Minimal Surfaces I: Global to Local Index
We show that in any triangulated
3-manifold, every index $n$ topologically minimal surface can be
transformed to a surface which has local indices (as computed in
each tetrahedron) that sum to at most $n$. This generalizes
classical theorems of Kneser and Haken, and more recent theorems
of Rubinstein and Stocking, and is the first step in a program
to show that every topologically minimal surface has a normal
form with respect to any triangulation.
Minimal Surfaces II: Disks
We show that a topologically minimal
disk in a tetrahedron with index $n$ is either a normal
triangle, a normal quadrilateral, or a normal helicoid with
boundary length $4(n+1)$. This mirrors geometric results of
Colding and Minicozzi.
Minimal Surfaces III: Bounded Combinatorics
We show that there are a finite number
of possible pictures for a surface in a tetrahedron with local
index $n$. Combined with previous results, this establishes that
any topologically minimal surface can be transformed into one
with a particular normal form with respect to any triangulation.
Heegaard structure respects
complicated JSJ-decompositions (joint with R. Derby-Talbot and
Let $M$ be a
3-manifold with torus boundary components $T_1$ and $T_2$. Let
$\phi:T_1 \to T_2$ be a homeomorphism, $M_\phi$ the manifold
obtained from $M$ by gluing $T_1$ to $T_2$ via the map $\phi$,
and $T$ the image of $T_1$ in $M_\phi$. We show that if $\phi$
is “sufficiently complicated” then any incompressible or
strongly irreducible surface in $M_\phi$ can be isotoped to be
disjoint from $T$. It follows that every Heegaard splitting of a
3-manifold admitting a “sufficiently complicated” JSJ
decomposition is an amalgamation of Heegaard splittings of the
components of the JSJ decomposition.
Surfaces that become
isotopic after Dehn filling (joint with R. Derby-Talbot and E.
We show that after
generic filling along a torus boundary component of a
3-manifold, no two closed, 2-sided, essential surfaces become
isotopic, and no closed, 2-sided, essential surface becomes
inessential. That is, the set of essential surfaces (considered
up to isotopy) survives unchanged in all suitably generic Dehn
fillings. Furthermore, for all but finitely many non-generic
fillings, we show that two essential surfaces can only become
isotopic in a constrained way.