The ricci flow is a geometric evolution equation of. I will discuss the history of this problem and sketch the proof of the differentiable sphere theorem. A mathematical interpretation of hawkings black hole. Schoen, year2010 this is a survey paper focusing on the interplay between the curvature and topology of a riemannian manifold. The ricci flow in riemannian geometry a complete proof. To prove the theorem, we use a geomag lemma which shows that every geodesic in g can be extended to an immersed 2graph s of positive curvature and must so be a. Buy ricci flow and the sphere theorem graduate studies in mathematics book online at best prices in india on. The strategy used in the proof of the differentiable sphere theorem is to deform. Ricci flow and the sphere theorem fields institute for.
In this paper, using perelmans no local collapsing theorem and the geometric interpretation of hamiltons harnack expressions along the ricci flow intr oduced by r. These include the topological sphere theorem of berger and klingenberg as well as the differentiable version obtained by the authors. An introduction to curveshortening and the ricci flow. Ricci flowbased spherical parameterization and surface. The ricci flow in riemannian geometry a complete proof of the.
We use euclidean ricci flow for the computation of spherical parameterization. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. This question has been studied by many authors during the past decades, a milestone being the topological sphere theorem of berger and klingenberg. Hamiltons ricci flow princeton math princeton university. This monograph gives an account of the recent proof of the famous. In particular, the fundamental group is virtually free, that is, there exists a free subgroup of finite index in. In particular the proof of the differentiable sphere theorem relies heavily on this work. This evolution equation is known as the ricci flow, and it has since been used widely and with great success, most notably in perelmans solution of the poincare conjecture. We use these results to prove the \original ricci ow theorem the 1982 theorem of richard hamilton that closed 3manifolds which admit metrics of strictly positive ricci curvature are di eomorphic to quotients of the round 3 sphere by nite groups of isometries acting freely. Anderson 184 noticesoftheams volume51, number2 introduction the classification of closed surfaces is a milestone in the development of topology, so much so that.
The ricci ow exhibits many similarities with the heat equation. A complete proof of the differentiable 14pinching sphere theorem. Visualizing ricci flow of manifolds of revolution, experimental mathematics v. The ricci flow, named after gregorio riccicurbastro, was first introduced by richard s. Therefore, it, unlike spherical ricci flow, converges to the global optimum. In this paper, we give a complete proof of the poincar. This work depends on the accumulative works of many geometric analysts in the past thirty years.
The metric inducing the target curvature is the unique global optimum of the ricci energy. The proof relies on the ricci flow method pioneered by richard hamilton. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as hamiltons ricci flow. A class of examples closely related to the 3 sphere are the 3dimensional spherical spaceforms, i. By contrast, in the plane the holonomy is always the identity, i. Ricci flow theorem hamilton 1982 for a closed surface of nonpositive euler characteristic,if the total area of the surface is preserved during the. A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean. There exists a unique connection ron tmthat is torsion free r xyr yx x.
Pdf on jan 1, 1991, bennett chow and others published the ricci flow on the 2 sphere find, read and cite all the research you need on researchgate. The ricci flow, named after gregorio ricci curbastro, was first introduced by richard s. Ricci flow and the poincare conjecture 5 the sphere is curved, and the amount by which we rotate depends on the curvature. Warped products in this section we prove theorem 1. Ricci flow and the sphere theorem graduate studies in. This work depends on the accumulative works of many geometric analysts. A class of examples closely related to the 3sphere are the 3dimensional spherical spaceforms, i. With the third part of their volume on techniques and applications of the theory, the authors give a presentation of hamiltons ricci flow for graduate students and mathematicians interested in working in the subject, with an emphasis on the geometric and analytic aspects.
Ricci flow and the sphere theorem simon brendle american mathematical society providence, rhode island. Hamilton, we present a mathematical interpretation of hawkings black hole theory in 1. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems. After describing the basic properties of, and intuition behind the ricci flow, core elements of the theory are discussed such as consequences of various forms of maximum principle, issues related to existence theory, and basic properties of singularities in the flow.
By contrast, the proof of the compactness theorem for the ricci flow is greatly aided by the fact that a. In the twosphere case, the hamiltonchow convergence proof. This is a survey paper focusing on the interplay between the curvature and topology of a riemannian manifold. This book focuses on hamiltons ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for riemannian manifolds, and perelmans noncollapsing results, and culminating in a detailed. It is wellknown that the ricci flow of a closed 3manifold containing an essential minimal 2 sphere will fail to exist after a finite time. In the second part, we sketch the proof of the differentiable sphere theorem, and discuss various related results. The ricci flow in riemannian geometry springerlink. Spheres are very nice spaces in that every point looks like every other point. An optimal differentiable sphere theorem for complete.
One famous example is the 14 pinching sphere theorem of klingernberg, berger and rauch. This book focuses on hamiltons ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for riemannian manifolds, and perelmans noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of bohm and wilking and. Math 230br advanced di erential geometry taught by karsten gimre notes by dongryul kim spring 2018 this course was taught by karsten gimre, on tuesdays and thursdays from. Hamilton in 1981 and is also referred to as the ricci hamilton flow. Estimating the trace free ricci tensor in ricci flow 3 main theorem. The set of zoll metrics is not preserved by the ricci flow. These tools also led to the proof of the differentiable sphere theorem by brendle. Furthermore, various convergence theorems have been established.
Buy ricci flow and the sphere theorem graduate studies in. Isotropic curvature and the ricci flow international. Schoen, year2010 this is a survey paper focusing on the interplay between the curvature and topology of a. Ricci flow and the sphere theorem 51 pinched in the global sense must be homeomorphic to the standard sphere sn. The ricci flow in riemannian geometry a complete proof of. An introduction to hamiltons ricci flow olga iacovlenco department of mathematics and statistics, mcgill university, montreal, quebec, canada abstract in this project we study the ricci ow equation introduced by richard hamilton in 1982. In 1956, milnor 8 had shown that there exist smooth manifolds which are homeomorphic but not diffeomorphic to s7 socalled exotic 7spheres. This paper, originally written in the early eighties, is published in the collected papers on ricci flow book edited by h. We use these results to prove the \original ricci ow theorem the 1982 theorem of richard hamilton that closed 3manifolds which admit metrics of strictly positive ricci curvature are di eomorphic to quotients of the round 3sphere by nite groups of isometries acting freely. Conversely, the ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. Geometrization of 3manifolds via the ricci flow michael t. Pdf curvature, sphere theorems, and the ricci flow.
This strategy has been used to prove the convergence of the ricci. The class was based on a series of papers, with brendles ricci flow and the sphere theorem as a main reference. A conformally invariant gap theorem characterizing. Sep 12, 2009 to prove that the positive isotropic curvature is preserved by the ricci flow, we will use a refined version the advanced maximum principle. The ricci flow preserves metric symmetries the group of isometries and the set of killing fields. Springer nature is making coronavirus research free. We need some basic concepts before returning to parallel transport and curvature. The ricci flow does not preserve the set of zoll metrics. A complete proof of the differentiable 14pinching sphere theorem lecture notes in mathematics, vol. Positive complex sectional curvature, ricci flow and the. Ricci flow and metric geometry by michael raymond robert co ey thesis submitted to the university of warwick for the degree of doctor of philosophy.
In 1960, marcel berger and wilhelm klingenberg proved the topological version of the sphere theorem with the optimal pinching constant. Hamilton in 1981 and is also referred to as the riccihamilton flow. Therefore, the ricci energy is convex and converges to the global minimum. The first part of the paper provides a background discussion, aimed at nonexperts, of hopfs pinching problem and the sphere theorem. It has natural applications in the analysis of singularities of the ricci ow by blowup which we will employ in the proof of the di erentiable sphere theorem. The ricci flow, named after gregorio riccicurbastro, was first introduced by richard hamilton in 1981 and is also referred to as the riccihamilton flow.
In fact, in addition to the conditions on f required for the. The ricci flow in riemannian geometry mathematical sciences. I believe that the ricci flow on orbifolds was first studied by hamilton in the paper threeorbifolds with positive ricci curvature, answering a question of thurston. Feb 15, 2010 buy ricci flow and the sphere theorem graduate studies in mathematics book online at best prices in india on. Tutorial on surface ricci flow, theory, algorithm and. This book focuses on hamiltons ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for riemannian manifolds, and perelmans noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of. In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the yau rigidity theory of submanifolds. Request pdf positive complex sectional curvature, ricci flow and the differential sphere theorem the paper provides a different proof of the result of brendleschoen on the differential sphere. The sphere theorem we say that the riemannian manifold m,g is. The ricci flow uses methods from analysis to study the geometry and topology of manifolds. The value of y is fixed it only depends on x, t, while we are free to take t as. Ricci flow for 3d shape analysis xianfeng gu 1sen wang junho kim yun zeng1 yang wang2 hong qin 1dimitris samaras 1stony brook university 2carnegie mellon university abstract ricci. A mathematical interpretation of hawkings black hole theory.
I discuss certain applications of the ricci flow in physics. In this paper, we give a survey of various sphere theorems in geometry. Hamilton introduced a nonlinear evolution equation for riemannian metrics with the aim of finding canonical metrics on manifolds. For zoll surfaces, given the global symmetries of the periodic structure in the tangent bundle, one might hope that the zoll structure was preserved under the ricci flow. A ricci flow is a family pgtqtpi of riemannian metrics on a smooth manifold. It is the primary tool used in grigori perelman s solution of the poincare conjecture, 1 as well as in the proof of the differentiable sphere theorem by simon brendle and richard. Ricci flow for 3d shape analysis carnegie mellon school. Keywords black hole, ricci flow, no local collapsing theorem, uncertainty principle. Uniformization theorem theuniformization theoremsays that any riemannian metric on a compact orientable surface is conformally equivalent to a constant curvature metric.
The paper used in this book is acid free and falls within the guidelines. In 1951, harry rauch showed that a simply connected manifold with curvature in 34,1 is homeomorphic to a sphere. The ricci flow on the sphere with marked points phong, d. Buy ricci flow and the sphere theorem graduate studies in mathematics on free shipping on qualified orders. Denote by mthe trace free part of the hessian of f, i. Know that ebook versions of most of our titles are still available and may be downloaded.
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