Geometric constructions related to isoparametric foliations Chao Qian Beijing Institute of Technology Based on the joint works with Prof. Peng and Tang 02. 06. 2019 Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Contents Introduction 1 Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties Related constructions 2 A generalization by Q.-Tang Pinkall-Thorbergsson Construction Curvature of leaves 3 Ricci curvature sectional curvature Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Contents Introduction 1 Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties Related constructions 2 A generalization by Q.-Tang Pinkall-Thorbergsson Construction Curvature of leaves 3 Ricci curvature sectional curvature Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Singular Riemannian foliation Let N be a connected complete Riemannian manifold. Definition A transnormal system F is a subdivision of N into submanifolds, the so-called leaves, such that geodesics perpendicular to one leaf stay perpendicular to all. Moreover, if there are vertical vector fields { X i ∈ Γ( TN ) | i ∈ I } such that T p L ( p ) = span { X i | p | i ∈ I } , (1) for any p ∈ N and L ( p ) the leaf through p , then ( N , F ) is said to be a singular Riemannian foliation . Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Singular Riemannian foliation Let N be a connected complete Riemannian manifold. Definition A transnormal system F is a subdivision of N into submanifolds, the so-called leaves, such that geodesics perpendicular to one leaf stay perpendicular to all. Moreover, if there are vertical vector fields { X i ∈ Γ( TN ) | i ∈ I } such that T p L ( p ) = span { X i | p | i ∈ I } , (1) for any p ∈ N and L ( p ) the leaf through p , then ( N , F ) is said to be a singular Riemannian foliation . Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric function Definition Let N be a closed Riemannian manifold. A smooth function f : N → R is said to be transnormal , if there is a smooth function b such that |∇ f | 2 = b ( f ) . (2) If moreover ∆ f = a ( f ) (3) for a function a , then f is called isoparametric . Write Im f = [ α, β ] . According to Wang [Math. Ann., 1987], ( Im f ) ◦ has only regular value, and each level set f − 1 ( t ) is a smooth submanifold. Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric function Definition Let N be a closed Riemannian manifold. A smooth function f : N → R is said to be transnormal , if there is a smooth function b such that |∇ f | 2 = b ( f ) . (2) If moreover ∆ f = a ( f ) (3) for a function a , then f is called isoparametric . Write Im f = [ α, β ] . According to Wang [Math. Ann., 1987], ( Im f ) ◦ has only regular value, and each level set f − 1 ( t ) is a smooth submanifold. Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric function Define M + = f − 1 ( β ) and M − = f − 1 ( α ) , the focal submanifolds . Actually, all the level sets form a singular Riemannian foliation of codimension 1, such that each regular leaf has CMC, the so-called isoparametric foliation (of codimension 1). Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric function Define M + = f − 1 ( β ) and M − = f − 1 ( α ) , the focal submanifolds . Actually, all the level sets form a singular Riemannian foliation of codimension 1, such that each regular leaf has CMC, the so-called isoparametric foliation (of codimension 1). Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric foliations in unit spheres Cartan : 1938: Isoparametric hypersurface in R n + 1 , S n + 1 or H n + 1 ⇐ ⇒ hypersurface with CPC. 1939-1940: g = 3 , tubes over Veronese embeddings of F P 2 in S 4 , S 7 , S 13 , S 25 for F = R , C , H , Ca . Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric foliations in unit spheres unzner : M n , an isoparametric hypersurface in S n + 1 with constant M¨ principal curvatures cot θ i , 0 < θ 1 < · · · < θ g < π , with multiplicities m i , then: 1). θ k = θ 1 + k − 1 g π ; 2). m k = m k + 2 (subscripts mod g ); 3). M : open subset of level hypersurface in S n + 1 of a homogeneous polynomial F of degree g on R n + 2 satisfying Cartan-M¨ unzner equations: � |∇ F | 2 = g 2 | x | 2 g − 2 , = m 2 − m 1 g 2 | x | g − 2 ; △ F 2 4). g = 1 , 2 , 3 , 4 , 6 . Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric foliations in unit spheres Cecil-Chi-Jensen , S. Immervoll , Q. Chi : For g=4, isoparametric hypersurfaces must be OT-FKM type or homogeneous with (2,2), (4,5). Dorfmeister-Neher , g=6, m=1, R. Miyaoka , g=6, m=2: For g=6, isoparametric hypersurfaces are homogenous. Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Isoparametric foliations in unit spheres Cecil-Chi-Jensen , S. Immervoll , Q. Chi : For g=4, isoparametric hypersurfaces must be OT-FKM type or homogeneous with (2,2), (4,5). Dorfmeister-Neher , g=6, m=1, R. Miyaoka , g=6, m=2: For g=6, isoparametric hypersurfaces are homogenous. Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties OT-FKM constructions For a symmetric Clifford system { P 0 , · · · , P m } on R 2 l , i.e. , P i ’s are symmetric matrices satisfying P i P j + P j P i = 2 δ ij I 2 l , Ferus-Karcher-M¨ unzner [Math. Z., 1981] constructed a m polynomial F on R 2 l defined by F ( x ) = | x | 4 − 2 � � P i x , x � 2 . Then i = 0 f = F | S 2 l − 1 is an isoparametric function,i.e., |∇ f | 2 = 16 ( 1 − f 2 ) , � △ f = 8 ( m 2 − m 1 ) − 4 ( 2 l + 2 ) f . Consequently, each level hypersurface has g = 4 and ( m 1 , m 2 ) = ( m , l − m − 1 ) , the so-called OT-FKM type. Denote focal submanifolds by M + = f − 1 ( 1 ) and M − = f − 1 ( − 1 ) , which have codimension m 1 + 1 and m 2 + 1 . Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties OT-FKM constructions For a symmetric Clifford system { P 0 , · · · , P m } on R 2 l , i.e. , P i ’s are symmetric matrices satisfying P i P j + P j P i = 2 δ ij I 2 l , Ferus-Karcher-M¨ unzner [Math. Z., 1981] constructed a m polynomial F on R 2 l defined by F ( x ) = | x | 4 − 2 � � P i x , x � 2 . Then i = 0 f = F | S 2 l − 1 is an isoparametric function,i.e., |∇ f | 2 = 16 ( 1 − f 2 ) , � △ f = 8 ( m 2 − m 1 ) − 4 ( 2 l + 2 ) f . Consequently, each level hypersurface has g = 4 and ( m 1 , m 2 ) = ( m , l − m − 1 ) , the so-called OT-FKM type. Denote focal submanifolds by M + = f − 1 ( 1 ) and M − = f − 1 ( − 1 ) , which have codimension m 1 + 1 and m 2 + 1 . Chao Qian Geometric constructions related to isoparametric foliations
Introduction Isoparametric foliation Related constructions Isoparametric foliation in unit spheres Curvature of leaves Miscellaneous properties Miscellaneous properties Q.-Tang-Yan [A.G.A.G., 2013], Xie [Acta Math. Sinica, 2015]: M ± are Willmore in unit spheres; Q.-Tang [P .L.M.S., 2016]: Focal maps from isoparametric hypersurfaces M to M ± are harmonic; Q.-Tang [P .L.M.S., 2016]: For g = 4 , M ± are minimal with σ ( M ± ) = max { | B ( X , X ) | 2 | X ∈ TM ± , | X | = 1 } = 1 , which provide infinitely many counterexamples to Conjecture A and Conjecture B of P. F. Leung [Bull. L.M.S., 1991]; Chao Qian Geometric constructions related to isoparametric foliations
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