5,216 research outputs found
On the topology and area of higher dimensional black holes
Over the past decade there has been an increasing interest in the study of
black holes, and related objects, in higher (and lower) dimensions, motivated
to a large extent by developments in string theory. The aim of the present
paper is to obtain higher dimensional analogues of some well known results for
black holes in 3+1 dimensions. More precisely, we obtain extensions to higher
dimensions of Hawking's black hole topology theorem for asymptotically flat
() black hole spacetimes, and Gibbons' and Woolgar's genus
dependent, lower entropy bound for topological black holes in asymptotically
locally anti-de Sitter () spacetimes. In higher dimensions the genus
is replaced by the so-called -constant, or Yamabe invariant, which is a
fundamental topological invariant of smooth compact manifolds.Comment: 15 pages, Latex2e; typos corrected, a convention clarified, resulting
in the simplification of certain formulas, other improvement
Constant mean curvature surfaces
In this article we survey recent developments in the theory of constant mean
curvature surfaces in homogeneous 3-manifolds, as well as some related aspects
on existence and descriptive results for -laminations and CMC foliations of
Riemannian -manifolds.Comment: 102 pages, 17 figure
Notes on a paper of Mess
These notes are a companion to the article "Lorentz spacetimes of constant
curvature" by Geoffrey Mess, which was first written in 1990 but never
published. Mess' paper will appear together with these notes in a forthcoming
issue of Geometriae Dedicata.Comment: 26 page
Minimal surfaces - variational theory and applications
Minimal surfaces are among the most natural objects in Differential Geometry,
and have been studied for the past 250 years ever since the pioneering work of
Lagrange. The subject is characterized by a profound beauty, but perhaps even
more remarkably, minimal surfaces (or minimal submanifolds) have encountered
striking applications in other fields, like three-dimensional topology,
mathematical physics, conformal geometry, among others. Even though it has been
the subject of intense activity, many basic open problems still remain. In this
lecture we will survey recent advances in this area and discuss some future
directions. We will give special emphasis to the variational aspects of the
theory as well as to the applications to other fields.Comment: Proceedings of the ICM, Seoul 201
Minimal surfaces in S^3: a survey of recent results
In this survey, we discuss various aspects of the minimal surface equation in
the three-sphere S^3. After recalling the basic definitions, we describe a
family of immersed minimal tori with rotational symmetry. We then review the
known examples of embedded minimal surfaces in S^3. Besides the equator and the
Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well
as a new family of examples found recently by Choe and Soret. We next discuss
uniqueness theorems for minimal surfaces in S^3, such as the work of Almgren on
the genus 0 case, and our recent solution of Lawson's conjecture for embedded
minimal surfaces of genus 1. More generally, we show that any minimal surface
of genus 1 which is Alexandrov immersed must be rotationally symmetric. We also
discuss Urbano's estimate for the Morse index of an embedded minimal surface
and give an outline of the recent proof of the Willmore conjecture by Marques
and Neves. Finally, we describe estimates for the first eigenvalue of the
Laplacian on a minimal surface.Comment: Published pape
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