Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Classification in S n + 1 g = 1 , M n must be a hypersphere S n ; g = 2 , M n must be S p ( r ) × S n − p ( s ) ( r 2 + s 2 = 1 , r , s > 0 ) . g = 3 , M n must be homogeneous, m 1 = m 2 = 1 , 2 , 4 or 8 . ( E.Cartan , 1930’s) g = 4 , U. Abresch (1983), Z.Z. Tang (1991), F.Q. Fang (1999), S. Stolz (1999), Cecil-Chi-Jensen , Ann. Math. (2007), S.Immervoll , Ann. Math. (2008), Q.S. Chi , Nagoya Math. (2011), J.Diff. Geom. (2013), J.Diff. Geom. to appear : Theorem For g = 4 , M n is either of OT-FKM type or homogeneous with ( m 1 , m 2 ) = ( 2 , 2 ) , ( 4 , 5 ) . g = 6 , m 1 = m 2 = 1 or 2 ( U. Abresch (1983)). These two cases are both homogeneous: Dorfmeister-Neher , Comm. Algebra, 1985 , R.Miyaoka , Ann. Math. 2013, 2016. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Classification in S n + 1 g = 1 , M n must be a hypersphere S n ; g = 2 , M n must be S p ( r ) × S n − p ( s ) ( r 2 + s 2 = 1 , r , s > 0 ) . g = 3 , M n must be homogeneous, m 1 = m 2 = 1 , 2 , 4 or 8 . ( E.Cartan , 1930’s) g = 4 , U. Abresch (1983), Z.Z. Tang (1991), F.Q. Fang (1999), S. Stolz (1999), Cecil-Chi-Jensen , Ann. Math. (2007), S.Immervoll , Ann. Math. (2008), Q.S. Chi , Nagoya Math. (2011), J.Diff. Geom. (2013), J.Diff. Geom. to appear : Theorem For g = 4 , M n is either of OT-FKM type or homogeneous with ( m 1 , m 2 ) = ( 2 , 2 ) , ( 4 , 5 ) . g = 6 , m 1 = m 2 = 1 or 2 ( U. Abresch (1983)). These two cases are both homogeneous: Dorfmeister-Neher , Comm. Algebra, 1985 , R.Miyaoka , Ann. Math. 2013, 2016. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Classification in S n + 1 g = 1 , M n must be a hypersphere S n ; g = 2 , M n must be S p ( r ) × S n − p ( s ) ( r 2 + s 2 = 1 , r , s > 0 ) . g = 3 , M n must be homogeneous, m 1 = m 2 = 1 , 2 , 4 or 8 . ( E.Cartan , 1930’s) g = 4 , U. Abresch (1983), Z.Z. Tang (1991), F.Q. Fang (1999), S. Stolz (1999), Cecil-Chi-Jensen , Ann. Math. (2007), S.Immervoll , Ann. Math. (2008), Q.S. Chi , Nagoya Math. (2011), J.Diff. Geom. (2013), J.Diff. Geom. to appear : Theorem For g = 4 , M n is either of OT-FKM type or homogeneous with ( m 1 , m 2 ) = ( 2 , 2 ) , ( 4 , 5 ) . g = 6 , m 1 = m 2 = 1 or 2 ( U. Abresch (1983)). These two cases are both homogeneous: Dorfmeister-Neher , Comm. Algebra, 1985 , R.Miyaoka , Ann. Math. 2013, 2016. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Classification in S n + 1 g = 1 , M n must be a hypersphere S n ; g = 2 , M n must be S p ( r ) × S n − p ( s ) ( r 2 + s 2 = 1 , r , s > 0 ) . g = 3 , M n must be homogeneous, m 1 = m 2 = 1 , 2 , 4 or 8 . ( E.Cartan , 1930’s) g = 4 , U. Abresch (1983), Z.Z. Tang (1991), F.Q. Fang (1999), S. Stolz (1999), Cecil-Chi-Jensen , Ann. Math. (2007), S.Immervoll , Ann. Math. (2008), Q.S. Chi , Nagoya Math. (2011), J.Diff. Geom. (2013), J.Diff. Geom. to appear : Theorem For g = 4 , M n is either of OT-FKM type or homogeneous with ( m 1 , m 2 ) = ( 2 , 2 ) , ( 4 , 5 ) . g = 6 , m 1 = m 2 = 1 or 2 ( U. Abresch (1983)). These two cases are both homogeneous: Dorfmeister-Neher , Comm. Algebra, 1985 , R.Miyaoka , Ann. Math. 2013, 2016. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Classification in S n + 1 g = 1 , M n must be a hypersphere S n ; g = 2 , M n must be S p ( r ) × S n − p ( s ) ( r 2 + s 2 = 1 , r , s > 0 ) . g = 3 , M n must be homogeneous, m 1 = m 2 = 1 , 2 , 4 or 8 . ( E.Cartan , 1930’s) g = 4 , U. Abresch (1983), Z.Z. Tang (1991), F.Q. Fang (1999), S. Stolz (1999), Cecil-Chi-Jensen , Ann. Math. (2007), S.Immervoll , Ann. Math. (2008), Q.S. Chi , Nagoya Math. (2011), J.Diff. Geom. (2013), J.Diff. Geom. to appear : Theorem For g = 4 , M n is either of OT-FKM type or homogeneous with ( m 1 , m 2 ) = ( 2 , 2 ) , ( 4 , 5 ) . g = 6 , m 1 = m 2 = 1 or 2 ( U. Abresch (1983)). These two cases are both homogeneous: Dorfmeister-Neher , Comm. Algebra, 1985 , R.Miyaoka , Ann. Math. 2013, 2016. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Isoparametric hypersurface of OT-FKM type For a symmetric Clifford system { P 0 , · · · , P m } on R 2 l , i.e. , P i = P t i , P i P j + P j P i = 2 δ ij I 2 l Following Ozeki-Takeuchi (Tohoku Math. 1975,1976), Ferus, Karcher and M¨ unzner (Math.Z., 1981) constructed a homogeneous polynomial F of degree 4 on R 2 l : � m F ( x ) = | x | 4 − 2 � P i x , x � 2 , i = 0 where l = k δ ( m ) , k is a positive integer, δ ( m ) is valued: 1 2 3 4 5 6 7 8 · · · m +8 m δ ( m ) 16 δ ( m ) 1 2 4 4 8 8 8 8 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Proposition (Ferus, Karcher and M¨ unzner (Math.Z., 1981)) Denote f = F | S 2 l − 1 . Then f is an isoparametric function on S 2 l − 1 corresponding to g = 4 , ( m 1 , m 2 ) = ( m , l − m − 1 ) f = F | S 2 l − 1 : isoparametric function of OT-FKM type, M 2 l − 2 = f − 1 ( t ) : isoparametric hypersurface of OT-FKM type, M + = f − 1 ( 1 ) and M − = f − 1 ( − 1 ) : focal submanifolds of OT-FKM type. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Isoparametric hypersurface on Sn + 1 Positive answer to Yau’s 100th problem in the isoparametric case Classification in Sn + 1 Besse’s problem on generalizations of Einstein condition Isoparametric hypersurface of OT-FKM type A sufficient condition for a hypersurface to be isoparametric Proposition (Ferus, Karcher and M¨ unzner (Math.Z., 1981)) Denote f = F | S 2 l − 1 . Then f is an isoparametric function on S 2 l − 1 corresponding to g = 4 , ( m 1 , m 2 ) = ( m , l − m − 1 ) f = F | S 2 l − 1 : isoparametric function of OT-FKM type, M 2 l − 2 = f − 1 ( t ) : isoparametric hypersurface of OT-FKM type, M + = f − 1 ( 1 ) and M − = f − 1 ( − 1 ) : focal submanifolds of OT-FKM type. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Outline Introduction of isoparametric foliation 1 Positive answer to Yau’s 100th problem in the isoparametric case 2 Besse’s problem on generalizations of Einstein condition 3 A sufficient condition for a hypersurface to be isoparametric 4 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric ( M n , g ) : closed Riemannian manifold. for f ∈ C ∞ ( M ) . ∆ f = − div ( ∇ f ) ∆ is an elliptic operator and has a discrete spectrum: { 0 = λ 0 ( M ) < λ 1 ( M ) � λ 2 ( M ) � · · · � λ k ( M ) , · · · , ↑ ∞} . λ 1 ( M ) : the first eigenvalue of M . Theorem (T. Takahashi, 1966) Let f : M n → S N be an isometric immersion with the canonical coordinate ( x 1 , x 2 , ..., x N + 1 ) . Then ⇒ ∆( x i ◦ f ) = n ( x i ◦ f ) , ∀ 1 ≤ i ≤ N + 1 . M n is minimal ⇐ Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Yau conjecture Consequently, for a minimal hypersurface M n in the unit sphere S n + 1 λ 1 ( M n ) � n . Yau conjecture (1982, Problem Section, the 100th problem) Let M n be a closed embedded minimal hypersurface in the unit sphere S n + 1 . Then λ 1 ( M n ) = n . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Yau’s conjecture and Lawson’s conjecture Lawson conjecture(proved by S.Brendle [Acta Math. 2013] The only embedded minimal torus in S 3 is the Clifford torus! Remark S. Montiel and A. Ros (Invent. Math., 1986) showed that for minimal surfaces, ⇒ Yau conjecture Lawson conjecture . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Yau’s conjecture and Lawson’s conjecture Lawson conjecture(proved by S.Brendle [Acta Math. 2013] The only embedded minimal torus in S 3 is the Clifford torus! Remark S. Montiel and A. Ros (Invent. Math., 1986) showed that for minimal surfaces, ⇒ Yau conjecture Lawson conjecture . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem (Choi & Wang, J. Diff. Geom. 1983) Let M n be a closed embedded minimal hypersurface in S n + 1 . Then λ 1 ( M n ) � n 2 . We consider a restricted problem of Yau’s conjecture. Problem Let M n be a closed minimal isoparametric hypersurface in S n + 1 . Is it true that λ 1 ( M n ) = n ? Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem (Choi & Wang, J. Diff. Geom. 1983) Let M n be a closed embedded minimal hypersurface in S n + 1 . Then λ 1 ( M n ) � n 2 . We consider a restricted problem of Yau’s conjecture. Problem Let M n be a closed minimal isoparametric hypersurface in S n + 1 . Is it true that λ 1 ( M n ) = n ? Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Homogeneous cases Let M n be a minimal isoparametric hypersurface in S n + 1 . For g = 1 , 2 . The problem is trivially true. For g = 3 , 4 , 6 . Theorem (Muto, Ohnita & Urakawa, Tˆ ohoku Math. J. 1984; Kotani, Tˆ ohoku Math. J. 1985) Let M be the minimal homogeneous hypersurface with g = 3 , 6 and g = 4 , ( m 1 , m 2 ) = ( 1 , k ) , ( 2 , 2 ) . Then λ 1 ( M ) = dim( M ) . Combining with the classification results above, for g = 3 , 6 , λ 1 ( M ) = dim( M ) . Remark For homogeneous hypersurfaces with g = 4 , ( m 1 , m 2 ) must be ( 1 , k ) , ( 2 , 2 ) , ( 2 , 2 k − 1 ) , ( 4 , 4 k − 1 ) , ( 4 , 5 ) or ( 6 , 9 ) . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric General cases for g = 4 Theorem (H. Muto, Math. Z., 1988) For g = 4 , ( m 1 , m 2 ) = ( 3 , 4 k ) , ( 4 , 4 k + 3 ) , ( 7 , 8 k ) , k = 1 , 2 , 3 , · · · ( 4 , 5 ) ( 5 , 10 ) , ( 5 , 18 ) , ( 5 , 26 ) , ( 5 , 34 ) ( 6 , 9 ) , ( 6 , 17 ) , ( 6 , 25 ) , ( 6 , 33 ) ( 8 , 15 ) , ( 8 , 23 ) , ( 8 , 31 ) , ( 8 , 39 ) ( 9 , 22 ) , ( 9 , 38 ) ( 10 , 21 ) , ( 10 , 53 ) the first eigenvalue of the minimal isoparametric hypersurface M n in S n + 1 is λ 1 ( M n ) = n . Remark No results in homogeneous cases with ( m 1 , m 2 ) = ( 2 , 2 k − 1 )( k = 2 , 3 , · · · ) , and many other cases of OT-FKM-type with ( m 1 , m 2 ) = ( m , k δ ( m ) − m − 1 ) . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric General cases for g = 4 Theorem (H. Muto, Math. Z., 1988) For g = 4 , ( m 1 , m 2 ) = ( 3 , 4 k ) , ( 4 , 4 k + 3 ) , ( 7 , 8 k ) , k = 1 , 2 , 3 , · · · ( 4 , 5 ) ( 5 , 10 ) , ( 5 , 18 ) , ( 5 , 26 ) , ( 5 , 34 ) ( 6 , 9 ) , ( 6 , 17 ) , ( 6 , 25 ) , ( 6 , 33 ) ( 8 , 15 ) , ( 8 , 23 ) , ( 8 , 31 ) , ( 8 , 39 ) ( 9 , 22 ) , ( 9 , 38 ) ( 10 , 21 ) , ( 10 , 53 ) the first eigenvalue of the minimal isoparametric hypersurface M n in S n + 1 is λ 1 ( M n ) = n . Remark No results in homogeneous cases with ( m 1 , m 2 ) = ( 2 , 2 k − 1 )( k = 2 , 3 , · · · ) , and many other cases of OT-FKM-type with ( m 1 , m 2 ) = ( m , k δ ( m ) − m − 1 ) . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric the first eigenvalue of minimal isoparametric case Theorem Let M n be the minimal isoparametric hypersurface in S n + 1 . Then λ 1 ( M n ) = n . Theorem (Tang and Y., J. Diff. Geom. , 2013) Let M n be the minimal isoparametric hypersurface in S n + 1 with g = 4 and m 1 , m 2 ≥ 2 . Then λ 1 ( M n ) = n with multiplicity n + 2 . Remark The case with ( m 1 , m 2 ) = ( 1 , k ) is homogeneous, which has been proved by Muto-Ohnita-Urakawa. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric the first eigenvalue of minimal isoparametric case Theorem Let M n be the minimal isoparametric hypersurface in S n + 1 . Then λ 1 ( M n ) = n . Theorem (Tang and Y., J. Diff. Geom. , 2013) Let M n be the minimal isoparametric hypersurface in S n + 1 with g = 4 and m 1 , m 2 ≥ 2 . Then λ 1 ( M n ) = n with multiplicity n + 2 . Remark The case with ( m 1 , m 2 ) = ( 1 , k ) is homogeneous, which has been proved by Muto-Ohnita-Urakawa. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric the first eigenvalue of minimal isoparametric case Theorem Let M n be the minimal isoparametric hypersurface in S n + 1 . Then λ 1 ( M n ) = n . Theorem (Tang and Y., J. Diff. Geom. , 2013) Let M n be the minimal isoparametric hypersurface in S n + 1 with g = 4 and m 1 , m 2 ≥ 2 . Then λ 1 ( M n ) = n with multiplicity n + 2 . Remark The case with ( m 1 , m 2 ) = ( 1 , k ) is homogeneous, which has been proved by Muto-Ohnita-Urakawa. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our method is also applicable to the case g = 6 : Theorem (Tang, Xie and Y., J. Funct. Anal. 2014) Let M 12 be the minimal isoparametric hypersurface in S 13 with g = 6 and ( m 1 , m 2 ) = ( 2 , 2 ) . Then λ 1 ( M 12 ) = 12 with multiplicity 14 . Remark Muto, Ohnita & Urakawa computed the first eigenvalue of the minimal homogeneous hypersurface M 12 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our method is also applicable to the case g = 6 : Theorem (Tang, Xie and Y., J. Funct. Anal. 2014) Let M 12 be the minimal isoparametric hypersurface in S 13 with g = 6 and ( m 1 , m 2 ) = ( 2 , 2 ) . Then λ 1 ( M 12 ) = 12 with multiplicity 14 . Remark Muto, Ohnita & Urakawa computed the first eigenvalue of the minimal homogeneous hypersurface M 12 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric The first eigenvalues of the focal submanifolds Fact. Both M + and M − are minimal in S n + 1 . Hence, λ 1 ( M ± ) ≤ dim( M ± ) . Question λ 1 ( M ± ) = dim( M ± )? For the case : g = 1 , 2 , 3 , it is not difficult. For g = 4 , Theorem (Tang & Y., J. Diff. Geom. , 2013) Let M + be a focal submanifold of codimension m 1 + 1 in S n + 1 with g = 4 . If dim M + � 2 3 n + 1 , i . e . , m 2 � 1 2 ( m 1 + 3 ) , then λ 1 ( M + ) = dim M + = m 1 + 2 m 2 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric The first eigenvalues of the focal submanifolds Fact. Both M + and M − are minimal in S n + 1 . Hence, λ 1 ( M ± ) ≤ dim( M ± ) . Question λ 1 ( M ± ) = dim( M ± )? For the case : g = 1 , 2 , 3 , it is not difficult. For g = 4 , Theorem (Tang & Y., J. Diff. Geom. , 2013) Let M + be a focal submanifold of codimension m 1 + 1 in S n + 1 with g = 4 . If dim M + � 2 3 n + 1 , i . e . , m 2 � 1 2 ( m 1 + 3 ) , then λ 1 ( M + ) = dim M + = m 1 + 2 m 2 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric The first eigenvalues of the focal submanifolds Fact. Both M + and M − are minimal in S n + 1 . Hence, λ 1 ( M ± ) ≤ dim( M ± ) . Question λ 1 ( M ± ) = dim( M ± )? For the case : g = 1 , 2 , 3 , it is not difficult. For g = 4 , Theorem (Tang & Y., J. Diff. Geom. , 2013) Let M + be a focal submanifold of codimension m 1 + 1 in S n + 1 with g = 4 . If dim M + � 2 3 n + 1 , i . e . , m 2 � 1 2 ( m 1 + 3 ) , then λ 1 ( M + ) = dim M + = m 1 + 2 m 2 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Conjecture of Tang & Y. Conjecture (Tang & Y., J. Diff. Geom. , 2013) Let M d be a closed minimal submanifold in S n + 1 . If the dimension d of M d satisfies d ≥ 2 3 n + 1 , then λ 1 ( M d ) = d . Theorem (Solomon, Math. Ann. 1992) On M − of OT-FKM type, there exists eigenfunctions with eigenvalue 4 m 1 . If m 1 < 1 2 m 2 ( ⇔ dim M − < 2 3 n ), then λ 1 ( M − ) ≤ 4 m 1 < m 2 + 2 m 1 = dim M − . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Conjecture of Tang & Y. Conjecture (Tang & Y., J. Diff. Geom. , 2013) Let M d be a closed minimal submanifold in S n + 1 . If the dimension d of M d satisfies d ≥ 2 3 n + 1 , then λ 1 ( M d ) = d . Theorem (Solomon, Math. Ann. 1992) On M − of OT-FKM type, there exists eigenfunctions with eigenvalue 4 m 1 . If m 1 < 1 2 m 2 ( ⇔ dim M − < 2 3 n ), then λ 1 ( M − ) ≤ 4 m 1 < m 2 + 2 m 1 = dim M − . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Outline Introduction of isoparametric foliation 1 Positive answer to Yau’s 100th problem in the isoparametric case 2 Besse’s problem on generalizations of Einstein condition 3 A sufficient condition for a hypersurface to be isoparametric 4 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Willmore surface Definition An immersed surface x : M 2 � S N is called a Willmore surface if it is a critical surface of the functional � M 2 ( S − nH 2 ) dv , W ( x ) = where H is the norm of the mean curvature vector, S is the square norm of the second fundamental form. ⇒ minimal surfaces in S N are Willmore surfaces. Euler equation = Ejiri [Indiana. Univ. Math. 1982] : The 1st non-minimal example of a flat Willmore surface in high codimension. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Willmore surface Definition An immersed surface x : M 2 � S N is called a Willmore surface if it is a critical surface of the functional � M 2 ( S − nH 2 ) dv , W ( x ) = where H is the norm of the mean curvature vector, S is the square norm of the second fundamental form. ⇒ minimal surfaces in S N are Willmore surfaces. Euler equation = Ejiri [Indiana. Univ. Math. 1982] : The 1st non-minimal example of a flat Willmore surface in high codimension. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Willmore surface Definition An immersed surface x : M 2 � S N is called a Willmore surface if it is a critical surface of the functional � M 2 ( S − nH 2 ) dv , W ( x ) = where H is the norm of the mean curvature vector, S is the square norm of the second fundamental form. ⇒ minimal surfaces in S N are Willmore surfaces. Euler equation = Ejiri [Indiana. Univ. Math. 1982] : The 1st non-minimal example of a flat Willmore surface in high codimension. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Willmore submanifold Definition An immersed submanifold x : M n � S N is called a Willmore submanifold if it is an extremal submanifold of the Willmore functional � n M n ( S − nH 2 ) 2 dv , W ( x ) = Remark This Willmore functional W ( x ) is clearly a natural extension of that in the definition of Willmore surface. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Willmore submanifold B.Y.Chen [Boll. Un. Math. Ital, 1974] , C.P.Wang [Manuscripta Math. 1998] The Willmore functional W ( x ) is a conformal invariant. Pedit-Willmore [Atti.Sem. Mat. Fis. Modena, 1988] Let x : M n � � M ( c ) be a minimal immersion into a space of constant curvature � M ( c ) such that the induced metric is Einstein. Then x is a Willmore submanifold. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Examples minimal, Einstein, Willmore submanifold Both focal submanifolds of each Cartan hypersurface are Willmore submanifolds in S n + p . minimal, non-Einstein, Willmore hypersurface The minimal Cartan hypersurface in S n + 1 is a Willmore hypersurface. non-minimal, non-Einstein, Willmore hypersurface One certain hypersurface in each of Nomizu’s isoparametric families is Willmore. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Examples minimal, Einstein, Willmore submanifold Both focal submanifolds of each Cartan hypersurface are Willmore submanifolds in S n + p . minimal, non-Einstein, Willmore hypersurface The minimal Cartan hypersurface in S n + 1 is a Willmore hypersurface. non-minimal, non-Einstein, Willmore hypersurface One certain hypersurface in each of Nomizu’s isoparametric families is Willmore. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Examples minimal, Einstein, Willmore submanifold Both focal submanifolds of each Cartan hypersurface are Willmore submanifolds in S n + p . minimal, non-Einstein, Willmore hypersurface The minimal Cartan hypersurface in S n + 1 is a Willmore hypersurface. non-minimal, non-Einstein, Willmore hypersurface One certain hypersurface in each of Nomizu’s isoparametric families is Willmore. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results Question : Are there any Willmore submanifolds in S N ( 1 ) which are minimal but not Einstein? Theorem (Tang-Y. [Ann. Glob. Anal. Geom. 2012], Qian-Tang-Y. [Ann. Glob. Anal. Geom. 2013]) Both focal submanifolds of every isoparametric hypersurface in S N ( 1 ) with g = 4 are Willmore. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results Question : Are there any Willmore submanifolds in S N ( 1 ) which are minimal but not Einstein? Theorem (Tang-Y. [Ann. Glob. Anal. Geom. 2012], Qian-Tang-Y. [Ann. Glob. Anal. Geom. 2013]) Both focal submanifolds of every isoparametric hypersurface in S N ( 1 ) with g = 4 are Willmore. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results Theorem (Qian-Tang-Y. [Ann. Glob. Anal. Geom. 2013]) For the focal submanifolds of an isoparametric hypersurface in S n + 1 with g = 4 , we have All the M − of OT-FKM type are not Einstein; the M + of OT-FKM 1 type is Einstein if and only if it is diffeomorphic to Sp ( 2 ) in the homogeneous case with ( m 1 , m 2 ) = ( 4 , 3 ) . For ( m 1 , m 2 ) = ( 2 , 2 ) , the one diffeomorphic to � G 2 ( R 5 ) is Einstein, 2 while the other one diffeomorphic to C P 3 is not. For ( m 1 , m 2 ) = ( 4 , 5 ) , both are not Einstein. 3 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Einstein-like property of focal submanifolds E : Rie. manifolds with constant Ricci curvatures (Einstein); S : Rie. manifolds with constant scalar curvatures; P : Rie. manifolds with parallel Ricci tensor. Clearly, Relations E ⊂ P ⊂ S . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Einstein-like property of focal submanifolds E : Rie. manifolds with constant Ricci curvatures (Einstein); S : Rie. manifolds with constant scalar curvatures; P : Rie. manifolds with parallel Ricci tensor. Clearly, Relations E ⊂ P ⊂ S . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Einstein-like property of focal submanifolds E : Rie. manifolds with constant Ricci curvatures (Einstein); S : Rie. manifolds with constant scalar curvatures; P : Rie. manifolds with parallel Ricci tensor. Clearly, Relations E ⊂ P ⊂ S . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Einstein-like property of focal submanifolds E : Rie. manifolds with constant Ricci curvatures (Einstein); S : Rie. manifolds with constant scalar curvatures; P : Rie. manifolds with parallel Ricci tensor. Clearly, Relations E ⊂ P ⊂ S . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric As further generalizations of the Einstein condition, A. Gray (Geom. Ded. 1978) introduced two significant classes: A : the Ricci tensor ρ is cyclic parallel ∇ i ρ jk + ∇ j ρ ki + ∇ k ρ ij = 0 ; B : the Ricci tensor ρ is a Codazzi tensor ∇ i ρ jk − ∇ j ρ ik = 0 . Remark In view of the second Bianchi identity, the class B coincides with those having harmonic curvatures. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric As further generalizations of the Einstein condition, A. Gray (Geom. Ded. 1978) introduced two significant classes: A : the Ricci tensor ρ is cyclic parallel ∇ i ρ jk + ∇ j ρ ki + ∇ k ρ ij = 0 ; B : the Ricci tensor ρ is a Codazzi tensor ∇ i ρ jk − ∇ j ρ ik = 0 . Remark In view of the second Bianchi identity, the class B coincides with those having harmonic curvatures. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric As further generalizations of the Einstein condition, A. Gray (Geom. Ded. 1978) introduced two significant classes: A : the Ricci tensor ρ is cyclic parallel ∇ i ρ jk + ∇ j ρ ki + ∇ k ρ ij = 0 ; B : the Ricci tensor ρ is a Codazzi tensor ∇ i ρ jk − ∇ j ρ ik = 0 . Remark In view of the second Bianchi identity, the class B coincides with those having harmonic curvatures. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric As further generalizations of the Einstein condition, A. Gray (Geom. Ded. 1978) introduced two significant classes: A : the Ricci tensor ρ is cyclic parallel ∇ i ρ jk + ∇ j ρ ki + ∇ k ρ ij = 0 ; B : the Ricci tensor ρ is a Codazzi tensor ∇ i ρ jk − ∇ j ρ ik = 0 . Remark In view of the second Bianchi identity, the class B coincides with those having harmonic curvatures. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric A. Gray also proved: Relations A ⊂ ⊂ E ⊂ P = A ∩ B A ∪ B ⊂ S ⊂ ⊂ B Remark A and B are the only classes between P and S from the view of group representations. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric A. Gray also proved: Relations A ⊂ ⊂ E ⊂ P = A ∩ B A ∪ B ⊂ S ⊂ ⊂ B Remark A and B are the only classes between P and S from the view of group representations. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric A natural question arises: Question Which focal submanifolds of isoparametric hypersurfaces are Ricci parallel, A -manifolds, or B -manifolds ? Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem (Tang-Y., [Adv. Math., 2015]; Li-Y. [Sci. China Math. 2015] All the focal submanifolds of isoparametric hypersurfaces in spheres with g = 4 are A -manifolds. Thus from the relation P = A ∩ B , it follows that M ± ∈ P ⇐ ⇒ M ± ∈ B . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem (Tang-Y., [Adv. Math., 2015]; Li-Y. [Sci. China Math. 2015] All the focal submanifolds of isoparametric hypersurfaces in spheres with g = 4 are A -manifolds. Thus from the relation P = A ∩ B , it follows that M ± ∈ P ⇐ ⇒ M ± ∈ B . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem (Tang-Y. [Adv. Math., 2015]) For the focal submanifolds of isoparametric hypersurfaces in spheres with g = 4 , we have For OT-FKM type, M + is Ricci parallel if and only if 1 ( m 1 , m 2 ) = ( 2 , 1 ) , ( 6 , 1 ) , or it is diffeomorphic to Sp ( 2 ) in the homogeneous case with ( m 1 , m 2 ) = ( 4 , 3 ) ; while M − is Ricci parallel if and only if ( m 1 , m 2 ) = ( 1 , k ) . For ( m 1 , m 2 ) = ( 2 , 2 ) , the one diffeomorphic to � G 2 ( R 5 ) is Ricci 2 parallel, while the other diffeomorphic to C P 3 is not. For ( m 1 , m 2 ) = ( 4 , 5 ) , both are not Ricci parallel. 3 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric J.E.D’Atri and H.K.Nickerson (J. Diff. Geom. 1969, 1974) D’Atri spaces: Riemannian manifolds ( M , g ) such that for each x ∈ M , the local geodesic symmetry at x (assigning exp x ( − X ) to exp x X , for X ∈ T x M close to 0 ) preserves the volume element. It is well known that { Normal homogeneous Rie. Manifolds } ⊂ D’Atri spaces ⊂ class A Thus The examples of A -manifolds are not rare in the literature, but mostly are (locally) homogeneous. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric In this regard, Besse ( ≪ Einstein manifolds ≫ , 16.56(i), pp.451) posed the following problem as one of “some open problems” : A problem of Besse on generalizations of Einstein condition Find examples of A -manifolds, which have non-parallel Ricci tensor; 1 are not locally homogeneous; 2 are not locally isometric to Riemannian products. 3 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric W. Jelonek (Polish Acad. Sci., 1995) and H. Pedersen and P . Tod (Diff. Geom. Appl. 1999) constructed A -manifolds on S 1 -bundles over locally non-homogeneous K¨ ahler-Einstein manifolds, and on S 1 -bundles over a K3 surface. However, their examples are not simply-connected. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Proposition (Tang-Y., Adv. Math., 2015) The focal submanifolds of isoparametric hypersurfaces in spheres with g = 4 and m 1 , m 2 > 1 are not Riemannian products. Proposition (Tang-Y., Adv. Math., 2015) The focal submanifolds M + of OT-FKM type with ( m 1 , m 2 ) = ( 3 , 4 k ) are not intrinsically homogeneous. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Proposition (Tang-Y., Adv. Math., 2015) The focal submanifolds of isoparametric hypersurfaces in spheres with g = 4 and m 1 , m 2 > 1 are not Riemannian products. Proposition (Tang-Y., Adv. Math., 2015) The focal submanifolds M + of OT-FKM type with ( m 1 , m 2 ) = ( 3 , 4 k ) are not intrinsically homogeneous. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Lemma (Z.Z.Tang, Chinese Sci. Bull. 1991) If m 1 > 1 (resp. m 2 > 1 ), the focal submanifold M − (resp. M + ) is simply-connected. Combining with theorems and propositions above, we conclude that Examples of Besse’s problem The focal submanifolds M + of OT-FKM type with ( m 1 , m 2 ) = ( 3 , 4 k ) are simply-connected examples to the Besse problem. Remark Much more examples to the problem of Besse can be obtained in this way. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Lemma (Z.Z.Tang, Chinese Sci. Bull. 1991) If m 1 > 1 (resp. m 2 > 1 ), the focal submanifold M − (resp. M + ) is simply-connected. Combining with theorems and propositions above, we conclude that Examples of Besse’s problem The focal submanifolds M + of OT-FKM type with ( m 1 , m 2 ) = ( 3 , 4 k ) are simply-connected examples to the Besse problem. Remark Much more examples to the problem of Besse can be obtained in this way. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Lemma (Z.Z.Tang, Chinese Sci. Bull. 1991) If m 1 > 1 (resp. m 2 > 1 ), the focal submanifold M − (resp. M + ) is simply-connected. Combining with theorems and propositions above, we conclude that Examples of Besse’s problem The focal submanifolds M + of OT-FKM type with ( m 1 , m 2 ) = ( 3 , 4 k ) are simply-connected examples to the Besse problem. Remark Much more examples to the problem of Besse can be obtained in this way. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Outline Introduction of isoparametric foliation 1 Positive answer to Yau’s 100th problem in the isoparametric case 2 Besse’s problem on generalizations of Einstein condition 3 A sufficient condition for a hypersurface to be isoparametric 4 Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Chern’s conjecture In 1968, S. S. Chern proposed the following conjecture: Chern’s conjecture M n � S n + 1 : compact minimal immersed hypersurface with S = constant ( S is the squared norm of second fundamental form) (or equivalently, constant scalar curvature: R M = constant ). ⇒ the possible values of S form a discrete set, ∀ n . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Chern’s conjecture This version of Chern’s conjecture is related with J. Simons (1968) M n � S n + 1 : compact minimal immersed hypersurface, if 0 ≤ S ≤ n ( S is not necessarily constant), then either S ≡ 0 or S ≡ n . S ≡ 0 : M n is the equatorial sphere, which is an isoparametric hypersurfaces with g = 1 . � S ≡ n : must be Clifford tori S r ( � r n − r n ) × S n − r ( n ) ( 0 < r < n ) , which are isoparametric hypersurfaces with g = 2 . (characterized by [Lawson, 1969] and [Chern-do Carmo-Kobayashi, 1970] independently) Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Chern’s conjecture This version of Chern’s conjecture is related with J. Simons (1968) M n � S n + 1 : compact minimal immersed hypersurface, if 0 ≤ S ≤ n ( S is not necessarily constant), then either S ≡ 0 or S ≡ n . S ≡ 0 : M n is the equatorial sphere, which is an isoparametric hypersurfaces with g = 1 . � S ≡ n : must be Clifford tori S r ( � r n − r n ) × S n − r ( n ) ( 0 < r < n ) , which are isoparametric hypersurfaces with g = 2 . (characterized by [Lawson, 1969] and [Chern-do Carmo-Kobayashi, 1970] independently) Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Chern’s conjecture This version of Chern’s conjecture is related with J. Simons (1968) M n � S n + 1 : compact minimal immersed hypersurface, if 0 ≤ S ≤ n ( S is not necessarily constant), then either S ≡ 0 or S ≡ n . S ≡ 0 : M n is the equatorial sphere, which is an isoparametric hypersurfaces with g = 1 . � S ≡ n : must be Clifford tori S r ( � r n − r n ) × S n − r ( n ) ( 0 < r < n ) , which are isoparametric hypersurfaces with g = 2 . (characterized by [Lawson, 1969] and [Chern-do Carmo-Kobayashi, 1970] independently) Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Pinching results Peng-Terng (1983) M n � S n + 1 : compact minimal immersed hypersurface, 1 S = constant and n ≤ S ≤ n + 12 n ⇒ S = n . Yang-Cheng(1998) M n � S n + 1 : compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + n 3 ⇒ S = n . Suh-Yang (2007) M n � S n + 1 : compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + 3 n 7 ⇒ S = n . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Pinching results Peng-Terng (1983) M n � S n + 1 : compact minimal immersed hypersurface, 1 S = constant and n ≤ S ≤ n + 12 n ⇒ S = n . Yang-Cheng(1998) M n � S n + 1 : compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + n 3 ⇒ S = n . Suh-Yang (2007) M n � S n + 1 : compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + 3 n 7 ⇒ S = n . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Pinching results Peng-Terng (1983) M n � S n + 1 : compact minimal immersed hypersurface, 1 S = constant and n ≤ S ≤ n + 12 n ⇒ S = n . Yang-Cheng(1998) M n � S n + 1 : compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + n 3 ⇒ S = n . Suh-Yang (2007) M n � S n + 1 : compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + 3 n 7 ⇒ S = n . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Pinching results In particular, for the case n = 3 , there is a sharp result: Peng-Terng (1983) M 3 � S 4 : compact minimal immersed hypersurface, S = constant and 3 ≤ S ≤ 6 ⇒ S = 3 or S = 6 . Open problem M n � S n + 1 ( n > 3 ) : compact minimal hypersurface, If S = constant and n ≤ S ≤ 2 n , then S = n or S = 2 n ? Without assuming S = constant , there are also results on pinching constants [Cheng-Ishikawa, 1999], [Wei-Xu, 2007] and [Ding-Xin, 2011] , etc. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Pinching results In particular, for the case n = 3 , there is a sharp result: Peng-Terng (1983) M 3 � S 4 : compact minimal immersed hypersurface, S = constant and 3 ≤ S ≤ 6 ⇒ S = 3 or S = 6 . Open problem M n � S n + 1 ( n > 3 ) : compact minimal hypersurface, If S = constant and n ≤ S ≤ 2 n , then S = n or S = 2 n ? Without assuming S = constant , there are also results on pinching constants [Cheng-Ishikawa, 1999], [Wei-Xu, 2007] and [Ding-Xin, 2011] , etc. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Pinching results In particular, for the case n = 3 , there is a sharp result: Peng-Terng (1983) M 3 � S 4 : compact minimal immersed hypersurface, S = constant and 3 ≤ S ≤ 6 ⇒ S = 3 or S = 6 . Open problem M n � S n + 1 ( n > 3 ) : compact minimal hypersurface, If S = constant and n ≤ S ≤ 2 n , then S = n or S = 2 n ? Without assuming S = constant , there are also results on pinching constants [Cheng-Ishikawa, 1999], [Wei-Xu, 2007] and [Ding-Xin, 2011] , etc. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Chern’s conjecture (strong version) Fact Isoparametric hypersurfaces are the only known examples of compact minimal hypersurfaces in S n + 1 with S = constant (or R M = constant ). In 1986, Verstraelen-Montiel-Ros-Urbano gave Chern’s conjecture (strong version) M n � S n + 1 : compact minimal immersed hypersurface with R M = constant . ⇒ M n is isoparametric. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Chern’s conjecture (strong version) Fact Isoparametric hypersurfaces are the only known examples of compact minimal hypersurfaces in S n + 1 with S = constant (or R M = constant ). In 1986, Verstraelen-Montiel-Ros-Urbano gave Chern’s conjecture (strong version) M n � S n + 1 : compact minimal immersed hypersurface with R M = constant . ⇒ M n is isoparametric. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric answer to Chern’s conjecture in n = 3 In 1993, S. P . Chang finally proved Chern’s conjecture in the case n = 3 : S. P . Chang (J. Diff. Geom., 1993) M 3 � S 4 : closed minimal immersed hypersurface with R M = constant ⇒ M 3 is isoparametric with g = 1 , 2 or 3 . When n > 3 , no more essentially affirmative answer to Chern’s conjecture. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric answer to a generalized version of Chern’s conjecture for n = 3 On the other hand, it is possible to prove a generalized version of Chern’s conjecture for n = 3 , where the hypersurface is not necessarily minimal: de Almeida-Brito (Duke. Math. J. 1990) M 3 � S 4 : closed hypersurface with constant mean curvature and constant scalar curvature R M ≥ 0 . ⇒ M 3 is isoparametric. [S.P. Chang, Comm. Anal. Geom. 1993] and [Q.M.Cheng, Geometry and Global Analysis, Sendai, 1993] independently removed the assumption R M ≥ 0 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric answer to a generalized version of Chern’s conjecture for n = 3 On the other hand, it is possible to prove a generalized version of Chern’s conjecture for n = 3 , where the hypersurface is not necessarily minimal: de Almeida-Brito (Duke. Math. J. 1990) M 3 � S 4 : closed hypersurface with constant mean curvature and constant scalar curvature R M ≥ 0 . ⇒ M 3 is isoparametric. [S.P. Chang, Comm. Anal. Geom. 1993] and [Q.M.Cheng, Geometry and Global Analysis, Sendai, 1993] independently removed the assumption R M ≥ 0 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Using the same approach of de Almeida-Brito, Lusala-Scherfner-Sousa (Asian J. Math. 2005) M 4 � S 5 : closed minimal Willmore hypersurface with constant scalar curvature R M ≥ 0 . Then M 4 is isoparametric. In fact, by H.Z.Li’s criterion, in this case � M 4 is Willmore ⇐ λ 3 ⇒ i = 0 ( λ i : principal curvatures ) [Deng-Gu-Wei, Adv. Math. 2017] removed the assumption R M ≥ 0 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Using the same approach of de Almeida-Brito, Lusala-Scherfner-Sousa (Asian J. Math. 2005) M 4 � S 5 : closed minimal Willmore hypersurface with constant scalar curvature R M ≥ 0 . Then M 4 is isoparametric. In fact, by H.Z.Li’s criterion, in this case � M 4 is Willmore ⇐ λ 3 ⇒ i = 0 ( λ i : principal curvatures ) [Deng-Gu-Wei, Adv. Math. 2017] removed the assumption R M ≥ 0 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Using the same approach of de Almeida-Brito, Lusala-Scherfner-Sousa (Asian J. Math. 2005) M 4 � S 5 : closed minimal Willmore hypersurface with constant scalar curvature R M ≥ 0 . Then M 4 is isoparametric. In fact, by H.Z.Li’s criterion, in this case � M 4 is Willmore ⇐ λ 3 ⇒ i = 0 ( λ i : principal curvatures ) [Deng-Gu-Wei, Adv. Math. 2017] removed the assumption R M ≥ 0 . Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Define for r ≥ 3 , � λ r f r := i . Using the same approach, Tang-Yang (2015) M 4 � S 5 : closed minimal hypersurface with constant scalar curvature R M ≥ 0 . If f 3 = constant and g = constant , ⇒ M 4 is isoparametric. = Scherfner-Vrancken-Weiss (2012) M 6 � S 7 : closed hypersurface with H = f 3 = f 5 = 0 , f 4 = constant , and R M = constant ≥ 0 . ⇒ M 6 is isoparametric. = Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem of de Almeida-Brito with more general setting The previous theorem of de Almeida-Brito is an application of de Almeida-Brito (Duke. Math. J. 1990) M 3 : closed Riemannian manifold. a : a smooth symmetric ( 0 , 2 ) tensor field on M 3 A : dual ( 1 , 1 ) tensor field of a . Suppose (i) R M ≥ 0 ; (ii) the field ∇ a of type ( 0 , 3 ) is symmetric; (iii) tr ( A ) , tr ( A 2 ) are constants. ⇒ tr ( A 3 ) is a constant, i.e., eigenvalues of A are all constants. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Theorem of de Almeida-Brito with more general setting The previous theorem of de Almeida-Brito is an application of de Almeida-Brito (Duke. Math. J. 1990) M 3 : closed Riemannian manifold. a : a smooth symmetric ( 0 , 2 ) tensor field on M 3 A : dual ( 1 , 1 ) tensor field of a . Suppose (i) R M ≥ 0 ; (ii) the field ∇ a of type ( 0 , 3 ) is symmetric; (iii) tr ( A ) , tr ( A 2 ) are constants. ⇒ tr ( A 3 ) is a constant, i.e., eigenvalues of A are all constants. Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results: g = n everywhere We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) � M n ( n > 3 ) : closed Riemannian manifold on which M R M ≥ 0 . a : smooth symmetric ( 0 , 2 ) tensor field on M n , A : dual ( 1 , 1 ) tensor field of a . Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ 1 , · · · , λ n ; (1.3) tr ( A k ) ( k = 1 , · · · , n − 1 ) are constants. Then (a) tr ( A n ) is a constant, i.e., λ 1 , · · · , λ n are constants; � M R M ≡ 0 . (b) Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results: g = n everywhere We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) � M n ( n > 3 ) : closed Riemannian manifold on which M R M ≥ 0 . a : smooth symmetric ( 0 , 2 ) tensor field on M n , A : dual ( 1 , 1 ) tensor field of a . Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ 1 , · · · , λ n ; (1.3) tr ( A k ) ( k = 1 , · · · , n − 1 ) are constants. Then (a) tr ( A n ) is a constant, i.e., λ 1 , · · · , λ n are constants; � M R M ≡ 0 . (b) Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results: g = n everywhere We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) � M n ( n > 3 ) : closed Riemannian manifold on which M R M ≥ 0 . a : smooth symmetric ( 0 , 2 ) tensor field on M n , A : dual ( 1 , 1 ) tensor field of a . Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ 1 , · · · , λ n ; (1.3) tr ( A k ) ( k = 1 , · · · , n − 1 ) are constants. Then (a) tr ( A n ) is a constant, i.e., λ 1 , · · · , λ n are constants; � M R M ≡ 0 . (b) Wenjiao Yan Isoparametric foliation and its applications
Introduction of isoparametric foliation Positive answer to Yau’s 100th problem in the isoparametric case Besse’s problem on generalizations of Einstein condition A sufficient condition for a hypersurface to be isoparametric Our results: g = n everywhere We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) � M n ( n > 3 ) : closed Riemannian manifold on which M R M ≥ 0 . a : smooth symmetric ( 0 , 2 ) tensor field on M n , A : dual ( 1 , 1 ) tensor field of a . Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ 1 , · · · , λ n ; (1.3) tr ( A k ) ( k = 1 , · · · , n − 1 ) are constants. Then (a) tr ( A n ) is a constant, i.e., λ 1 , · · · , λ n are constants; � M R M ≡ 0 . (b) Wenjiao Yan Isoparametric foliation and its applications
Recommend
More recommend
