Celebratio Mathematica — Uhlenbeck

Karen Uhlenbeck's contributions to gauge theoretic analysis

Works connected to Jonathan Sacks

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J. Sacks and K. Uhlenbeck: “The existence of minimal immersions of two-spheres,” Bull. Am. Math. Soc. 83 : 5 (1977), pp. 1033–1036. A related article with almost the same title was published in Ann. Math. 113:1 (1981). MR 448408 Zbl 0375.49016 article

J. Sacks and K. Uhlenbeck: “The existence of minimal immersions of 2-spheres,” Ann. Math. (2) 113 : 1 (January 1981), pp. 1–24. A related article with almost the same title was published in Bull. Am. Math. Soc. 83:5 (1977). MR 604040 Zbl 0462.58014 article

In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of isolated points, where the structure is that of a branch point. We obtain an existence theory for harmonic maps of orientable surfaces into Riemannian manifolds via a complete existence theory for a perturbed variational problem. Convergence of the critical maps of the pertured problem is sufficient to produce at least one harmonic map of the sphere into the Riemannian manifold. A harmonic map from a sphere is in fact a conformal branched minimal immersion.

J. Sacks and K. Uhlenbeck: “Minimal immersions of closed Riemann surfaces,” Trans. Am. Math. Soc. 271 : 2 (1982), pp. 639–652. MR 654854 Zbl 0527.58008 article

Let \( M \) be a closed orientable surface of genus larger than zero and \( N \) a compact Riemannian manifold. If \( u:M \to N \) 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 conformal branched minimal immersion \( s:M \to N \) having least area among all branched immersions with the same action on \( \pi_1(M) \) 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 conformal 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.