Jacob Bernstein's Homepage

Various Documents

• My CV
• My Research Statement
• My Teaching Statement
• Helicoid-Like Minimal Disks (Joint with C.Breiner)
  Abstract:
  We show that an embedded minimal disk in R^3 with large curvature is bi-Lipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided.
• Slides for a talk based on the preceding are here.
• Distortions of the Helicoid (Joint with C.Breiner)
  Abstract:
  Colding and Minicozzi have shown that an embedded minimal disk $0\in\Sigma\subset B_R$ in $\Real^3$ with large curvature at $0$ looks like a helicoid on the scale of $R$. Near $0$, this can be sharpened: on the scale of $|A|^{-1}(0)$, $\Sigma$ is close, in a Lipschitz sense, to a piece of a helicoid. We use surfaces constructed by Colding and Minicozzi to see this description cannot hold on the scale $R$.
• Conformal Structure of Minimal Surfaces with Finite Topology (Joint with C.Breiner)
  Abstract:
  In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if $g$ is the stereographic projection of the Gauss map, then in a neighborhood of the puncture, $g(p) = \exp(i\alpha z(p) + F(p))$, where $\alpha \in \Real$, $z=x_3+ix_3^*$ is a holomorphic coordinate defined in this neighborhood and $F(p)$ is holomorphic in the neighborhood and extends over the puncture with a zero there. This further implies that the end is actually Hausdorff close to a helicoid.