Welcome to the math webpage of Alex Mramor!
About me: My current position is as a J.J. Sylvester Assistant Prof of Mathematics at Johns Hopkins University, where I will be until 2022. Before I was a graduate student at UC Irvine as a student of Rick Schoen. My research thus far is concentrated in the mean curvature flow but my interests in geometric analysis are broad.
Seminars and colloquia and JHU: I help out with the following at JHU:
The JHU analysis and PDE seminar: weekly seminar talks cover a broad spectrum of both analysis and geometry.
Junior colloquium : Biweekly informal, general audience talks during the department wine and cheese time given by postdocs, professors, and occasionally visitors.
Honors Analysis 2
Intro to Lie Groups and Algebras
Publications, preprints, and conference proceedings:
1. "On the unknoteddness of self shrinkers in R3" in Proceedings of the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, May 29–June 1, 2018 link
2. Compactness and finiteness theorems for rotationally symmetric self shrinkers. Preprint. link
In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to S1×Sn−1 for each n≥2 in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely that the profile curve is convex, we remove the entropy assumption. Supposing the profile curve is additionally reflection symmetric we show there are only finitely many such shrinkers up to rigid motion.
3. On the construction of closed nonconvex nonsoliton ancient mean curvature flows (joint with Theodora Bourni and Mat Langford). Preprint. link
In this article Theo, Mat, and myself construct closed, embedded, ancient mean curvature flows in each dimension n≥2 with the topology of S1×Sn−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)
4. Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces (joint with Alec Payne). Preprint. link
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 and we prove a partial uniqueness theorem concerning these.
5. Nonconvex Surfaces which Flow to Round Points (joint with Alec Payne). Preprint. link
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
6. Low Entropy and the Mean Curvature Flow with Surgery (joint with Shengwen Wang). Preprint. link
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
7. Regularity and stability results for the level set flow via the mean curvature flow with surgery. To appear in Communications in Analysis and Geometry. link
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.
8. On the topological rigidity of self shrinkers in R^3 (joint with Shengwen Wang). Published in International Mathematics Research Notices. link
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).
9. Entropy and generic mean curvature in curved ambient spaces. Published in the Proceedings of the AMS. link
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.
10. A finiteness theorem via the mean curvature flow with surgery. Published in the Journal of Geometric Analysis. link
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.