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Renormalization in quantum field theory and the RiemannHilbert problem. II. The βfunction, diffeomorphisms and the renormalization group
 Comm. Math. Phys
"... We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann–Hilbert problem. Given a loop γ(z), z  = 1 of elements of a complex Lie group G the general procedure is given by evalu ..."
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Cited by 332 (39 self)
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by evaluation of γ+(z) at z = 0 after performing the Birkhoff decomposition γ(z) = γ−(z) −1 γ+(z) where γ±(z) ∈ G are loops holomorphic in the inner and outer domains of the Riemann sphere (with γ−(∞) = 1). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop
The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature.
 Comm. Pure Appli. Math.
, 1980
"... The purpose of this paper is to study minimal surfaces in threedimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian threemanifold N, then the condition that M be stable is expressed analytically by the requirement that o n a ..."
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Cited by 192 (1 self)
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stable minimal surface M is a plane (Corollary 4). The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement
Variational theory for the total scalar curvature functional for Riemannian metrics and related topics
 in Topics in Calculus of Variations (Montecatini
, 1987
"... The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We ..."
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Cited by 177 (2 self)
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compute the first and secol~d variation and observe the distinction which arises between conformal directions and their orthogonal complements. We discuss variational characterizations of constant curvalure m trics, and give a proof of 0bata's uniqueness theorem. Much of the material in this section
SPHERE THEOREMS IN GEOMETRY
, 2009
"... In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques ..."
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Cited by 5 (1 self)
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In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface
Separators for spherepackings and nearest neighbor graphs
 J. ACM
, 1997
"... Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the ..."
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Cited by 100 (8 self)
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Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those
The Extremal Spheres Theorem
, 2009
"... Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two em ..."
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Cited by 1 (0 self)
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empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in ddimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres. In this paper, we show
Proof of the Riemannian Penrose inequality using the positive mass theorem
 MR MR1908823 (2004j:53046) MATHEMATICAL GENERAL RELATIVITY 73
"... We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3manifolds with nonnegative scalar curvature which contain minimal sphe ..."
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Cited by 119 (14 self)
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spheres. In particular, if we consider a Riemannian 3manifold as a totally geodesic submanifold of a spacetime in the context of general relativity, then outermost minimal spheres with total area A correspond to apparent horizons of black holes contributing a mass A/16π, scalar curvature corresponds
Theorems
, 2003
"... We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a MaxwellChern Simons field. For constant dilaton, the theory appea ..."
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Cited by 1 (1 self)
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appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can
The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign
 Math
, 1994
"... Abstract. We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries. 1. ..."
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Cited by 33 (3 self)
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Abstract. We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries. 1.
Results 1  10
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