Low Dimensional Topology

October 2, 2008

Minimal vs. normal surfaces

Filed under: 3-manifolds,Heegaard splittings,Triangulations — Jesse Johnson @ 10:19 am

I was very excited to see a paper, [1], appear on the arXiv with a combinatorial version of a result I mentioned a couple of weeks ago.  (I assume the authors know each other, since they’re both based in Bangalore, India.)  The original paper (the first paper mentioned in that post) shows that an emedded surface in a 3-manifold is incompressible if and only if it is isotopic to a minimal area surface in its homotopy class for every Riemannian metric on the 3-manifold.  The second paper shows that an embedded surface is incompressible if and only if it is isotopic to a least (combinatorial) area normal surface in every triangulation of the 3-manifold.  (Recall that an embedded surface is normal with respect to a triangulation if it intersects each tetrahedron in triangles and quadrelaterals.)

This analogy between minimal surfaces in Riemannian metrics and normal surfaces in triangulations is not new. For example, it shows up in work of Hyam Rubinstein.  He first showed, with Pitts [2] that a strongly irreducible Heegaard splitting is always isotopic to an index-1 minimal surface, or double covers an index-0 minimal surface, in a “generic” Riemannian metric.  (Actually, the reference below may not be the correct one.) Later, he showed in [3] (independently from Stocking [4], who gave a more detailed account of a similar proof) that every strongly irreducible Heegaard splitting is isotopic to an almost normal surface in any triangulation having a property called 0-efficiency, which I won’t define here.  

An almost normal surface is one whose intersections with the tetrahedra consist of triangles and quadrelaterals, plus one “almost normal” piece.  You can look up the exact definition of the almost normal pieces if you’re interested, but the basic idea is that you can isotope or compress the almost normal pieces in two distinct ways to reduce them.  These two distinct ways are analogous to pushing an unstable (say, index-1) minimal surface in two different ways, each of which allows you to minimize it further.

[2] Pitts, Jon T.; Rubinstein, J. H, Existence of minimal surfaces of bounded topological type in three-manifolds. Miniconference on geometry and partial differential equations (Canberra, 1985), 163–176,  Proc. Centre Math. Anal. Austral. Nat. Univ., 10, Austral. Nat. Univ., Canberra, 1986. 

[3] Rubinstein, J. H.(5-MELB) Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$-dimensional manifolds. Geometric topology (Athens, GA, 1993), 1–20,  AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. 

[4]  Stocking, Michelle Almost normal surfaces in $3$-manifolds. Trans. Amer. Math. Soc. 352 (2000), no. 1, 171–207.

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