Gromov-Lawson-Schoen-Yau theory Introduction of Rie. . . . and - PowerPoint PPT Presentation

Gromov-Lawson-Schoen-Yau theory Introduction of Rie. . . . and isoparametric foliations Gromov-Lawson theory The “double” manifold . . . Tang Zizhou ↔ ✴ ✭ ➩ ↕ Home Page School of Mathematical Sciences, Beijing Normal University Title Page zztang@bnu.edu.cn ◭◭ ◮◮ Joint work with Xie Y.Q. ↔ ✜ ④ ➣ ↕ & Yan W.J. ↔ ☎ ➞ ✝ ↕ ◭ ◮ Available at arXiv: 1107.5234 Page 1 of 21 Go Back Full Screen Close Quit

1 Introduction Introduction of Rie. . . . Gromov-Lawson theory The “double” manifold . . . Definition 1.1. A Riemannian manifold M is said to carry a metric of positive scalar curvature R M if Home Page R M ≥ 0 and R M ( p ) > 0 at some point p ∈ M. Title Page ✷ ◭◭ ◮◮ Denote by R M > 0 if M carries a metric of positive scalar curvature ( p.s.c. ). ◭ ◮ Page 2 of 21 Question: Which compact manifolds admit Riemannian metrics of p.s.c. ? Go Back Full Screen Close Quit

Theorem (A. Lichnerowicz, 1963) For a Rie. manifold X 4 k , which is com- pact and Spin ⇒ � R X > 0 = A ( X ) = 0 . Introduction of Rie. . . . Gromov-Lawson theory ✷ The “double” manifold . . . A ( C P 2 k ) = ( − 1) k 2 − 4 k � 2 k � Remark For example: C P 2 k is not Spin, but � � = 0 . k Home Page Title Page Theorem (N. Hitchin, 1974) There is a ring homomorphism ◭◭ ◮◮ α : Ω spin → KO − n ( pt ) ◭ ◮ − ∗ Page 3 of 21 α = � A if dim = 4 k . For X compact spin, R X > 0 ⇒ α ( X ) = 0 . ✷ Go Back For example There exist 8 k + 1 and 8 k + 2 dimensional exotic spheres with Full Screen α � = 0 . Thus, these exotic spheres admit no metrics of p.s.c. Close Quit

Theorem Introduction of Rie. . . . Gromov-Lawson theory (Gromov-Lawson, [Ann. of Math. 1980]; The “double” manifold . . . Schoen-Yau, [Manuscripta Math. 1979]) Let M be a manifold obtained from a compact Riemannian manifold N by Home Page surgeries of codim ≥ 3 . Then Title Page R N > 0 = ⇒ R M > 0 . ◭◭ ◮◮ ✷ ◭ ◮ Page 4 of 21 Go Back Full Screen Close Quit

2 Gromov-Lawson theory around a point Let X be a Rie. manifold of dimension n with R X > 0 . Fix p ∈ X with R X ( p ) > 0 . D n := { x ∈ X n : | x | ≤ r } : a small normal ball centered at p . Consider a hypersurface of D n × R : M n := { ( x, t ) ∈ D n × R : ( | x | , t ) ∈ γ } Introduction of Rie. . . . Gromov-Lawson theory The “double” manifold . . . where | x | = dist ( x, p ) , and γ is a curve in the ( r, t ) -plane as pictured below: Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 21 Go Back Full Screen Close N : the unit exterior normal vector of M . The curve γ begins with a vertical line segment t = 0 , r 1 ≤ r ≤ ¯ r , and ends with a horizontal line segment Quit r = r ∞ > 0 , with r ∞ small enough.

Fix q = ( x, t ) ∈ M corresponding to ( r, t ) ∈ γ . orthonormal basis on T q M ← → principal curvatures of M e 1 , e 2 , ..., e n − 1 , e n ← → λ 1 , λ 2 , ..., λ n − 1 , λ n := k. � �� � =( − 1 Introduction of Rie. . . . r + O ( r )) sinθ Gromov-Lawson theory The “double” manifold . . . where e n is the tangent vector to γ , k ≥ 0 is the curvature of the plane curve γ . By Gauss equuation: ij = K D × R K M + λ i λ j , ij Home Page Since D × R has the product metric, Title Page K D × R = K D 1 ≤ i, j ≤ n − 1 ◭◭ ◮◮ ij , ij K D × R = K D ∂r ,j cos 2 θ, ◭ ◮ ∂ n,j Page 6 of 21 ⇒ R M = R D − 2 Ric D ( ∂ ∂r, ∂ ∂r ) sin 2 θ + ( n − 1)( n − 2)( 1 Go Back r 2 + O (1)) sin 2 θ = Full Screen + 2 ( n − 1)( − 1 r + O ( r )) ksinθ Close Quit

3 The “double” manifold on isoparamet- ric foliation Assumptions: X n ( n ≥ 3 ) compact, connected, ∂X = ∅ . Y n − 1 : a compact, connected embedding hypersurface in X , ( ⇒ ∃ a unit normal vector field ξ on Y ), with trivial normal bundle Introduction of Rie. . . . ( ⇒ Y n − 1 separates X n into two components, X n Gromov-Lawson theory + , X n and π 0 ( X − Y ) � = 0 − ). The “double” manifold . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 21 Go Back Full Screen ξ on Y � a unit normal v.f. in a neighborhood of Y, still denoted by ξ. Close D ( X ± ) := the double of X ± , the manifold obtained by gluing X ± with itself Quit along the boundary Y .

Define a continuous function r : X n − → R � dist ( x, Y ) if x ∈ X + x �→ − dist ( x, Y ) if x ∈ X − where dist ( x, Y ) is the distance from x to the hypersurface Y . Introduction of Rie. . . . Gromov-Lawson theory Let Y r := { x ∈ X | r ( x ) = r } , r > 0 small. Consider a manifold The “double” manifold . . . M n := { ( x, t ) ∈ X n × R | ( | r ( x ) | , t ) ∈ γ, | r ( x ) | ≤ ¯ r } where γ is the plane curve as before. Home Page Title Page Fix q = ( x, t ) ∈ M ∩ ( X + × R ) , corresponding to ( r, t ) ∈ γ ( r > 0 ). ◭◭ ◮◮ Choose an o.n. basis e 1 , e 2 , ..., e n − 1 on T x Y r such that ◭ ◮ A ξ e i = µ i e i for i = 1 , ..., n − 1 , Page 8 of 21 where A ξ is the shape operator of the hypersurface Y r in X . Go Back Principal curvatures of M in X + × R : Full Screen λ i = µ i sinθ for i = 1 , ..., n − 1 , where sinθ := � N, ξ � Close λ n := k. Quit

We obtain: k � ij = R X + 2 Asin 2 θ + 2 kH ( r ) sinθ K M R M = (1) i � = j Introduction of Rie. . . . Gromov-Lawson theory where The “double” manifold . . . n − 1 � � µ i µ j − Ric X ( ξ, ξ ) , A := H ( r ) = µ i ( r ) : mean curvature of Y r . Home Page i<j ≤ n − 1 i =1 Title Page ◭◭ ◮◮ Gromov and Lawson computed the scalar curvature of M constructed from a submanifold with trivial normal bundle. Their formula is expressed in form ◭ ◮ of estimate, losing a factor 2 and one item related to the second fundamental Page 9 of 21 form of the submanifold. But this mistake would result in the missing of the Go Back item H ( r ) in our formula (1), which is essential for our research. Rosenberg and Stolz [Ann. Math. Studies, 2001] modified Gromov- Full Screen Lawson’s expression, but they also lost the second fundamental form. Close Quit

From now on, we deal with X n = S n (1) , and Y n − 1 is a minimal isoparamet- ric hypersurface in S n (1) , i.e. , minimal hypersurface with constant principal curvatures, separating S n into S n + ( r ≥ 0 ) and S n − ( r ≤ 0 ). Gauss equation implies Introduction of Rie. . . . Gromov-Lawson theory S = ( n − 1)( n − 2) − R Y The “double” manifold . . . where S is norm square of the second fundamental form. Home Page Peng and Terng :( [Annals of Math. Studies, 1983] ) Title Page If Y is a minimal isoparametric hypersurface in S n , then ◭◭ ◮◮ ◭ ◮ S = ( g − 1)( n − 1) , Page 10 of 21 where g is the number of distinct principal curvatures of Y . Go Back Therefore, R Y ≥ 0 , and Full Screen R N = 0 ⇐ ⇒ ( m + , m − ) = (1 , 1) . Close Quit

Theorem 3.1 Let Y n − 1 be a minimal isoparametric hypersurface in S n (1) , n ≥ 3 . Then each of doubles D ( S n + ) and D ( S n − ) has a metric of positive scalar curvature. Moreover, there is still an isoparametric foliation in D ( S n + ) (or D ( S n − ) ) . ✷ Introduction of Rie. . . . Gromov-Lawson theory Outline of proof. The scalar curvature of M restricted to Y r is The “double” manifold . . . R M | Y r = n ( n − 1) cos 2 θ +( n − g − 1)( n − 1) sin 2 θ + a ( r ) sin 2 θ +2 kH ( r ) sinθ, Home Page where H ( r ) has the property that Title Page H (0) = 0 and H ( r ) > 0 for any r > 0 , ◭◭ ◮◮ and a ( r ) satisfies ◭ ◮ lim r → 0 a ( r ) = 0 Page 11 of 21 In fact, a ( r ) is identically 0 when n − 1 − g = 0 . Go Back In each of two cases n − 1 − g > 0 and n − 1 − g = 0 , we can control the Full Screen “bending angle” of the curve γ , so that R M | Y r > 0 . Close Quit

Let Y be a compact minimal isoparametric hypersurface in S n with focal sub- manifolds M + and M − . Introduction of Rie. . . . Gromov-Lawson theory Proposition 3.2 Let the ring of coefficient R = Z if M + and M − are both The “double” manifold . . . orientable and R = Z 2 , otherwise. Then for the cohomology groups, we have isomorphisms:  Home Page + )) ∼ H 0 ( D ( S n = R    + )) ∼  H 1 ( D ( S n = H 1 ( M + ) Title Page  + )) ∼ = H q − 1 ( M − ) ⊕ H q ( M + ) H q ( D ( S n for 2 ≤ q ≤ n − 2 ◭◭ ◮◮  + )) ∼  H n − 1 ( D ( S n = H n − 2 ( M − )   ◭ ◮  + )) ∼ H n ( D ( S n = R Page 12 of 21 For D ( S n − ) , similar identities hold. ✷ Go Back Full Screen Close Quit