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**A piecewise linear theory of minimal surfaces in 3-manifolds.**
*(English)*
Zbl 0596.53007

Geometry and partial differential equations, Miniconf. Canberra/Aust. 1985, Proc. Cent. Math. Anal. Aust. Natl. Univ. 10, 99-110 (1986).

[For the entire collection see Zbl 0583.00013.]

The existence of embedded minimal surfaces in three dimensional manifolds has been investigated by many authors and provides a useful tool in the study of the topological properties of such manifolds. The authors introduce the concept of piecewise linear (PL) minimal surfaces in 3- manifolds being more easier to handle than the analytic minimal surfaces but powerful enough to reprove certain theorems on 3-manifolds: Referring to a triangulation of the 3-manifold M a proper immersed surface \(f: F\to M\) is said to be normal if f(F) intersects each 3-simplex in a finite set of disks being of the type of the seven properly embedded disks in a 3- simplex. The PL-area \(\ell (f)\) of f is just the total length of all the arcs in which f(F) meets 2-simplices. Finally, the PL-minimal surfaces are the stationary points of \(\ell (f)\) (w.r.t. small normal variations), and a normal surface f is of PL-least area if \(\ell (f)\leq \ell (g)\) for all surfaces g being normally homotopic to f.

In Theorem 1, 2 they prove existence and uniqueness of PL-minimal surfaces in a given normal homotopy class [f] assuming that f is not the boundary of a regular neighborhood of a vertex. Further results concern the existence of incompressible PL-least area surfaces in homotopy classes and (under certain restrictions on M) the existence of essential PL-least area disks. As an application the authors discuss a theorem of W. Meeks III, L. Simon and S. T. Yau [Ann. Math., II. Ser. 116, 621-659 (1982; Zbl 0521.53007)] saying that any covering of a \(P^ 2\)-irreducible 3-manifold is \(P^ 2\)-irreducible.

The existence of embedded minimal surfaces in three dimensional manifolds has been investigated by many authors and provides a useful tool in the study of the topological properties of such manifolds. The authors introduce the concept of piecewise linear (PL) minimal surfaces in 3- manifolds being more easier to handle than the analytic minimal surfaces but powerful enough to reprove certain theorems on 3-manifolds: Referring to a triangulation of the 3-manifold M a proper immersed surface \(f: F\to M\) is said to be normal if f(F) intersects each 3-simplex in a finite set of disks being of the type of the seven properly embedded disks in a 3- simplex. The PL-area \(\ell (f)\) of f is just the total length of all the arcs in which f(F) meets 2-simplices. Finally, the PL-minimal surfaces are the stationary points of \(\ell (f)\) (w.r.t. small normal variations), and a normal surface f is of PL-least area if \(\ell (f)\leq \ell (g)\) for all surfaces g being normally homotopic to f.

In Theorem 1, 2 they prove existence and uniqueness of PL-minimal surfaces in a given normal homotopy class [f] assuming that f is not the boundary of a regular neighborhood of a vertex. Further results concern the existence of incompressible PL-least area surfaces in homotopy classes and (under certain restrictions on M) the existence of essential PL-least area disks. As an application the authors discuss a theorem of W. Meeks III, L. Simon and S. T. Yau [Ann. Math., II. Ser. 116, 621-659 (1982; Zbl 0521.53007)] saying that any covering of a \(P^ 2\)-irreducible 3-manifold is \(P^ 2\)-irreducible.

Reviewer: M.Fuchs