The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in <i>M</i> that satisfy the required properties. The loop theorem respects the action of the group <i>G</i> in a suitable manner. The chapter puts a metric on <i>M</i> so that the group <i>G</i> acts isometrically and so that <i>M</i> is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk <i>D</i>₁ is demonstrated, in <i>M</i> whose boundary <i> D₁</i>, represents a nontrivial element in ₁(<i>S</i>) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary

this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension. We consider a vector-valued function $u : \Omega \subset \math{R}^n \to \math{R}^m$ that satisfies the minimal surface system, see equation (1.1) in §1. The graph of u is then a minimal submanifold of $\math{R}^{n+m}$. We prove an a priori gradient bound under the assumption that the Jacobian of $ du: \math{R}^n \to \math{R}^m$ on any two dimensional subspace of $\mtah{R}^n$ is less than or equal to one. This assumption is automatically satisfied when $du$ is of rank one and thus the estimate covers the case when m=1, i.e., the original minimal surface equation. This is applied to Bernstein type theorems for minimal submanifolds of higher codimension.

THE MAIN PURPOSE of this paper is to give a simple and unified new proof of the Witten rigidity theorems, which were conjectured by Witten and first proved by Taubes [25], Bott-Taubes [S], Hirzebruch Cl23 and Krichever [15]. Our proof shows that the modular invariance, which is the intrinsic symmetry of elliptic genera, actually implies their rigidity. Some new properties of elliptic genera and their relationships with theta-functions are also discussed. We remark that our proof makes essential uses of the new feature of loop groups and loop spaces, the modular invariance. We note that, with the help of the modular group, we can catch the topological information on loop space by simply working on finite-dimensional manifold. By developing this idea further, in [21] we have proved the rigidity of the Dirac operator on loop space twisted by higher-level loop group representations, while the Witten ridigity theorems are the special cases of level 1. Many topological vanishing theorems are also derived in [21] by refining the argument in this paper, especially an &amp;-vanishing theorem for loop space. In [19] modular invariance is used again to establish a general miraculous cancellation formula, relating the Hirzebruch L-form to certain twisted &amp;-forms, which has many interesting topological results as consequences. These results were announced in [20].

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Hartman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of complete surfaces with constant curvature in R 3.

Making use of the relations among 3-Sasakian manifolds, hypercomplex manifolds and quaternionic Khler orbifolds, we prove that complete 3-Sasakian manifolds are rigid.