Richard M. Schoen

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The purpose of this paper is to study minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian three-manifold N, then the condition that M be stable is expressed analytically by the requirement that o n any compact domain of M, the first eigenvalue of the operator(More)
We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, ∆u + n(n−2) 4 u n+2 n−2 = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of(More)
One of the basic problems of Riemannian geometry is the classification of manifolds of positive sectional curvature. The known examples include the spherical space forms which carry constant curvature metrics and the rank 1 symmetric spaces whose canonical metrics have sectional curvatures at each point varying between 1 and 4. In 1951, H.E. Rauch [26](More)
A classical theorem due to M. Berger [2] and W. Klingenberg [11] states that a simply connected Riemannian manifold whose sectional curvatures all lie in the interval [1, 4] is either isometric to a symmetric space or homeomorphic to Sn (see also [12], Theorems 2.8.7 and 2.8.10). In this paper, we provide a classification, up to diffeomorphism, of all(More)
In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques. The Penrose inequality in general relativity is closely related to the(More)
Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain(More)