Celebratio Mathematica

article M. Freedman, J. Hass, and P. Scott: “Closed geodesics on surfaces,” Bull. London Math. Soc. 14 : 5 (1982), pp. 385–391. MR 0671777 Zbl 0476.53026

Let \( M^2 \) be a closed Riemannian 2-manifold, and let \( \alpha \) denote a non-trivial element of \( \pi_1(M) \). The set of all loops in \( M \) which represent a has a shortest element \( f:\mathbb{S}^1 \rightarrow M \), which can be assumed smooth and which will be a closed geodesic. (We say a loop represents \( \alpha \) when it represents any conjugate of \( \alpha \). Such a loop need not pass through the base point of \( M \).) The map \( f \) cannot be unique, because \( f \) is not necessarily parametrised by arc length and because there is no base point. In general, even the image set of a shortest loop is not unique. In this note, we prove the following result.

Let \( M^2 \) be a closed, Riemannian 2-manifold and let \( \alpha \) denote a non-trivial element of \( \pi_1M \) which is represented by a two-sided embedded loop \( C \). Then any shortest loop \( f:\mathbb{S}^1 \rightarrow M \) representing \( \alpha \) is either an embedding or a double cover of a one-sided embedded curve \( K \). In the second case, \( C \) bounds a Moebius band in \( M \) and \( K \) is isotopic to the centre of this band.

Joel Hass	Related
M. H. Freedman	Related	Volume

article M. Freedman, J. Hass, and P. Scott: “Least area incompressible surfaces in 3-manifolds,” Invent. Math. 71 : 3 (1983), pp. 609–642. MR 0695910 Zbl 0482.53045

Let \( M \) be a Riemannian manifold and let \( F \) be a closed surface. A map \( f:F\rightarrow M \) is called least area if the area of \( f \) is less than the area of any homotopic map from \( F \) to \( M \). Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function.

In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly.

Joel Hass	Related
M. H. Freedman	Related	Volume

J. Hass and P. Scott: “Intersections of curves on surfaces,” Israel J. Math. 51 : 1–2 (1985), pp. 90–120. MR 804478 Zbl 0576.57009 article

The authors consider curves on surfaces which have more intersections than the least possible in their homotopy class.

Let \( f \) be a general position arc or loop on an orientable surface \( F \) which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on \( F \) bounded by part of the image of \( f \).

Let \( f \) be a general position arc or loop on an orientable surface \( F \) which has excess self-intersection. Then there is a singular 1-gon or 2-gon on \( F \) bounded by part of the image of \( f \).

Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.

J. Hass and P. Scott: “The existence of least area surfaces in 3-manifolds,” Trans. Am. Math. Soc. 310 : 1 (November 1988), pp. 87–114. MR 965747 Zbl 0711.53008 article

J. Hass, H. Rubinstein, and P. Scott: “Compactifying coverings of closed 3-manifolds,” J. Differ. Geom. 30 : 3 (1989), pp. 817–832. MR 1021374 Zbl 0693.57011 article

J. Hyam Rubinstein	Related	Volume
Joel Hass	Related

J. Hass and P. Scott: “Homotopy equivalence and homeomorphism of 3-manifolds,” Topology 31 : 3 (July 1992), pp. 493–517. MR 1174254 Zbl 0771.57007 article

J. Hass and P. Scott: “Homotopy and isotopy in dimension three,” Comment. Math. Helv. 68 : 3 (1993), pp. 341–364. MR 1236759 Zbl 0805.57008 article

J. Hass and P. Scott: “Shortening curves on surfaces,” Topology 33 : 1 (January 1994), pp. 25–43. MR 1259513 Zbl 0798.58019 article

J. Hass and P. Scott: “Simplicial energy and simplicial harmonic maps,” Asian J. Math. 19 : 4 (2015), pp. 593–636. MR 3423736 Zbl 1332.57024 ArXiv 1206.2574 article

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G. Peter Scott

An interview with Joel Hass on his collaboration with Godfrey Peter Scott

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