Least area surfaces can have excess triple points

Abstract

LET F be a closed surface and ,tl a Riemannian 3-manifold. A map$ F-+M is incompressible if 1,: rl(F)-+nl(_lI) is injective. A smooth incompressible immersionfi F-+M is said to be a least clren map if the area offis less than the area of any map from F to M which is freely homotopic to& If 41 is a closed 3-manifold, with rrZ(M) zero, and if F is not S’ or P’, then a theorem of Schoen and Yau [12] asserts that any incompressible map from F to M is homotopic to a least area map. In [2], Freedman et nl. proved some results on the intersections and selfintersections of two-sided least area surfaces, which they summarized by the slogan “least area surfaces intersect least”. (Two-sided means that the map has trivial normal bundle.) In this article we give some examples to show that least area surfaces need not have the minimal possible number of triple points. This is the result of the title. These examples suggest that the results of [2] are the best that can be obtained in general. Iffi F-+Jl is a general position immersion, the only self-intersections will be double curves and triple points. and the usual invariant associated to this self-intersection is the complesity. which is the pair (t, d) where t is the number of triple points and n is the number of double curves. Sow a least area immersion need not be in general position, so Freedman et al. defined a new inv-ariant D(f) of any incompressible immersionfi F+M which, whenfis in general position. is closely related to the number of double curves d(J). They showed that a two-sided least area immersionfminimizes the invariant D(f) among all homotopic maps. They also gave an example to show that, in general, a least area immersion need not minimize the number of double curves d(f), which was another reason for defining a new invariant. However, in the case when the surface involved is the torus T, they showed that a least area mapfi T-+,LI was always self-transverse and that D(J) equals d(J). Thus f has the least number of double curves achievable by any map homotopic to& The main example in this paper is of a Riemannian 3-manifold M and a least area map f: T-+-.11 which has triple points but is homotopic to a general position immersion without triple points. Thusfdoes not minimize the complexity (t, d). At this point, we should mention an important unsolved problem. We stated above that a least area mapfi T+M must be selftransverse. This means that any two sheets off(T) cross transversely, but it does not mean thatf‘is in general position. The mapfcould have curves of triple points (or of even higher multiplicity). X more worrying possibility is thatfcould have a countably infinite set of triple points. An example of a local picture with this property is the intersection in ?z~ of the planes -_=O and y=r with the surface ~=e1’XL sin (l/.x). This last surface is not minimal, but Guliiver has constructed an example of three minimal surfaces which intersect with a countable set of triple points. If M has an analytic metric, thenfis also analytic [6-83 and it