The ricci flow, named after gregorio ricci curbastro, was first introduced by richard s. Visualizing ricci flow of manifolds of revolution rubinstein, j. Let m be a compact simplyconnected manifold admitting a riemannian metric whose sectional curvatures satis. Ricci flow and the sphere theorem 51 pinched in the global sense must be homeomorphic to the standard sphere sn. Analyzing the ricci flow of homogeneous geometries 8 5. Ricci flow and the poincare conjecture 5 the sphere is curved, and the amount by which we rotate depends on the curvature. Geometrization of 3manifolds via the ricci flow michael t. We present a new curvature condition that is preserved by the ricci flow in higher dimensions. The book is dedicated almost entirely to the analysis of the ricci flow, viewed first as a heat type equation hence its consequences, and later from the more recent developments due to perelmans monotonicity formulas and the blowup analysis of the flow which was made thus possible. Also, brendle and schoen 7 proved the differentiable sphere theorem by using ricci flow. Manifolds with bakryemery ricci curvature bounded below. Finally, we assume that the curvature tensor of m,g0 lies in the cone c for all points p. Ricci flow and the sphere theorem about this title.
Ricci flowbased spherical parameterization and surface. This work depends on the accumulative works of many geometric analysts in the past thirty years. An optimal differentiable sphere theorem for complete. The precise statement of the theorem is as follows. The theorem is that for d bigger than 1, every connected positive curvature dgraph is a d sphere. The ricci flow of a geometry with isotropy so 2 15 7. Stability of the ricci flow 743 if one has determined the stability of ricci flow convergence for metrics near a specified flat metric go theorem 3. Mains results the following theorem is a similar theorem proved in and 5 and is a generaliz4 ation of myers theorem. Conversely, the ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. A mathematical interpretation of hawkings black hole theory. This question has been studied by many authors during the past decades, a milestone being the topological sphere theorem of berger and klingenberg. This evolution equation is known as the ricci flow, and.
This book gives a concise introduction to the subject with the hindsight of perelmans breakthroughs from 20022003. Ricci flow and the sphere theorem fields institute for. A sphere theorem for manifolds of positive ricci curvature by katsuhiro shiohama dedicated to professor s. A new differentiable sphere theorem is obtained from the view of submanifold geometry. Ln2pinching theorem for submanifolds in a sphere xu, huiqun, kodai mathematical journal, 2007 differentiable pinching theorems for submanifolds via ricci flow huang, fei, xu, hongwei, and zhao, entao, tohoku mathematical journal, 2015.
In riemannian geometry, the sphere theorem, also known as the quarterpinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The ricci flow in riemannian geometry a complete proof. I will discuss the history of this problem and sketch the proof of the differentiable sphere theorem. The ricci flow on the 2 sphere article pdf available in journal of differential geometry 331991 january 1991 with 845 reads how we measure reads. The ricci flow on 2orbifolds was originally studied by langfang wu, who considered the case of a positively curved initial metric on a closed 2orbifold with positive euler characteristic, proving global existence and convergence to a shrinking gradient ricci soliton metric after adjustment by diffeomorphisms. The ricci flow on the sphere with marked points phong, d. Ricci flow and the poincare conjecture siddhartha gadgil and harish seshadri the eld of topology was born out of the realisation that in some fundamental sense, a sphere and an ellipsoid resemble each other but di er from a torus the surface of a rubber tube or a doughnut. This is an invited contribution for the bulletin of the am. By contrast, in the plane the holonomy is always the identity, i. In 1956, milnor 8 had shown that there exist smooth manifolds which are homeomorphic but not diffeomorphic to s7 socalled exotic 7spheres. 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. Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that ricm 1, km.
Download it once and read it on your kindle device, pc, phones or tablets. Buy ricci flow and the sphere theorem graduate studies in. A striking instance of this can be seen by imagining water. Anderson 184 noticesoftheams volume51, number2 introduction the classification of closed surfaces is a milestone in the development of topology, so much so that it is now taught to most mathematics undergraduates as an introduction to topology. Hamilton in 1981 and is also referred to as the riccihamilton flow. In the second part, we sketch the proof of the differentiable sphere theorem, and discuss various related results.
Ricci flow and the sphere theorem graduate studies in. The ricci flow, named after gregorio riccicurbastro, was first introduced by richard s. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as hamiltons ricci flow. Buy ricci flow and the sphere theorem graduate studies in mathematics book online at best prices in india on. The proofs of the poincare conjecture and the closely related 3dimensional spherical spaceform conjecture are then immediate. 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. This is a survey paper focusing on the interplay between the curvature and topology of a riemannian manifold. In 1960, marcel berger and wilhelm klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.
The ricci flow is a technique first exploited by richard hamiton back in the early 80s to study various invariant gemoetric properties of manifolds. This evolution equation is known as the ricci flow, and it has since been used widely and with great success, most notably in. This book gives a concise introduction to the subject with the hindsight. It has proven to be a very useful tool in understanding the topology of such manifolds. Cylinder to sphere rule ricci flow mathematics stack exchange. But avoid asking for help, clarification, or responding to other answers. It forms the heart of the proof via ricci flow of thurstons geometrization conjecture. Hamilton in 1981 and is also referred to as the ricci hamilton flow. These results employ a variety of methods, including geodesic and minimal surface techniques as well as hamiltons ricci flow. For initial metrics satisfying this condition, we establish a higher dimensional version of hamiltons necklike curvature pinching estimate. Hyam and sinclair, robert, experimental mathematics, 2005. Pdf curvature, sphere theorems, and the ricci flow. As the main technical tool used in the book is the maximum principle, familiar to any undergraduate, this book would make a fine reading course for advanced undergraduates or postgraduates and, in particular, is an excellent. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
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. The first part of the paper provides a background discussion, aimed at nonexperts, of hopfs pinching problem and the sphere theorem. In 1956, milnor 8 had shown that there exist smooth manifolds which are homeomorphic but not diffeomorphic to s7 socalled exotic 7. The ricci flow of a geometry with maximal isotropy so 3 11 6. If m is a complete, simplyconnected, ndimensional riemannian manifold with sectional curvature taking values in the interval, then m is homeomorphic. Hamiltons ricci flow has attracted considerable attention since its introduction in 1982, owing partly to its promise in addressing the poincare conjecture and thurstons geometrization conjecture. 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. The ricci flow on the 2sphere article pdf available in journal of differential geometry 331991 january 1991 with 845 reads how we measure reads. 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 with surgery in higher dimensions annals of.
Ricci flow theorem hamilton 1982 for a closed surface of nonpositive euler characteristic,if the total area of the surface is preserved during the. 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. It is highly recommended for both researchers and students interested in differential geometry, topology and ricci flow. Spheres are very nice spaces in that every point looks like every other point. Hamilton set about constructing a programme to use ricci ow to prove the thurston geometrisation conjecture see 41, which classi es threemanifolds, and contains the famous poincar e conjecture as a special case. Use features like bookmarks, note taking and highlighting while reading the ricci flow in. The ricci flow of a geometry with trivial isotropy 17 notes and commentary 19 chapter 2. 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. A mathematical interpretation of hawkings black hole. Ancient solutions to the ricci flow 543 corollary 7 let. Hamilton introduced a nonlinear evolution equation for riemannian metrics with the aim of finding canonical metrics on manifolds. The existence of ricci flow with surgery has application to 3manifolds far beyond the poincare conjecture.