Minimal immersions of closed Riemann surfaces

Abstract

Abstract. Let M be a closed orientable surface of genus larger than zero and N a compact Riemannian manifold. If u: M -> N is a continuous map, such that the map induced by it between the fundamental groups of M and N contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformai branched minimal immersion s: M -N having least area among all branched immersions with the same action on it¡(M) as u. Uniqueness within the homotopy class of u fails in general: It is shown that for certain 3-manifolds which fiber over the circle there are at least two geometrically distinct conformai branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those 3-manifolds for which uniqueness fails.Introduction. In a previous paper we obtained results on minimal immersions of the two-sphere into compact Riemannian manifolds [18]. Here we extend some of these results to surfaces of higher topological type. The main idea is to reduce the minimal area problem for such surfaces to a variational problem on a moduli space for conformai structures on the surface. The main technical difficulty to overcome is the fact that the moduli spaces are not compact. We overcome this by using a standard compactification of the Riemann moduli space (see Bers [5, 6] and Abikoff [2]) and controlling the behaviour of our variational problem near the boundary points.Our main result, Theorem 4.4, proves the existence of a conformai branched minimal immersion of a surface M with genus larger than zero corresponding to every homotopy class of maps u: M -* N which induces an injection u^: ttx(M) -» 77,(jV). Here N is any compact Riemannian manifold of dimension larger than two.

Compactification of moduli space of harmonic mappings

“…The important work of Sacks-Uhlenbeck [SU1] asserts the existence of a non-constant harmonic map from S 2 into a compact Riemannian manifold N with non-contractible universal cover. Sacks-Uhlenbeck [SU2] and Schoen-Yau [SU] proved independently the following beautiful result on incompressible minimal surfaces: if φ : π 1 (Σ) → π 1 (M ) is an injective homomorphism, then there exists a branched minimal immersion from Σ to M which minimizes area among maps in the same conjugacy class of φ. Note that this does not say that each homotopy class of maps from Σ (even S 2 ) to N contains a harmonic representative.…”

“…In the given homotopy class, first find minimizing harmonic nodal maps for the given conformal structures; then minimize the energy among conformal structures. The existence of minimal surfaces has been extensively studied by many people for a very long time and many important results have been obtained (see [Do], [Gu], [Hi], [J2], [Mo], [MY], [MSY1,2], [Os], [SU1], [SU2], [SY], [TT] for example).…”

A counterexample to the energy identity for sequences ofα-harmonic maps

“…In fact, by discussing the convergence of α-harmonic map sequences, Sacks and Uhlenbeck developed an existence theory on minimal surfaces in [17,18]. In particular, they established in [17] the well-known ǫ-regularity theorem on α-harmonic maps and removal singularity theorem on harmonic maps, which will be used repeatedly in the present paper.…”

Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case

“…Hence, by the standard covering argument (c.f. [17] and Theorem 2.3 in [18]), there exist finitely many points {x 1 , x 2 , . .…”