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The entropy formula for the Ricci flow and its geometric applications
TLDR
In this article, a monotonic expression for Ricci flow, valid in all dimensions and without curvature assumptions, is presented, interpreted as an entropy for a certain canonical ensemble.Abstract:
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.read more
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On the geometry of metric measure spaces. II
TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.
Posted Content
Ricci flow with surgery on three-manifolds
TL;DR: In this article, the Ricci flow with surgeries was constructed, and a lower bound on the volume of maximal horns and the smoothness of solutions was established. But this lower bound was later shown to be unjustified and irrelevant for the other conclusions.
Journal ArticleDOI
Geometrically Accurate Topology-Correction of Cortical Surfaces Using Nonseparating Loops
TL;DR: The proposed method is a wholly self-contained topology correction algorithm, which determines geometrically accurate, topologically correct solutions based on the magnetic resonance imaging (MRI) intensity profile and the expected local curvature.
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Membranes at Quantum Criticality
Petr Hořava,Petr Hořava +1 more
TL;DR: In this paper, a quantum theory of membranes designed such that the ground-state wavefunction of the membrane with compact spatial topology reproduces the partition function of the bosonic string on worldsheet Σh was proposed.
Book
The Ricci Flow: An Introduction
Bennett Chow,Dan Knopf +1 more
TL;DR: The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci Flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities as discussed by the authors.
References
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On the parabolic kernel of the Schrödinger operator
Peter Li,Shing-Tung Yau +1 more
TL;DR: Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x, t)=0 sur une variete riemannienne generale as discussed by the authors.
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Asymptotic-behavior for singularities of the mean-curvature flow
TL;DR: In this paper, the authors study the singularities of (1) which can occur for nonconvex initial data and characterize the asymptotic behavior of the hypersurface Mt near a singularity using rescaling techniques.
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On the structure of spaces with Ricci curvature bounded below. I
Jeff Cheeger,Tobias H. Colding +1 more
TL;DR: In this article, the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound was studied.
Journal ArticleDOI
Four-manifolds with positive curvature operator
Book
Quantum Fields and Strings: A Course for Mathematicians
TL;DR: The first truly comprehensive introduction to quantum field theory and perturbative string theory aimed at a mathematics audience can be found in this article, which offers a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum fields and string theory.