Three-manifolds with positive Ricci curvature

“…from which we infer, in view of [6,Lemma 14.2], that the metrics (g ij (s)) are all uniformly equivalent in I and converge to a positive definite metric when s → b. Hence, the limit metric belongs to M andγ stays in a compact subset of M .…”

Section: Lemma 42 the Hypersurface M = M (X) Is A Cauchy Hypersurfamentioning

confidence: 69%

“…Since the metrics g ij are also looked at as the section of a fiber bundle, we define the bundle E as a fiber bundle with base space S 0 and fibers 6) where the fibers are a metric product.…”

Section: Gravity Interacting With a Scalar Fieldmentioning

confidence: 99%

The quantization of gravity in globally hyperbolic spacetimes

Gerhardt

2013

Advances in Theoretical and Mathematical Physics

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We apply the Arnowitt-Deser-Misner approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) we eliminate the diffeomorphism constraints, (ii) we replace the densities √ g by a function ϕ(x, g ij ) with the help of a fixed metric χ such that the Lagrangian and hence the Hamiltonian are functions, (iii) we consider the Lagrangian to be defined in a fiber bundle with base space S 0 and fibers F (x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic, and (iv) the Hamiltonian operator H is a normally hyperbolic operator in the bundle acting only in the fibers and the Wheeler-DeWitt equation Hu = 0 is a hyperbolic equation in the bundle. Since the corresponding Cauchy problem can be solved for arbitrary smooth data with compact support, we then apply the standard techniques of Algebraic Quantum Field Theory (QFT) which can be naturally modified to work in the bundle.e-print archive:

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“…As an example, we mention developments of new geometric tools in nonlinear PDEs [98,37,210,164], fully nonlinear elliptic equations, optimal transportation and Navier-Stokes equations [57,29,218,139] as examples which should pave the way for future developments in numerical methods.…”

Section: Mathematical Models and Modern Mathematical Toolsmentioning

confidence: 99%

“…We conclude with an example from topology. The Ricci flow, introduced by Hamilton [98] and used by Perelman [164,153] to solve the Poincaré conjecture, takes the form (2.10)…”

Section: Examples Of Time-dependent Pdes Atomic Physics Is Dominatedmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Three-manifolds with positive Ricci curvature

Cited by 2,362 publications

References 4 publications

On the weak Kähler-Ricci flow

On the weak Kähler-Ricci flow

The quantization of gravity in globally hyperbolic spacetimes

A review of numerical methods for nonlinear partial differential equations

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