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. 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 site is like a library, use search box in the widget to get ebook. 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. In the second part, we sketch the proof of the differentiable sphere theorem, and discuss various related results. Definitions in this section, we recall the socalled reduced geometry of the ricci.
It forms the heart of the proof via ricci flow of thurstons geometrization conjecture. I will discuss the history of this problem and sketch the proof of the differentiable sphere theorem. This site is like a library, use search box in the widget to get ebook that you want. A conformally invariant gap theorem characterizing. In this paper, we give a survey of various sphere theorems in geometry. 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 proofs of the poincare conjecture and the closely related 3dimensional spherical spaceform conjecture are then immediate. Let m be a compact simplyconnected manifold admitting a riemannian metric whose sectional curvatures satis. These results employ a variety of methods, including geodesic and minimal surface techniques as well as hamiltons ricci flow. If g is any metric on a riemann surface, then under. These include the topological sphere theorem of berger and klingenberg as well as the differentiable version obtained by the authors. In this short note we give a new proof of a theorem of hamilton and chow on the ricci flow on the 2sphere.
Uniformization theorem theuniformization theoremsays that any riemannian metric on a compact orientable surface is conformally equivalent to a constant curvature metric. In 1960, marcel berger and wilhelm klingenberg proved the topological version of the sphere theorem with the optimal pinching constant. Pdf on jan 1, 1991, bennett chow and others published the ricci flow on the 2sphere find, read and cite all the research you need on. Ricci flow and the sphere theorem fields institute for. The first part of the paper provides a background discussion, aimed at nonexperts, of hopfs pinching problem and the sphere theorem. Ricci flow and the sphere theorem 51 pinched in the global sense must be homeomorphic to the standard sphere sn. 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. It is wellknown that the ricci flow of a closed 3manifold containing an essential minimal 2sphere will fail to exist after a finite time. Ancient solutions to the ricci flow in higher dimensions 11 shrinking sphere by corollary 3. To prove the poincare conjecture, perelman 25,26 used ricci flow and its surgery. It is a process that deforms the metric of a riemannian manifold in a way formally analogous to the diffusion of heat. The poincar conjecture download ebook pdf, epub, tuebl, mobi. Heuristically speaking, at every point of the manifold the ricci flow shrinks directions of.
This is a survey paper focusing on the interplay between the curvature and topology of a riemannian manifold. Also, brendle and schoen 7 proved the differentiable sphere theorem by using ricci flow. The ricci flow in riemannian geometry a complete proof of. In the mathematical field of differential geometry, the ricci flow. The ricci flow on the sphere with marked points phong, d.
Hamiltons ricci flow, g converges to a metric of constant curvature. Ricci flow and the poincare conjecture john morgan, gang. In 1956, milnor 8 had shown that there exist smooth manifolds which are homeomorphic but not diffeomorphic to s7 socalled exotic 7. A complete proof of the differentiable 14pinching sphere theorem find, read and. The proof relies on the ricci flow method pioneered by richard hamilton. This work depends on the accumulative works of many geometric analysts in the past thirty years. It develops hamiltons ricci flow from the ground up leading to brendle and schoens proof of the differentiable sphere theorem and also provides a very good. Conversely, the ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. This is an invited contribution for the bulletin of the am. Differentiable pinching theorems for submanifolds via ricci flow huang, fei, xu, hongwei, and zhao, entao, tohoku mathematical journal, 2015. Ricci flow is the gradient flow of the action functional of dilaton gravity. To help you succeed, whether you are teaching or conducting research remotely, or just looking for a good book to read at home, all ams ebooks will be 50% off list price through may 31. In the twosphere case, the hamiltonchow convergence proof.
Download fulltext pdf the ricci flow on the 2 sphere article pdf available in journal of differential geometry 331991 january 1991 with 845 reads. The sphere theorem we say that the riemannian manifold m,g is. The ricci flow in riemannian geometry mathematical sciences. Ln2pinching theorem for submanifolds in a sphere xu, huiqun, kodai mathematical journal, 2007. Pdf curvature, sphere theorems, and the ricci flow. Curvature, sphere theorems, and the ricci flow by s. D minimal twospheres and the topology of manifolds with curvature on totally isotropic twoplanes. S is the euler characteristic number of the surface s, a0 is the total area at. 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. A complete proof of the differentiable 14pinching sphere theorem. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required.
Positive complex sectional curvature, ricci flow and the. Tutorial on surface ricci flow, theory, algorithm and. Geometrization of 3manifolds via the ricci flow michael t. It develops hamiltons ricci flow from the ground up leading to brendle and schoens proof of the differentiable sphere theorem and also provides a very good overview of the required geometry in the first chapter. 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. Cylinder to sphere rule ricci flow mathematics stack exchange. 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. 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. Download our spring pdf catalog for a look at our latest releases. 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. Buy ricci flow and the sphere theorem graduate studies in. In 1956, milnor 8 had shown that there exist smooth manifolds which are homeomorphic but not diffeomorphic to s7 socalled exotic 7spheres. Enter your mobile number or email address below and well send you a link to download the free kindle app.
The proofs of the poincara conjecture and the closely related 3dimensional spherical spaceform conjecture are then immediate. An application of the gaussbonnet theorem allows us to show that ancient solutions must have backwards limit an equator. The existence of ricci flow with surgery has application to 3manifolds far beyond the poincara conjecture. 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. Download focusing on sobolev inequalities and their applications to analysis on manifolds and ricci flow, sobolev inequalities, heat kernels under ricci flow, and the poincare conjecture introduces the field of analysis on riemann manifolds and uses the tools of sobolev imbedding and heat kernel estimates to study ricci flows, especially with surgeries.
As mentioned earlier, we intend to develop the reduced geometry in more general situation. The ricci flow does not preserve volume, so to be more careful, in applying the ricci flow to uniformization and geometrization one needs to normalize the ricci flow to obtain a flow which preserves volume. If one fails to do this, the problem is that for example instead of evolving a given threedimensional manifold into one of thurstons. To help you succeed, whether you are teaching or conducting research remotely, or just looking for a good book to read at home, all.
To prove that the positive isotropic curvature is preserved by the ricci flow, we will use a refined version the advanced maximum principle. Furthermore, various convergence theorems have been established. Click download or read online button to get the poincar conjecture book now. In 1951, harry rauch showed that a simply connected manifold with curvature in 34,1 is homeomorphic to a sphere. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as hamiltons ricci flow. Ricci flow and the sphere theorem graduate studies in. The ricci flow in riemannian geometry a complete proof. 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. The existence of ricci flow with surgery has application to 3manifolds far beyond the poincare conjecture. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The ams bookstore is open, but rapid changes related to the spread of covid19 may cause delays in delivery services for print products.
In this paper, we give a complete proof of the poincar. 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. Ben andrews and others published ricci flow in riemannian geometry. An alexandrov reflection argument then completes the proof.
If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized ricci flow on the two sphere. The ricci flow in riemannian geometry a complete proof of the. This question has been studied by many authors during the past decades, a milestone being the topological sphere theorem of berger and klingenberg.
Hamilton introduced a nonlinear evolution equation for riemannian metrics with the aim of finding canonical metrics on manifolds. Buy ricci flow and the sphere theorem graduate studies in mathematics book online at best prices in india on. A striking instance of this can be seen by imagining water. Hamiltons ricci flow princeton math princeton university. Isotropic curvature and the ricci flow international. Spheres are very nice spaces in that every point looks like every other point. We prove that the only ancient solutions to the curve shortening flow on the 2sphere are families of shrinking geodesic circles.
19 249 1315 604 21 81 1068 824 507 1095 167 1160 1465 694 3 928 623 881 711 1034 397 845 594 869 561 580 1196 776 725