers with an entropy bound, then for self shrinkers with convex profile curve but no other assumption, and finally a finiteness theorem assuming reflection symmetry. Essentially due to rotational symmetry, the results are not confined in small dimension in contrast to many related compactness results for minimal surfaces. These results give some further positive evidence on the question of uniqueness of Angenent’s donut (the shrinker version of Lawson’s conjecture), at least amongst rotationally symmetric shrinkers and subclasses.

    • On the construction of closed nonconvex nonsoliton ancient mean curvature flows (joint with Theodora Bourni and Mat Langford). To appear in IMRN.

    In this article Theo, Mat, and myself construct closed, embedded, ancient mean curvature flows in each dimension n>1 with the topology of S1xSn-1. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Angenent’s self shrinking torus when n=2). We take a soft, ad hoc approach by proving curvature estimates by blowup analysis, using crucially that there are self shrinking donuts of entropy less than 2. We later learned that center manifold analysis can also in fact be applied, and in particular these type of ideas can be used to give a proof of the conjecture we discuss at the end — see Chodosh, Choi, Mantoulidis, Schulze and Choi, Mantoulidis. Note that an approach with some similarity to the one in this paper was used in my paper with Alec below but there center manifold techniques (at least as developed so far) don’t seem to apply immediately (or at least at the time of its writing) because we consider ancient flows from necessarily noncompact surfaces.

    • Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces (joint with Alec Payne). To appear in Math Annalen.

    In this article Alec and I construct ancient and eternal flows which flow “out” of certain classes of unstable minimal hypersurfaces in Rn+1. Special attention is given to the rotationally symmetric cases where one flows out of a catenoid – in this case we show there in fact exists an eternal solution (called the reapernoid, a portmanteau of grim reaper and catenoid that was too hard to pass up) and we prove a partial uniqueness theorem concerning these.

    • Nonconvex Surfaces which Flow to Round Points (joint with Alec Payne). To appear in Communications in Analysis and Geometry.

    In this article we extend Huisken’s theorem that convex surfaces flow to round points by mean curvature flow. We will construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that exist on a uniform time interval, have uniformly bounded diameter, and shrink to round points, yet the sequence of initial surfaces has no subsequence converging in the Gromov-Hausdorff sense. Moreover, we construct such a sequence of flows where the initial surfaces converge to a space-filling surface. Also constructed are surfaces of arbitrarily large area which are close in Hausdorff distance to the round sphere yet shrink to round points.

    In this article, my collaborator Shengwen and I extend the mean curvature flow with surgery to mean convex hypersurfaces with low entropy. In particular, 2-convexity is not assumed. We then show that smooth n-dimensional closed self shrinkers with entropy less than that of the round (n-2) sphere are isotopic to the(round) n-sphere.

    • Regularity and stability results for the level set flow via the mean curvature flow with surgery. To appear in Communications in Analysis and Geometry.

    In this article I use the mean curvature flow with surgery to get regularity results for the level set flow that go beyond Brakke regularity theorem. I also show a stability result for the plane under the level set flow. This is another article (in reference to my first one) where I tried to explore how the surgery flow could be used, taking the perspective this time that it could be used to understand other weak notions of the flow (in particular the associated level set flow) “better” because the surgery flow across singular times is very well understood. The theme of the statements shown is that small patches of 2-convex “fuzz” should quickly dissipate/flatten out under the flow — it seems perhaps a barrier argument (using a bowl soliton) combined with one sided minimization could be used to prove similar results without refering to the surgery flow.

    • On the topological rigidity of self shrinkers in R^3 (joint with Shengwen Wang). Published in International Mathematics Research Notices.

    In this article my collaborator Shengwen and I show that self shrinkers in R^3 are “topologically standard” in that they are ambiently isotopic to standard genus g surfaces. In particular, self shrinking tori are unknotted (in the obvious sense). There are a couple small technical issues in the article to do with nonproperness of a certain shrinker found in the course of the argument along with some (in)stability issues which I discuss in detail in the unknottedness article for shrinkers with one end above (therein I also reprove the result in this article), but as I discuss there these issues can be fixed/explained without much effort.

    In this article I show that surfaces that shrink to points do so generically to round points, in the sense of Colding and Minicozzi. The main point is to overcome a lack of good monotonicity formula for mean curvature flow in curved ambient spaces.

    In this article I use the mean curvature flow with surgery to construct an ambient isotopy of a 2-convex hypersurface to a “skeleton,” or a number of embedded S^1s connected by intervals. Then I can estimate the number of skeletons up to isotopy (given certain conditions on the original class of hypersurfaces) to establish an extrinsic finiteness theorem in the spirit of Cheeger’s finiteness theorem. It seems reasonable that other approaches to this result are possible (using more “by hand” deformations) given the assumptions in question — but the surgery flow is nice to use because it does it for you.