Isoparametric foliation and its applications Wenjiao Yan Beijing - - PowerPoint PPT Presentation

Isoparametric foliation and its applications Wenjiao Yan

Beijing Normal University based on joint work with C.Qian, Z.Z.Tang and D.Y.Wei

Workshop on the Isoparametric Theory

• 2019. 06. 06

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

Content

1

Introduction of isoparametric foliation

2

Positive answer to Yau’s 100th problem in the isoparametric case

3

Besse’s problem on generalizations of Einstein condition

4

A sufficient condition for a hypersurface to be 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Outline

1

Introduction of isoparametric foliation

2

Positive answer to Yau’s 100th problem in the isoparametric case

3

Besse’s problem on generalizations of Einstein condition

4

A sufficient condition for a hypersurface to be 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric function

Definition A smooth function f : (Nn+1, ds2) → R is called isoparametric , if it satisfies:

• |∇f|2 = b(f),

(1) ∆f = a(f). (2) where b ∈ C∞(R) and a ∈ C0(R). (1) ⇐ ⇒ Mt := f −1(t) are parallel, Under this assumption, (2) ⇐ ⇒ Mt has constant mean curvature, for each regular value t.

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric function

Definition A smooth function f : (Nn+1, ds2) → R is called isoparametric , if it satisfies:

• |∇f|2 = b(f),

(1) ∆f = a(f). (2) where b ∈ C∞(R) and a ∈ C0(R). (1) ⇐ ⇒ Mt := f −1(t) are parallel, Under this assumption, (2) ⇐ ⇒ Mt has constant mean curvature, for each regular value t.

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric function

Definition A smooth function f : (Nn+1, ds2) → R is called isoparametric , if it satisfies:

• |∇f|2 = b(f),

(1) ∆f = a(f). (2) where b ∈ C∞(R) and a ∈ C0(R). (1) ⇐ ⇒ Mt := f −1(t) are parallel, Under this assumption, (2) ⇐ ⇒ Mt has constant mean curvature, for each regular value t.

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric hypersurface: Mn = f −1(t0) for a regular value t0 (cf. Q.M. Wang [Math. Ann., 1987]). Focal submanifolds: Ms = f −1(s), for a singular value s ∈ Im(f). When Nn+1 is closed, M+ := f −1(max);

• codim. := m1 + 1 in Nn+1;

M− := f −1(min),

• codim. := m2 + 1 in Nn+1

When Nn+1 = Sn+1, the focal submanifolds are minimal (Nomizu (1973)); When Nn+1 is closed, the focal submanifolds are minimal (Q.M.Wang (1987), Ge-Tang [Asian J. Math. 2013])

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric hypersurface: Mn = f −1(t0) for a regular value t0 (cf. Q.M. Wang [Math. Ann., 1987]). Focal submanifolds: Ms = f −1(s), for a singular value s ∈ Im(f). When Nn+1 is closed, M+ := f −1(max);

• codim. := m1 + 1 in Nn+1;

M− := f −1(min),

• codim. := m2 + 1 in Nn+1

When Nn+1 = Sn+1, the focal submanifolds are minimal (Nomizu (1973)); When Nn+1 is closed, the focal submanifolds are minimal (Q.M.Wang (1987), Ge-Tang [Asian J. Math. 2013])

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric hypersurface

Theorem (E.Cartan, 1939, 1940) A closed hypersurface Mn in a real space form Nn+1(c) is isoparametric if it has constant principal curvatures. Nn+1 = Rn+1, Hn+1: isoparametric hypersurfaces are completely

• classified. (E.Cartan et al)

Nn+1 = Sn+1: classification is recently completed.

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric hypersurface

Theorem (E.Cartan, 1939, 1940) A closed hypersurface Mn in a real space form Nn+1(c) is isoparametric if it has constant principal curvatures. Nn+1 = Rn+1, Hn+1: isoparametric hypersurfaces are completely

• classified. (E.Cartan et al)

Nn+1 = Sn+1: classification is recently completed.

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric hypersurface on Sn+1

M¨ unzner([Math. Ann. 1980, 1981]): Mn: an isoparametric hypersurface in Sn+1 with principal curvatures cot θi, 0 < θ1 < · · · < θg < π, with multiplicities mi, then: 1). θk = θ1 + k−1

g π;

2). mk = mk+2(subscripts mod g); 3). M: open subset of level hypersurface in Sn+1 of a homogeneous polynomial F of degree g on Rn+2 satisfying Cartan-M¨ unzner equations: |∇F|2 = g2|x|2g−2, △F = g2 2 (m2 − m1)|x|g−2. 4). g = 1, 2, 3, 4, 6.

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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Classification in Sn+1

g = 1, Mn must be a hypersphere Sn; g = 2, Mn must be Sp(r) × Sn−p(s) (r2 + s2 = 1, r, s > 0). g = 3, Mn must be homogeneous, m1 = m2 = 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, Mn is either of OT-FKM type or homogeneous with (m1, m2) = (2, 2), (4, 5). g = 6, m1 = m2 = 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 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Classification in Sn+1

g = 1, Mn must be a hypersphere Sn; g = 2, Mn must be Sp(r) × Sn−p(s) (r2 + s2 = 1, r, s > 0). g = 3, Mn must be homogeneous, m1 = m2 = 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, Mn is either of OT-FKM type or homogeneous with (m1, m2) = (2, 2), (4, 5). g = 6, m1 = m2 = 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 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Classification in Sn+1

g = 1, Mn must be a hypersphere Sn; g = 2, Mn must be Sp(r) × Sn−p(s) (r2 + s2 = 1, r, s > 0). g = 3, Mn must be homogeneous, m1 = m2 = 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, Mn is either of OT-FKM type or homogeneous with (m1, m2) = (2, 2), (4, 5). g = 6, m1 = m2 = 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 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Classification in Sn+1

g = 1, Mn must be a hypersphere Sn; g = 2, Mn must be Sp(r) × Sn−p(s) (r2 + s2 = 1, r, s > 0). g = 3, Mn must be homogeneous, m1 = m2 = 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, Mn is either of OT-FKM type or homogeneous with (m1, m2) = (2, 2), (4, 5). g = 6, m1 = m2 = 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 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Classification in Sn+1

g = 1, Mn must be a hypersphere Sn; g = 2, Mn must be Sp(r) × Sn−p(s) (r2 + s2 = 1, r, s > 0). g = 3, Mn must be homogeneous, m1 = m2 = 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, Mn is either of OT-FKM type or homogeneous with (m1, m2) = (2, 2), (4, 5). g = 6, m1 = m2 = 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 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Classification in Sn+1

g = 1, Mn must be a hypersphere Sn; g = 2, Mn must be Sp(r) × Sn−p(s) (r2 + s2 = 1, r, s > 0). g = 3, Mn must be homogeneous, m1 = m2 = 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, Mn is either of OT-FKM type or homogeneous with (m1, m2) = (2, 2), (4, 5). g = 6, m1 = m2 = 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 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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Isoparametric hypersurface of OT-FKM type

For a symmetric Clifford system {P0, · · · , Pm} on R2l, i.e., Pi = Pt

i, PiPj + PjPi = 2δijI2l

Following Ozeki-Takeuchi (Tohoku Math. 1975,1976), Ferus, Karcher and M¨ unzner (Math.Z., 1981) constructed a homogeneous polynomial F of degree 4 on R2l: F(x) = |x|4 − 2

m

• i=0

Pix, x2, where l = kδ(m), k is a positive integer, δ(m) is valued: m 1 2 3 4 5 6 7 8 · · · m+8 δ(m) 1 2 4 4 8 8 8 8 16δ(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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Proposition (Ferus, Karcher and M¨ unzner (Math.Z., 1981)) Denote f = F|S2l−1. Then f is an isoparametric function on S2l−1 corresponding to g = 4, (m1, m2) = (m, l − m − 1) f = F|S2l−1: isoparametric function of OT-FKM type, M2l−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 Isoparametric hypersurface on Sn+1 Classification in Sn+1 Isoparametric hypersurface of OT-FKM type

Proposition (Ferus, Karcher and M¨ unzner (Math.Z., 1981)) Denote f = F|S2l−1. Then f is an isoparametric function on S2l−1 corresponding to g = 4, (m1, m2) = (m, l − m − 1) f = F|S2l−1: isoparametric function of OT-FKM type, M2l−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

1

Introduction of isoparametric foliation

2

Positive answer to Yau’s 100th problem in the isoparametric case

3

Besse’s problem on generalizations of Einstein condition

4

A sufficient condition for a hypersurface to be 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

(Mn, g): closed Riemannian manifold. ∆f = −div(∇f) for f ∈ C∞(M). ∆ is an elliptic operator and has a discrete spectrum: {0 = λ0(M) < λ1(M) λ2(M) · · · λk(M), · · · , ↑ ∞}. λ1(M): the first eigenvalue of M. Theorem (T. Takahashi, 1966) Let f : Mn → SN be an isometric immersion with the canonical coordinate (x1, x2, ..., xN+1). Then Mn is minimal ⇐ ⇒ ∆(xi ◦ f) = n(xi ◦ f), ∀ 1 ≤ i ≤ N + 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

Yau conjecture

Consequently, for a minimal hypersurface Mn in the unit sphere Sn+1 λ1(Mn) n. Yau conjecture (1982, Problem Section, the 100th problem) Let Mn be a closed embedded minimal hypersurface in the unit sphere Sn+1. Then λ1(Mn) = 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 S3 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 S3 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 Mn be a closed embedded minimal hypersurface in Sn+1. Then λ1(Mn) n 2. We consider a restricted problem of Yau’s conjecture. Problem Let Mn be a closed minimal isoparametric hypersurface in Sn+1. Is it true that λ1(Mn) = 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 Mn be a closed embedded minimal hypersurface in Sn+1. Then λ1(Mn) n 2. We consider a restricted problem of Yau’s conjecture. Problem Let Mn be a closed minimal isoparametric hypersurface in Sn+1. Is it true that λ1(Mn) = 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 Mn be a minimal isoparametric hypersurface in Sn+1. For g = 1, 2. The problem is trivially true. For g = 3, 4, 6. Theorem (Muto, Ohnita & Urakawa, Tˆ

• hoku Math. J. 1984;

Kotani, Tˆ

• hoku Math. J. 1985)

Let M be the minimal homogeneous hypersurface with g = 3, 6 and g = 4, (m1, m2) = (1, k), (2, 2). Then λ1(M) = dim(M). Combining with the classification results above, for g = 3, 6, λ1(M) = dim(M). Remark For homogeneous hypersurfaces with g = 4, (m1, m2) must be (1, k), (2, 2), (2, 2k − 1), (4, 4k − 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, (m1, m2) = (3, 4k), (4, 4k + 3), (7, 8k), 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 Mn in Sn+1 is λ1(Mn) = n. Remark No results in homogeneous cases with (m1, m2) = (2, 2k − 1)(k = 2, 3, · · · ), and many other cases of OT-FKM-type with (m1, m2) = (m, kδ(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, (m1, m2) = (3, 4k), (4, 4k + 3), (7, 8k), 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 Mn in Sn+1 is λ1(Mn) = n. Remark No results in homogeneous cases with (m1, m2) = (2, 2k − 1)(k = 2, 3, · · · ), and many other cases of OT-FKM-type with (m1, m2) = (m, kδ(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 Mn be the minimal isoparametric hypersurface in Sn+1. Then λ1(Mn) = n. Theorem (Tang and Y., J. Diff. Geom., 2013) Let Mn be the minimal isoparametric hypersurface in Sn+1 with g = 4 and m1, m2 ≥ 2. Then λ1(Mn) = n with multiplicity n + 2. Remark The case with (m1, m2) = (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 Mn be the minimal isoparametric hypersurface in Sn+1. Then λ1(Mn) = n. Theorem (Tang and Y., J. Diff. Geom., 2013) Let Mn be the minimal isoparametric hypersurface in Sn+1 with g = 4 and m1, m2 ≥ 2. Then λ1(Mn) = n with multiplicity n + 2. Remark The case with (m1, m2) = (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 Mn be the minimal isoparametric hypersurface in Sn+1. Then λ1(Mn) = n. Theorem (Tang and Y., J. Diff. Geom., 2013) Let Mn be the minimal isoparametric hypersurface in Sn+1 with g = 4 and m1, m2 ≥ 2. Then λ1(Mn) = n with multiplicity n + 2. Remark The case with (m1, m2) = (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 M12 be the minimal isoparametric hypersurface in S13 with g = 6 and (m1, m2) = (2, 2). Then λ1(M12) = 12 with multiplicity 14. Remark Muto, Ohnita & Urakawa computed the first eigenvalue of the minimal homogeneous hypersurface M12.

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 M12 be the minimal isoparametric hypersurface in S13 with g = 6 and (m1, m2) = (2, 2). Then λ1(M12) = 12 with multiplicity 14. Remark Muto, Ohnita & Urakawa computed the first eigenvalue of the minimal homogeneous hypersurface M12.

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 Sn+1. Hence, λ1(M±) ≤ dim(M±). Question λ1(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 m1 + 1 in Sn+1 with g = 4. If dim M+ 2

3n + 1, i.e., m2 1 2(m1 + 3), then

λ1(M+) = dim M+ = m1 + 2m2.

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 Sn+1. Hence, λ1(M±) ≤ dim(M±). Question λ1(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 m1 + 1 in Sn+1 with g = 4. If dim M+ 2

3n + 1, i.e., m2 1 2(m1 + 3), then

λ1(M+) = dim M+ = m1 + 2m2.

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 Sn+1. Hence, λ1(M±) ≤ dim(M±). Question λ1(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 m1 + 1 in Sn+1 with g = 4. If dim M+ 2

3n + 1, i.e., m2 1 2(m1 + 3), then

λ1(M+) = dim M+ = m1 + 2m2.

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 Md be a closed minimal submanifold in Sn+1. If the dimension d of Md satisfies d ≥ 2

3n + 1, then

λ1(Md) = d. Theorem (Solomon, Math. Ann. 1992) On M− of OT-FKM type, there exists eigenfunctions with eigenvalue 4m1. If m1 < 1

2m2 (⇔ dim M− < 2 3n), then

λ1(M−) ≤ 4m1<m2 + 2m1 = 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 Md be a closed minimal submanifold in Sn+1. If the dimension d of Md satisfies d ≥ 2

3n + 1, then

λ1(Md) = d. Theorem (Solomon, Math. Ann. 1992) On M− of OT-FKM type, there exists eigenfunctions with eigenvalue 4m1. If m1 < 1

2m2 (⇔ dim M− < 2 3n), then

λ1(M−) ≤ 4m1<m2 + 2m1 = 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

1

Introduction of isoparametric foliation

2

Positive answer to Yau’s 100th problem in the isoparametric case

3

Besse’s problem on generalizations of Einstein condition

4

A sufficient condition for a hypersurface to be 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

Willmore surface

Definition An immersed surface x : M2 SN is called a Willmore surface if it is a critical surface of the functional W(x) =

• M2(S − nH2)dv,

where H is the norm of the mean curvature vector, S is the square norm of the second fundamental form. Euler equation = ⇒ minimal surfaces in SN are Willmore surfaces. 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 : M2 SN is called a Willmore surface if it is a critical surface of the functional W(x) =

• M2(S − nH2)dv,

where H is the norm of the mean curvature vector, S is the square norm of the second fundamental form. Euler equation = ⇒ minimal surfaces in SN are Willmore surfaces. 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 : M2 SN is called a Willmore surface if it is a critical surface of the functional W(x) =

• M2(S − nH2)dv,

where H is the norm of the mean curvature vector, S is the square norm of the second fundamental form. Euler equation = ⇒ minimal surfaces in SN are Willmore surfaces. 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 : Mn SN is called a Willmore submanifold if it is an extremal submanifold of the Willmore functional W(x) =

• Mn(S − nH2)

n 2 dv,

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 : Mn 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 Sn+p. minimal, non-Einstein, Willmore hypersurface The minimal Cartan hypersurface in Sn+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 Sn+p. minimal, non-Einstein, Willmore hypersurface The minimal Cartan hypersurface in Sn+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 Sn+p. minimal, non-Einstein, Willmore hypersurface The minimal Cartan hypersurface in Sn+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 SN(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 SN(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 SN(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 SN(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 Sn+1 with g = 4, we have

1

All the M− of OT-FKM type are not Einstein; the M+ of OT-FKM type is Einstein if and only if it is diffeomorphic to Sp(2) in the homogeneous case with (m1, m2) = (4, 3).

2

For (m1, m2) = (2, 2), the one diffeomorphic to G2(R5) is Einstein, while the other one diffeomorphic to CP3 is not.

3

For (m1, m2) = (4, 5), both are not Einstein.

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 ∇iρjk + ∇jρki + ∇kρij = 0; B: the Ricci tensor ρ is a Codazzi tensor ∇iρjk − ∇jρik = 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 ∇iρjk + ∇jρki + ∇kρij = 0; B: the Ricci tensor ρ is a Codazzi tensor ∇iρjk − ∇jρik = 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 ∇iρjk + ∇jρki + ∇kρij = 0; B: the Ricci tensor ρ is a Codazzi tensor ∇iρjk − ∇jρik = 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 ∇iρjk + ∇jρki + ∇kρij = 0; B: the Ricci tensor ρ is a Codazzi tensor ∇iρjk − ∇jρik = 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 E ⊂ P = A ∩ B ⊂ A ⊂ B ⊂ ⊂ A ∪ B ⊂ S 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 E ⊂ P = A ∩ B ⊂ A ⊂ B ⊂ ⊂ A ∪ B ⊂ S 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

1

For OT-FKM type, M+ is Ricci parallel if and only if (m1, m2) = (2, 1), (6, 1), or it is diffeomorphic to Sp(2) in the homogeneous case with (m1, m2) = (4, 3); while M− is Ricci parallel if and only if (m1, m2) = (1, k).

2

For (m1, m2) = (2, 2), the one diffeomorphic to G2(R5) is Ricci parallel, while the other diffeomorphic to CP3 is not.

3

For (m1, m2) = (4, 5), both are not Ricci parallel.

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 expx(−X) to expxX, for X ∈ TxM 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

1

have non-parallel Ricci tensor;

2

are not locally homogeneous;

3

are not locally isometric to Riemannian products.

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 S1-bundles over locally non-homogeneous K¨ ahler-Einstein manifolds, and on S1-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 m1, m2 > 1 are not Riemannian products. Proposition (Tang-Y., Adv. Math., 2015) The focal submanifolds M+ of OT-FKM type with (m1, m2) = (3, 4k) 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 m1, m2 > 1 are not Riemannian products. Proposition (Tang-Y., Adv. Math., 2015) The focal submanifolds M+ of OT-FKM type with (m1, m2) = (3, 4k) 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 m1 > 1 (resp. m2 > 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 (m1, m2) = (3, 4k) 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 m1 > 1 (resp. m2 > 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 (m1, m2) = (3, 4k) 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 m1 > 1 (resp. m2 > 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 (m1, m2) = (3, 4k) 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

1

Introduction of isoparametric foliation

2

Positive answer to Yau’s 100th problem in the isoparametric case

3

Besse’s problem on generalizations of Einstein condition

4

A sufficient condition for a hypersurface to be 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

In 1968, S. S. Chern proposed the following conjecture: Chern’s conjecture Mn Sn+1: compact minimal immersed hypersurface with S = constant (S is the squared norm of second fundamental form) (or equivalently, constant scalar curvature: RM = 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)

Mn Sn+1: compact minimal immersed hypersurface, if 0 ≤ S ≤ n (S is not necessarily constant), then either S ≡ 0 or S ≡ n. S ≡ 0: Mn is the equatorial sphere, which is an isoparametric hypersurfaces with g = 1. S ≡ n: must be Clifford tori Sr( r

n) × Sn−r(

• 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)

Mn Sn+1: compact minimal immersed hypersurface, if 0 ≤ S ≤ n (S is not necessarily constant), then either S ≡ 0 or S ≡ n. S ≡ 0: Mn is the equatorial sphere, which is an isoparametric hypersurfaces with g = 1. S ≡ n: must be Clifford tori Sr( r

n) × Sn−r(

• 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)

Mn Sn+1: compact minimal immersed hypersurface, if 0 ≤ S ≤ n (S is not necessarily constant), then either S ≡ 0 or S ≡ n. S ≡ 0: Mn is the equatorial sphere, which is an isoparametric hypersurfaces with g = 1. S ≡ n: must be Clifford tori Sr( r

n) × Sn−r(

• 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) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n +

1 12n ⇒ S = n.

Yang-Cheng(1998) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + n

3 ⇒ S = n.

Suh-Yang (2007) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + 3n

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) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n +

1 12n ⇒ S = n.

Yang-Cheng(1998) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + n

3 ⇒ S = n.

Suh-Yang (2007) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + 3n

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) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n +

1 12n ⇒ S = n.

Yang-Cheng(1998) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + n

3 ⇒ S = n.

Suh-Yang (2007) Mn Sn+1: compact minimal immersed hypersurface, S = constant and n ≤ S ≤ n + 3n

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) M3 S4: compact minimal immersed hypersurface, S = constant and 3 ≤ S ≤ 6 ⇒ S = 3 or S = 6. Open problem Mn Sn+1 (n > 3): compact minimal hypersurface, If S = constant and n ≤ S ≤ 2n, then S = n or S = 2n ? 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) M3 S4: compact minimal immersed hypersurface, S = constant and 3 ≤ S ≤ 6 ⇒ S = 3 or S = 6. Open problem Mn Sn+1 (n > 3): compact minimal hypersurface, If S = constant and n ≤ S ≤ 2n, then S = n or S = 2n ? 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) M3 S4: compact minimal immersed hypersurface, S = constant and 3 ≤ S ≤ 6 ⇒ S = 3 or S = 6. Open problem Mn Sn+1 (n > 3): compact minimal hypersurface, If S = constant and n ≤ S ≤ 2n, then S = n or S = 2n ? 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 Sn+1 with S = constant (or RM = constant). In 1986, Verstraelen-Montiel-Ros-Urbano gave Chern’s conjecture (strong version) Mn Sn+1: compact minimal immersed hypersurface with RM = constant. ⇒ Mn 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 Sn+1 with S = constant (or RM = constant). In 1986, Verstraelen-Montiel-Ros-Urbano gave Chern’s conjecture (strong version) Mn Sn+1: compact minimal immersed hypersurface with RM = constant. ⇒ Mn 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) M3 S4: closed minimal immersed hypersurface with RM = constant ⇒ M3 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) M3 S4: closed hypersurface with constant mean curvature and constant scalar curvature RM ≥ 0. ⇒ M3 is isoparametric. [S.P. Chang, Comm. Anal. Geom. 1993] and [Q.M.Cheng, Geometry and Global Analysis, Sendai, 1993] independently removed the assumption RM ≥ 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) M3 S4: closed hypersurface with constant mean curvature and constant scalar curvature RM ≥ 0. ⇒ M3 is isoparametric. [S.P. Chang, Comm. Anal. Geom. 1993] and [Q.M.Cheng, Geometry and Global Analysis, Sendai, 1993] independently removed the assumption RM ≥ 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) M4 S5 : closed minimal Willmore hypersurface with constant scalar curvature RM ≥ 0. Then M4 is isoparametric. In fact, by H.Z.Li’s criterion, in this case M4 is Willmore ⇐ ⇒

• λ3

i = 0

(λi : principal curvatures ) [Deng-Gu-Wei, Adv. Math. 2017] removed the assumption RM ≥ 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) M4 S5 : closed minimal Willmore hypersurface with constant scalar curvature RM ≥ 0. Then M4 is isoparametric. In fact, by H.Z.Li’s criterion, in this case M4 is Willmore ⇐ ⇒

• λ3

i = 0

(λi : principal curvatures ) [Deng-Gu-Wei, Adv. Math. 2017] removed the assumption RM ≥ 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) M4 S5 : closed minimal Willmore hypersurface with constant scalar curvature RM ≥ 0. Then M4 is isoparametric. In fact, by H.Z.Li’s criterion, in this case M4 is Willmore ⇐ ⇒

• λ3

i = 0

(λi : principal curvatures ) [Deng-Gu-Wei, Adv. Math. 2017] removed the assumption RM ≥ 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, fr :=

• λr

i.

Using the same approach, Tang-Yang (2015) M4 S5 : closed minimal hypersurface with constant scalar curvature RM ≥ 0. If f3 = constant and g = constant, = ⇒ M4 is isoparametric. Scherfner-Vrancken-Weiss (2012) M6 S7: closed hypersurface with H = f3 = f5 = 0, f4 = constant, and RM = constant ≥ 0. = ⇒ M6 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) M3: closed Riemannian manifold. a: a smooth symmetric (0, 2) tensor field on M3 A: dual (1, 1) tensor field of a. Suppose (i) RM ≥ 0; (ii) the field ∇a of type (0, 3) is symmetric; (iii) tr(A), tr(A2) are constants. ⇒ tr(A3) 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) M3: closed Riemannian manifold. a: a smooth symmetric (0, 2) tensor field on M3 A: dual (1, 1) tensor field of a. Suppose (i) RM ≥ 0; (ii) the field ∇a of type (0, 3) is symmetric; (iii) tr(A), tr(A2) are constants. ⇒ tr(A3) 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) Mn (n > 3): closed Riemannian manifold on which

• M RM ≥ 0.

a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ1, · · · , λn; (1.3) tr(Ak) (k = 1, · · · , n − 1) are constants. Then (a) tr(An) is a constant, i.e., λ1, · · · , λn are constants; (b)

• M RM ≡ 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

Our results: g = n everywhere

We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) Mn (n > 3): closed Riemannian manifold on which

• M RM ≥ 0.

a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ1, · · · , λn; (1.3) tr(Ak) (k = 1, · · · , n − 1) are constants. Then (a) tr(An) is a constant, i.e., λ1, · · · , λn are constants; (b)

• M RM ≡ 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

Our results: g = n everywhere

We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) Mn (n > 3): closed Riemannian manifold on which

• M RM ≥ 0.

a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ1, · · · , λn; (1.3) tr(Ak) (k = 1, · · · , n − 1) are constants. Then (a) tr(An) is a constant, i.e., λ1, · · · , λn are constants; (b)

• M RM ≡ 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

Our results: g = n everywhere

We generalize Theorem of de Almeida-Brito to higher dimension: Theorem 1 (Tang-Wei-Y., 2018) Mn (n > 3): closed Riemannian manifold on which

• M RM ≥ 0.

a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (1.1) a is Codazzian; (1.2) A has n distinct eigenvalues λ1, · · · , λn; (1.3) tr(Ak) (k = 1, · · · , n − 1) are constants. Then (a) tr(An) is a constant, i.e., λ1, · · · , λn are constants; (b)

• M RM ≡ 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

Our results: g = n everywhere

Corollary 1 (Tang-Wei-Y., 2018) Mn Sn+1 (n > 3): closed hypersurface. Suppose (2.1)

• M RM ≥ 0;

(2.2) the principal curvatures λ1, · · · , λn are distinct; (2.3)

n

• i=1

λk

i (k = 1, · · · , n − 1) are constants,

⇒ Mn is isoparametric and RM ≡ 0. More precisely, Mn can be only one of: (a) M4 S5: Cartan’s example of isoparametric hypersurface with g = 4; (b) M6 S7: the isoparametric hypersurface with g = 6.

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

Corollary 1 (Tang-Wei-Y., 2018) Mn Sn+1 (n > 3): closed hypersurface. Suppose (2.1)

• M RM ≥ 0;

(2.2) the principal curvatures λ1, · · · , λn are distinct; (2.3)

n

• i=1

λk

i (k = 1, · · · , n − 1) are constants,

⇒ Mn is isoparametric and RM ≡ 0. More precisely, Mn can be only one of: (a) M4 S5: Cartan’s example of isoparametric hypersurface with g = 4; (b) M6 S7: the isoparametric hypersurface with g = 6.

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

Corollary 1 (Tang-Wei-Y., 2018) Mn Sn+1 (n > 3): closed hypersurface. Suppose (2.1)

• M RM ≥ 0;

(2.2) the principal curvatures λ1, · · · , λn are distinct; (2.3)

n

• i=1

λk

i (k = 1, · · · , n − 1) are constants,

⇒ Mn is isoparametric and RM ≡ 0. More precisely, Mn can be only one of: (a) M4 S5: Cartan’s example of isoparametric hypersurface with g = 4; (b) M6 S7: the isoparametric hypersurface with g = 6.

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

Corollary 1 (Tang-Wei-Y., 2018) Mn Sn+1 (n > 3): closed hypersurface. Suppose (2.1)

• M RM ≥ 0;

(2.2) the principal curvatures λ1, · · · , λn are distinct; (2.3)

n

• i=1

λk

i (k = 1, · · · , n − 1) are constants,

⇒ Mn is isoparametric and RM ≡ 0. More precisely, Mn can be only one of: (a) M4 S5: Cartan’s example of isoparametric hypersurface with g = 4; (b) M6 S7: the isoparametric hypersurface with g = 6.

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

Remark For an isoparametric hypersurface in the unit sphere with simple principal curvatures (not necessarily minimal), RM ≡ 0. This is interesting and different from the case that some principal curvature has multiplicity greater than 1 ([Tang-Xie-Y., Comm. Anal.

• Geom. 2012]).

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

Other cases: g = constant(< n)

Chang (Pacific J. Math. 1994) Mn Sn+1: closed hypersurface with constant mean curvature and constant scalar curvature, which has g = 3 everywhere. ⇒ Mn is isoparametric. de Almeida-Brito-Scherfner-Weiss (Adv. Geom. 2018) Mn Sn+1 (n > 3): closed CMC hypersurface with constant Gauss-Kronecker curvature and g = 3 everywhere. ⇒ Mn 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

Our results: g = constant(< n)

When the number of distinct eigenvalues is less than n, we obtain easily: Proposition 1 (Tang-Wei-Y., 2018) Mn: closed Riemannian manifold. a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (3.1) the number g of distinct eigenvalues of A is a constant and g < n; (3.2) tr(Ak) (k = 1, · · · , g) are constants; ⇒ the 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 = constant(< n)

When the number of distinct eigenvalues is less than n, we obtain easily: Proposition 1 (Tang-Wei-Y., 2018) Mn: closed Riemannian manifold. a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (3.1) the number g of distinct eigenvalues of A is a constant and g < n; (3.2) tr(Ak) (k = 1, · · · , g) are constants; ⇒ the 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 = constant(< n)

When the number of distinct eigenvalues is less than n, we obtain easily: Proposition 1 (Tang-Wei-Y., 2018) Mn: closed Riemannian manifold. a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (3.1) the number g of distinct eigenvalues of A is a constant and g < n; (3.2) tr(Ak) (k = 1, · · · , g) are constants; ⇒ the 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 = constant(< n)

When the number of distinct eigenvalues is less than n, we obtain easily: Proposition 1 (Tang-Wei-Y., 2018) Mn: closed Riemannian manifold. a: smooth symmetric (0, 2) tensor field on Mn, A: dual (1, 1) tensor field of a. Suppose (3.1) the number g of distinct eigenvalues of A is a constant and g < n; (3.2) tr(Ak) (k = 1, · · · , g) are constants; ⇒ the 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 = constant(< n)

Again, considering a as the second fundamental form, we obtain immediately Corollary 2 (Tang-Wei-Y., 2018) Mn Sn+1: closed hypersurface. Suppose (4.1) the number g of principal curvatures is a constant and g < n; (4.2) the k-th (k = 1, · · · , g) power sum of principal curvatures are constants, ⇒ Mn 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

Our results: g = constant(< n)

Again, considering a as the second fundamental form, we obtain immediately Corollary 2 (Tang-Wei-Y., 2018) Mn Sn+1: closed hypersurface. Suppose (4.1) the number g of principal curvatures is a constant and g < n; (4.2) the k-th (k = 1, · · · , g) power sum of principal curvatures are constants, ⇒ Mn 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

Our results: g = constant(< n)

Again, considering a as the second fundamental form, we obtain immediately Corollary 2 (Tang-Wei-Y., 2018) Mn Sn+1: closed hypersurface. Suppose (4.1) the number g of principal curvatures is a constant and g < n; (4.2) the k-th (k = 1, · · · , g) power sum of principal curvatures are constants, ⇒ Mn 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

Thank you !

wjyan@bnu.edu.cn

Wenjiao Yan Isoparametric foliation and its applications