Celebratio Mathematica

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.

M. H. Freedman	Related	Volume
Peter Scott	Related	Volume

J. Hass: “Bounded 3-manifolds admit negatively curved metrics with concave boundary,” J. Diff. Geom. 40 : 3 (1994), pp. 449–459. MR 1305977 Zbl 0821.53035 article

J. Hass, J. C. Lagarias, and N. Pippenger: “The computational complexity of knot and link problems,” J. ACM 46 : 2 (March 1999), pp. 185–211. A preliminary report was published in Proceedings: 38th annual symposium on the foundations of computer science (1997). MR 1693203 Zbl 1065.68667 article

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, i.e., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

J. Hass and R. Schlafly: “Double bubbles minimize,” Ann. Math. (2) 151 : 2 (March 2000), pp. 459–515. MR 1765704 Zbl 0970.53008 article

J. Hass and P. Koehl: “Comparing shapes of genus-zero surfaces,” J. Appl. Comput. Topol. 1 : 1 (2017), pp. 57–87. MR 3975549 article

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Joel Hass

Joel Hass: A short biography

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