Following the work of Hamilton, Ilmanen and the first author [4], in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability (also called ν-stability in this paper) restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more ν-stable and ν-unstable examples than previously known and also the first ν-stable examples, other than the standard spheres, whose second variations are negative definite.

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

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

Guy Bonneau has kindly pointed out two errors in [1]. The first is that the manifolds M (k) and M (k)/Z2 do not admit U (2)-invariant Einstein-Weyl structures for k 2; thus the last four entries in Table 4 (page 422) do not occur. The error is on pages 429430. The analysis there is correct, except that the critical line to consider in Lemma 7.5 and the subsequent calculation is = , instead of = 2/k, because of the constraint (D, ](equivalently, &gt; ) occurring in the definition of . On this line, one has

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.