Jacob Bernstein






Research Interests

My work focuses on geometric analysis, with a particular emphasis on minimal surface theory and mean curvature flow. Minimal surfaces are mathematical idealizations of soap films. In addition to being a beautiful subject in itself, the theory of minimal surfaces interacts with many branches of mathematics and has found important applications in geometry, topology and theoretical physics. Likewise, the mean curvature flow has had important applications in materials science and in computer imaging.

Published Papers and Preprints

  • Topological uniqueness for self-expanders of small entropy (with L. Wang). ArXiv.
  • For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.

  • An integer degree for asymptotically conical self-expanders (with L. Wang). ArXiv.
  • We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the three-dimensional Euclidean space for which each cone in the set has a strictly unstable self-expanding annuli asymptotic to it.

  • Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (with L. Wang). ArXiv.
  • We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set self-expanders of low entropy and for the set mean convex self-expanders with strictly mean convex asymptotic cones. From this we deduce that the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones is proper for these classes.

  • The level set flow of a hypersurface in $\mathbb R^4$ of low entropy does not disconnect (with S. Wang). ArXiv.
  • We show that if $\Sigma\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $\lambda(\Sigma)\leq \lambda(\mathbb{S}^2\times \mathbb R)$, then the level set flow of $\Sigma$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in $\mathbb R^4$ of low entropy.

  • The space of asymptotically conical self-expanders of mean curvature flow (with L. Wang). ArXiv.
  • We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain sense.

  • Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends. ArXiv.
  • We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone.

  • Hausdorff Stability of the Round Two-Sphere Under Small Perturbations of the Entropy (with L. Wang). ArXiv.
  • We show that if a closed surface in R^3 has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.

  • Topology of Closed Hypersurfaces of Small Entropy (with L. Wang). Submitted. ArXiv.
  • We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in euclidean four space with entropy less than or equal to that of the round cylinder, are diffeomorphic to the three-sphere.

  • A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy (with L. Wang). To Appear: Duke Math J. ArXiv.
  • For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all non-flat two-dimensional self-shrinkers. This confirms a conjecture of Colding-Ilmanen-Minicozzi-White in dimension two.

  • A Sharp Lower Bound for the Entropy of Closed Hypersurfaces up to Dimension Six (with L. Wang). To Appear: Invent. Math. ArXiv.
  • In [5], Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in ℝn+1, the entropy is uniquely minimized at the round sphere. They conjectured that, for 2≤n≤6, the round sphere minimizes the entropy among all closed smooth hypersurfaces. Using an appropriate weak mean curvature flow, we prove their conjecture. For these dimensions, our approach also gives a new proof of the main result of [5] and extends its conclusions to compact singular self-shrinkers.

  • One-dimensional Projective Structures, Convex Curves and the Ovals of Benguria & Loss (with T. Mettler). Commun. Math. Phys. 336 (2015), no. 2, 933-952. ArXiv.
  • Benguria and Loss conjectured that, amongst all smooth closed curves in $\Real^2$ of length $2\pi$, the lowest possible eigenvalue of the operator $L=-\Delta+\kappa^2$ was $1$. They observed that this value was achieved on a two-parameter family, $\mathcal{O}$, of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in $\mathcal{O}$ as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all three problems.

  • A Remark on a Uniqueness Property of High Multiplicity Tangent Flows in Dimension Three (with L. Wang). Int. Math. Res. Not. 2015 (2015), no. 15, 6286-6294.
  • In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive integer multiple of a shrinking plane, cylinder or sphere, then, modulo rotations, all tangent flows at the point are the same.

  • Topological Type of Limit Laminations of Embedded Minimal Disks (with G. Tinaglia). J. Differential Geom. 102 (2016), no. 1, 1-23. ArXiv.
  • We consider two natural classes of minimal laminations in three-manifolds. Both classes may be thought of as limits -- in different senses -- of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations are topologically either disks, annuli or M\"obius bands. This answers a question posed by Hoffman and White.

  • Two-Dimensional Gradient Ricci Solitons Revisited (with T. Mettler). Int. Math. Res. Not. 2015 (2015), no. 1, 78-98. ArXiv.
  • We complete the classification of local two-dimensional gradient ricci solitons and in so doing answer some questions of Chow.

  • Characterizing classical minimal surfaces via the entropy differential (with T. Mettler). To Appear: J. Geom. Anal. ArXiv.
  • In this paper we introduce a geometrically natural meromorphic quadratic differential on minimal surfaces in R^3 and use it to classify some classical surfaces. A novel compactness result is also shown.

  • Some Singular Limit Laminations of Embedded Minimal Planar Domains. Int. Math. Res. Not. 2012 (2012), no. 18, 4301–4324. ArXiv.
  • I construct two sequences of embedded minimal planar domains in R^3. These sequences converge to singular laminations and provide some useful examples when considering problems arising from Colding and Minicozzi's work on compactness theory of embedded minimal surfaces with genus bounds. In particular, one of the sequences shows that embedded minimal planar domains do not admit a uniform chord arc bound (unlike the case for minimal disks) which has ramifications for the embedded Calabi-Yau problem for surfaces of infinite topology.

  • A Variational Characterization of the Catenoid (with C. Breiner). Calc. Var. and PDE. 49 (2014), no. 1-2, 215-232. ArXiv.
  • In this paper Christine Breiner and myself show that the pieces of the catenoid minimize area amongst all minimal annuli which span two parallel planes in R^3. That is we consider double connected minimal surfaces with one boundary component in one plane and the other boundary component in a second parallel plane and show that amongst all such surfaces the surface with least area is an explicit (maximally symmetric) part of a catenoid. We also show some sharp lower bounds for the lengths of pairs of curves in parallel planes so that the curves span a connected minimal surface.

  • Symmetry of Embedded Genus 1 Helicoids (with C. Breiner). Duke Math. J. 159 (2011), no. 1, 83–97. ArXiv.
  • Using the López-Ros deformation, we show that any embedded genus 1 helicoid must admit an orientation preserving isometry given by rotation by 180 degrees about a normal line. We also show this symmetry holds for embedded genus $g$ helicoids provided their underlying conformal structure is hyperelliptic. This partial answers a question of Bobenko.

  • Conformal Structure of Minimal Surfaces with Finite Topology (with C. Breiner). Comment. Math. Helv. 86 (2011), no. 2, 353–381. ArXiv.
  • We show that any once punctured, complete, properly embedded minimal surface of finite genus in R^3 must have the conformal type of a once punctured compact surface and be asymptotic to a helicoid. This completes the classification of the conformal type and asymptotic geometry of complete, properly embedded minimal surfaces of finite topology.

  • Distortions of the Helicoid (with C. Breiner) Geom. Dedicata. 137 (2008), no. 1, 143-147. ArXiv.
  • We study the scale at which an embedded minimal disk is well approximated by a helicoid. Using examples constructed by Colding and Minicozzi, we show that the scale of the curvature is in a sense the largest possible scale on which an embedded minimal disk can be well approximated by a helicoid.

  • Helicoid-like Embedded Minimal Disks (with C. Breiner). J. Reine Angew. Math. 655 (2011), 129-146. ArXiv.
  • We give an alternate proof of the uniqueness of the helicoid. That is, the helicoid and plane are the only complete properly embedded minimal disks in R^3 -- a result previously shown by Meeks and Rosenberg. Our proof makes more direct use of Colding-Minicozzi theory.

  • Conformal and Asymptotic Properties of Embedded Genus-g Minimal Surfaces with One End.
  • A copy of my thesis.

    Fall 2013 -- Department of Mathematics, Johns Hopkins University.