I'm a sixth year graduate student at UCI advised by Richard Schoen. My research interests are in geometric analysis, more particularly mean curvature flow (and weak solutions thereof), minimal surfaces, and mathematical general relativity.
Starting Fall 2019 I'll be a J.J. Sylvester assistant professor at Johns Hopkins.
My email is mramora at uci.edu
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.
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
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.
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).
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.
Spring 2019: No teaching duties this quarter
My wife Alicia and I have turned into avid hikers -it's a great way to relax a little and the outdoors can be a great place to get new math ideas too! UCI is in a prime location for hiking with many mountains relatively nearby. Two of our favorite peaks are San Jacinto and San Gorgonio about an hour and a half away:
Here are Alicia and I on San Gorgonio.
Here is a great view of San Jacinto from San Gorgonio.
In the background is San Gorgonio viewed from San Jacinto, on the way up to Long valley on the cactus to clouds hike.
Finally, here is (what I believe to be) San Gorgonio and San Jacinto in one shot from a plane.
Last Updated: May 8, 2019