Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book contains a self-consistent treatment of a geometric averaging technique, induced by the Ricci flow, that allows comparing a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold. This is a case study where two vibrant areas of research in geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. The interaction is of great relevance for applications in relativistic cosmology, allowing a mathematically rigorous approach to the initial data set averaging problem, at least when data sets are given on a closed space-like hypersurface. The book does not assume an a priori knowledge of Ricci flow theory, and considerable space is left for introducing the necessary techniques. These introductory parts gently evolve to a detailed discussion of the more advanced results concerning a Fourier-mode expansion and a sophisticated heat kernel representation of the Ricci flow, both of which are of independent interest in Ricci flow theory. This work is intended for advanced students in mathematical physics and researchers alike. .

Ricci flow. --- Flow, Ricci --- Evolution equations --- Global differential geometry

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Ricci Flow for Shape Analysis and Surface Registration introduces the beautiful and profound Ricci flow theory in a discrete setting. By using basic tools in linear algebra and multivariate calculus, readers can deduce all the major theorems in surface Ricci flow by themselves. The authors adapt the Ricci flow theory to practical computational algorithms, apply Ricci flow for shape analysis and surface registration, and demonstrate the power of Ricci flow in many applications in medical imaging, computer graphics, computer vision and wireless sensor network. Due to minimal pre-requisites, this book is accessible to engineers and medical experts, including educators, researchers, students and industry engineers who have an interest in solving real problems related to shape analysis and surface registration. .

Algorithms. --- Mathematical physics. --- Riemannian manifolds. --- Ricci flow --- Computer vision --- Discrete groups --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Computer science --- Computer vision. --- Discrete groups. --- Geometry. --- Mathematics. --- Math --- Groups, Discrete --- Machine vision --- Vision, Computer --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Computer graphics. --- Computer mathematics. --- Convex geometry. --- Discrete geometry. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Computational Mathematics and Numerical Analysis. --- Convex and Discrete Geometry. --- Science --- Euclid's Elements --- Infinite groups --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Optical data processing. --- Convex geometry . --- Combinatorial geometry --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Optical equipment --- Ricci flow. --- Evolution equations.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.   The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman’s celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation). As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman’s ideas: the Kähler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman’s surgeries.

Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematics. --- Partial differential equations. --- Functions of complex variables. --- Differential geometry. --- Several Complex Variables and Analytic Spaces. --- Partial Differential Equations. --- Differential Geometry. --- Kählerian structures. --- Ricci flow. --- Flow, Ricci --- Evolution equations --- Global differential geometry --- Structures, Kählerian --- Complex manifolds --- Geometry, Differential --- Hermitian structures --- Differential equations, partial. --- Global differential geometry. --- Partial differential equations --- Differential geometry --- Complex variables --- Elliptic functions --- Functions of real variables