We describe partial results by schoenyau, shi, zhu, meekssimonyau, anonovburagozalgaller, andersonrodriguez and zhu. Since these manifolds have special holonomy, one might ask whether compact manifolds with nonnegative ricci curvature and generic holonomy admit a. Minimal surfaces section that every compact riemannian 3manifold ad. Nonsingular solutions of the ricci flow on three manifolds richard s. The presented results have been obtained in joint work with lucas ambrozio, alessandro carlotto, and ben sharp. For example, any solution to the ricci flow on a compact threemanifold with positive ricci curvature is nonsingular, as are the equivariant solutions on torus. Sectional curvature is a further, equivalent but more geometrical, description of the curvature of riemannian manifolds. Geometrization of 3manifolds via the ricci flow michael t. The study of manifolds with lower ricci curvature bound has experienced tremendous progress in the past. A sphere theorem for three dimensional manifolds with integral.
Cohomogeneity one manifolds with positive ricci curvature 3 which we also record as h. For lower dimensional manifolds, we have a positive answer. Summer school and conference on geometry and topology of 3. Pdf examples of manifolds of positive ricci curvature. If the torsion invariant c is critical, the webster curvature cf. To explain the interest of the ow, let us recall the main result of that paper. In riemannian geometry, the natural framework for the study of spaces with positive curvature seems to be a lower bound on ricci curvature see e. Suppose that m, g is an ndimensional riemannian manifold, equipped with its levicivita connection the riemannian curvature tensor of m is the 1, 3tensor defined by. This is a weakly parabolic system and hamil ton showed that on a three dimensional manifold any initial metric of positive ricci curvature flows into a metric of constant positive curvature when evolved by equation 1. Construction of manifolds of positive ricci curvature with. Abstract in this paper we address the issue of uniformly positive scalar curvature on noncompact 3manifolds. Annals of mathematics, 116 1982, 621659 embedded minimal surfaces, exotic spheres, and manifolds with positive ricci curvature by william meeks iii, leon simon and shingtung yau let n be a three dimensional riemannian manifold. We show for such spaces, that a solution to ricci flow exists for a short time, and that the solution is smooth for all positive times and that it has nonnegative ricci curvature.
An analogous result restricting the ricci curvature of g was obtained in 4. In all cases, a ginvariant metric on m is determined by its restriction to the regular part. This is a weakly parabolic system and hamil ton showed that on a three dimensional manifold any initial metric of positive riccicurvature flows into a metric of constant positive curvature when evolved by equation 1. Since these manifolds have special holonomy, one might ask whether compact manifolds with nonnegative ricci curvature and generic holonomy admit a metric with positive ricci curvature. It is shown that a connected sum of an arbitrary number of complex projective planes carries a metric of positive ricci curvature with diameter one and, in contrast with the earlier examples of shayang and. Although individually, the weyl tensor and ricci tensor do not in general determine the full curvature tensor, the riemann curvature tensor can be decomposed into a weyl part and a ricci part.
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. The main idea in hamiltons approach is to control the positivity of the curvature tensor under the ricci ow using a form of parabolic maximum principle for tensors. Ricci flow with surgery on four manifolds with positive isotropic curvature chen, binglong and zhu, xiping, journal of differential geometry, 2006. Threemanifolds of positive ricci curvature and convex weakly. Ricci curvature is also special that it occurs in the einstein equation and in the ricci. The problem is analogous to yamabes problem on the conformed transformation of riemannian manifolds most recently, r.
Manifolds with constant ricci curvature are called einstein manifolds, and not very much is known about which obstructions there are for a manifold with ric. Construction of manifolds of positive ricci curvature with big volume and large betti numbers g. It remains to show that all manifolds in iv carry a metric with nonnegative ricci curvature, and to determine which manifolds in ii and iv carry one with positive ricci curvature. If r 2, then m admits a contact metric of positive ricci curvature. Curvature of riemannian manifolds uc davis mathematics. Scalar curvature is a function on any riemannian manifold, usually denoted by sc. Finite extinction time for the solutions to the ricci. For scalar curvature the situation is fairly well understood by comparison. Nonsingular solutions of the ricci flow on threemanifolds. The proof uses the ricci flow with surgery, the conformal method, and the connected sum construction of gromov and lawson. In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3manifold is. Quite a lot is known about manifolds with nonnegative or positive ricci curvature.
Ricci curvature of metric spaces university of chicago. More recently, marques mar12, using ricci ow with surgeries, proved the pathconnectedness of the space of metrics with positive scalar curvature on three manifolds. Ricci flow with surgery on fourmanifolds with positive isotropic curvature chen, binglong and zhu, xiping, journal of differential geometry, 2006. One can show that each class of kcrsik k positive scalar curvature using hp4 and manifolds in our proof and in 7. For a complete noncompact 3manifold with nonnegative ricci curvature, we prove that either it is diffeomorphic to.
Ricci flow of almost nonnegatively curved three manifolds. Hamilton showed that if the ricci curvature of a threedimensional manifold was initially positive, then one had. In section 5 we present schoenyaus proof that three manifolds with ricci 0 are diffeomorphic to 3. In four dimensions it is an open question to date whether there are.
This curvature scalar is a measure of how the area of an infinitesimal surface differs on a curved manifold as compared to the same surface in flat space. Rn rn denote the ricci tensor of r and ric0 the traceless part of ric. Positive ricci curvature on highly connected manifolds crowley, diarmuid and wraith, david j. Let t p m denote the tangent space of m at a point p. This approach was worked out in the classical paper 8 for 3manifolds with positive ricci curvature by proving a series of striking a priori estimates for solutions of the ricci. The proof uses the ricci ow with surgery, the conformal method, and the. This study uses a subriemannian generalization of the classical riemannian curvature. The generalized ricci curvature was introduced by the rst author in the 90s for some special cases including the three dimensional contact sub. More recently, marques mar12, using ricci ow with surgeries, proved the pathconnectedness of the space of metrics with positive scalar curvature on threemanifolds.
Geometrization of 3 manifolds via the ricci flow michael t. The work of perelman on hamiltons ricci flow is fundamental. His approach started a systematic study of the socalled ricci. Deforming threemanifolds with positive scalar curvature. Ricci flow with surgery on fourmanifolds with positive isotropic curvature chen, binglong and zhu, xiping, journal of differential geometry, 2006 positive ricci curvature on highly connected manifolds crowley, diarmuid and wraith, david j. Recall that npositive ricci curvature is positive scalar curvature and one. Special surgery constructions as in sy, wr and bundle constructions as in na have resulted in a large number of interesting manifolds with positive ricci curvature. Then it is a question of basic interest to see whether one. Hamilton launched a new program in order to prove the geometrization conjecture. Compactness of the space of embedded minimal surfaces with free boundary in threemanifolds with nonnegative ricci curvature and convex boundary fraser, ailana and li, martin manchun, journal of differential geometry, 2014. Manifolds with positive curvature operators 1081 ric0 are the curvature operators of traceless ricci type.
Open manifolds with asymptotically nonnegative curvature bazanfare, mahaman, illinois journal of mathematics, 2005. Fourmanifolds with positive isotropic curvature 3 corollary 1. This decomposition is known as the ricci decomposition, and plays an important role in the conformal geometry of riemannian manifolds. In higher dimensions it turns out that the ricci curvature is more complicated than the scalar curvature. In particular we show that the whitehead manifold lacks such a. Ricci curvature also appears in the ricci flow equation, where a timedependent riemannian metric is deformed in the direction of minus its ricci curvature. In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact threemanifold is pathconnected. Geometric analysis, submanifolds and geometry of pdes. In 16, schoen and yau proved that a complete noncompact 3manifold with positive ricci curvature is diffeomorphic to r3, they also announced the classi. A ricci curvature bound is weaker than a sectional curvature bound but stronger than a scalar curvature bound. Aspects of ricci curvature 87 one should compare these three steps with the corresponding three steps in the proof of theorem 1. February 1, 2008 in our previous paper we constructed complete solutions to the ricci.
In order to have any metric of positive ricci curvature we must have. Manifolds with a lower ricci curvature bound 207 definition 3. Estimate of distances and angles for positive ricci curvature. Introduction to ricci curvature and the convergence theory. Large portions of this survey were shamelessly stolen. In this paper we study the evolution of almost nonnegatively curved possibly singular three dimensional metric spaces by ricci flow.
Throughout we assume that m,g is an ndimensional riemannian manifold with n. Oct 24, 2012 for a complete noncompact 3manifold with nonnegative ricci curvature, we prove that either it is diffeomorphic to. This allows us to classify the topological type and the differential structure of the limit manifold in view of hamiltons theorem on closed three manifolds with. To prove this result, hamilton considered the evolution of the metric under the ricci ow and showed that it converges to a metric of constant positive sectional curvature. The curvature tensor can be decomposed into the part which depends on the ricci curvature, and the weyl tensor. Using ricci ow on closed threemanifolds, hamilton ham82 showed that the space of metrics with positive ricci curvature is pathconnected. Large manifolds with positive ricci curvature springerlink. Using ricci ow on closed three manifolds, hamilton ham82 showed that the space of metrics with positive ricci curvature is pathconnected. In particular we show that the whitehead manifold lacks such a metric, and in fact that r3 is the only contractible noncompact 3manifold with a metric of uniformly positive scalar curvature. Manifolds with positive curvature operators are space forms. If a compact, simply connected three manifold has positive ricci curvature, the metric deforms under the ricci. In dimensions 2 and 3 weyl curvature vanishes, but if the dimension n 3 then the second part can be nonzero.
T, and the volume one rescalings gt of gt converge to a constant curvature metric as t. Manifolds of low cohomogeneity and positive ricci curvature. Ricci curvature and fundamental group of complete manifolds. Abstract in this paper we address the issue of uniformly positive scalar curvature on noncompact 3 manifolds. Given a curvature operator r we let ri and rric 0 denote the projections onto i and ric0, respectively. Manifolds with a lower ricci curvature bound international press. Theorem 1 if m3 is a threedimensional contractible manifold with a complete metric of.
Only 3spheres have constant positive curvature the only simply connected, compact three manifolds carrying. On static threemanifolds with positive scalar curvature. Deforming threemanifolds with positive scalar curvature by fernando cod a marques abstract in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact threemanifold is pathconnected. Hamilton, on riemannian metrics adapted to threedimensional contact. The result does not depend on the choice of orthonormal basis. Hermitian curvature flow and curvature positivity conditions. M4 is a compact fourmanifold with positive isotropic curvature, then a if ixi 1,m4 is diffeomorphic to s4 b if ixi z2,m4 is diffeomorphic to rp4 c if tti z, m4 is diffeomorphic to s3 x 51 if it is oriented, and to sxs1 if it is not. Introduction to manifolds, curvature, connections, the covariant derivative, the riemann tensor, and the ricci tensor. We say that a nonprincipal orbit gk is exceptional if dimgk dimgh or equivalently kh s0. Given 0 and 0 0 such that, for any m of dimension nwith ricm n. Observe also that ifg 0 denotes the identity component ofg,theng 0 acts by cohomogeneity one on m as well, but generally mg. This system of partial differential equations is a nonlinear analog of the heat equation, and was first introduced by richard s.
Positive ricci curvature is also very relevant from a probabilistic or analytic point of view, as illustrated by the works of gromov 6 and bakry and emery 2,3 on concentration of. Li concerning noncompact manifolds with nonnegative ricci curvature and maximal volume. We show that if the initial manifold has positive ricci curvature and the boundary is convex nonnegative second. Manifolds of positive scalar curvature lenny ng 18. Nonsingular solutions of the ricci flow on threemanifolds 697 c the solution collapses. Deforming three manifolds with positive scalar curvature by fernando cod a marques abstract in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is pathconnected. Hamilton showed that if the ricci curvature of a threedimensional manifold was initially. The main results of this paper are that if n is a complete manifold of positive ricci.
371 170 1027 175 1173 1124 415 558 244 1256 127 1350 1220 1079 606 930 680 1205 344 604 1358 1060 1092 3 1155 1482 699 524 417 830 1035 910 1284 964 1222 1314 340 636 260 376 1073 1471 32 306 1356 735