Christine Breiner
Johns Hopkins University
Department of Mathematics
105 Krieger Hall
3400 N. Charles Street
Baltimore, MD 21218
Office: 105 Krieger Hall
Email: cbreiner"at"math"dot"jhu"dot"edu
Research
Brief Synopsis: Colding and Minicozzi have shown that the space of complete, embedded
minimal surfaces with finite topology and one end is not compact. Having
proven a conformality result for such surfaces, I am now interested in
determining what subspaces of this space are compact. I am connecting the
conformal structure, the Weierstrass data, and the geometry of the genus to
better answer this question. I am also considering the question of uniqueness of
the genus one helicoid of Weber, Hoffman, and Wolf.
Advisor: Bill Minicozzi
Publications and Preprints
Helicoid-Like Minimal Disks and Uniqueness.
Joint with Jacob Bernstein. Preprint.
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.
Distortions of the Helicoid. Joint with Jacob Bernstein. Geometriae Dedicata 137 (2008), no. 1, 143-147.
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 Jacob Bernnstein. Preprint.
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.
Teaching
Intro to Calculus - Fall 2005 (Taught course)
Calculus I - Fall 2004, Spring 2005
Calculus II - Spring 2007, Spring 2008
Calculus III - Spring 2006, Fall 2006
Linear Algebra - Spring 2009
Analysis and Honors Analysis - Fall 2007