The existence of least area surfaces in 3-manifolds. (English) Zbl 0711.53008

Hass, Joel; Scott, Peter

Trans. Am. Math. Soc. 310, No. 1, 87-114 (1988).

In this interesting paper, an elegant simplified approach is given to existence results for least area surfaces F in Riemannian 3-manifolds M. If F is embedded or immersed in M, we say that it is least area if F realizes the infimum of area in a nontrivial isotopy or homotopy class of maps. Such surfaces are of great utility in 3-dimensional topology and important existence results for homotopy classes of 2-spheres and closed orientable \(\pi_ 1\)-injective surfaces have been given by J. Sacks and K. K. Uhlenbeck [Ann. Math., II. Ser. 113, No.1, 1-24 (1981; Zbl 0462.58014)] and R. M. Schoen and S. T. Yau [ibid. 110, No.1, 127-142 (1979; Zbl 0431.53051)], respectively. Least area embedded surfaces in isotopy classes were constructed by W. H. Meeks III, L. Simon and S. T. Yau [ibid. 116, No.3, 621-659 Zbl 0521.53007)]. C. B. Morrey jun. [ibid. 49, 807-851 (1948; Zbl 0033.396)] showed that if a simple closed curve in a homogeneously regular manifold bounds a disk of finite area then it spans a least area disk. W. H. Meeks III and S. T. Yau [Topology 21, No.4, 409-442 (1982; Zbl 0489.57002)] proved that if M is a compact 3-manifold with a strictly convex boundary then a simple closed curve in the boundary which shrinks in M bounds an embedded least area disk in M.
Suppose that \(S_ i\) is a minimizing sequence of embedded surfaces, i.e. \(area(S_ i)\) converges to the infimum in some nontrivial isotopy class. The key idea is to replace the intersections of \(S_ i\) with small balls by least area disks, using the results of Morrey and Meeks-Yau. The new minimizing sequence obtained this way is shown to converge by establishing a convergence result for sequences of least area disks in balls. The Sachs-Uhlenbeck and Schoen-Yau results on least area immersions in homotopy classes are then deduced by using cut-and-paste techniques in suitable covering spaces. Finally, analogous results for noncompact manifolds and surfaces with boundary are also considered.

MSC:

53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
57M35	Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)
57N10	Topology of general \(3\)-manifolds (MSC2010)

Keywords:

least area surfaces; Riemannian 3-manifolds; 3-dimensional topology; homotopy classes

Citations:

Zbl 0462.58014; Zbl 0431.53051; Zbl 0521.53007; Zbl 0033.396; Zbl 0489.57002