pinching theorems on minimal submanifolds of …zlu/talks/2011-uci-geometry/...prelude introduction...

. . . . . .

Prelude Introduction The results Applications The proof

Pinching Theorems on minimalsubmanifolds of sphere

Zhiqin Lu (陆志勤)

University of California at Irvine

May 31st, 2011

Zhiqin Lu UC IrvinePinching Theorems

. . . . . .

Böttcher-Wenzel Conjecture

.Conjecture..

Let X,Y be n × n real matrices. Then

||[X,Y]||2 ≤ 2||X||2||Y||2.

Here the norm is the Hilbert-Schmidt norm:

||X||2 = tr(A∗A)

. . . . . .

...1 The conjecture was proved by L in 2007 (posted on arXivon 11/22/2007); after that there are three differentproofs by Vong-Jin (黄锡荣－金小庆 ),Böttcher-Wenzel, and Audenaert.

...2 Obviously, the result can be generalized to operators onseparable Hilbert spaces

...3 We sharpened the Schrödinger-Robertson relation, whichwas

||[A,B]|| ≤ 2||A||||B||.The optimal constant is

. . . . . .

Equality case

LetX =

||[X,Y]|| =√2||X|| · ||Y||.

. . . . . .

A result of Chern-do Carmo-Kobayashi (in 70’s).Theorem..

If A,B are symmetric matrices, then

||[A,B]||2 ≤ 2||A||2||B||2

Since the inequality is invariant under orthogonaltransformation, we can assume that A is diagonalized.Then the inequality becomes∑

(λi − λj)2b2

ij ≤ 2||A||2||B||2.

(λi − λj)2 ≤ 2(

∑λ2

. . . . . .

A result of Chern-do Carmo-Kobayashi (in 70’s).Theorem..

If A,B are symmetric matrices, then

||[A,B]||2 ≤ 2||A||2||B||2

Since the inequality is invariant under orthogonaltransformation, we can assume that A is diagonalized.Then the inequality becomes∑

(λi − λj)2b2

ij ≤ 2||A||2||B||2.

(λi − λj)2 ≤ 2(

∑λ2

. . . . . .

Proof of the BW conjecture.

LetX = Q1ΛQ2

be the singular decomposition of X, where Q1,Q2 areorthogonal matrices and Λ is a diagonal matrix. Let

B = Q2YQ−12 , C = Q−1

1 YQ1.

Then we have

||[X,Y]||2 = ||ΛB − CΛ||2, expect to be ≤ 2||Λ||2||B||2

. . . . . .

...1 If b11 = 0, then the conjecture can be proved

...2 We can make b11 = 0

. . . . . .

...1 If b11 = 0, then the conjecture can be proved

...2 We can make b11 = 0

. . . . . .

Since b11 = 0, we have(Λ = diag(s1, · · · , sn),

∑s2j = 1, s1 is the biggest)

||ΛB−CΛ||2 = c211s21+n∑

(sibi1−s1ci1)2+

n∑j=2

(s1b1j−sjc1j)2+∆1,

where∆1 =

n∑i,j=2

(sibij − sjcij)2.

Since s22 ≤ 1/2, we have

∆1 ≤n∑

(b2ij + c2ij).

. . . . . .

The conjecture is implied by the following inequality

c211s21+n∑

(sibi1−s1ci1)2+

n∑j=2

(s1b1j−sjc1j)2 ≤ ∆+

n∑i=2

n∑j=2

where∆ =

n∑i=2

b21i +

n∑j=2

c2j1 + c211.

. . . . . .

We consider the matrix∆ −b12c12 − b21c21 · · · −b1nc1n − bn1cn1

−b12c12 − b21c21 b221 + c212... . . .

−b1nc1n − bn1cn1 b2n1 + c21n

The conjecture is equivalent to that the maximum eigenvalueof the above matrix is no more than ∆+

∑ni=2 b2

i1 +∑n

j=2 c21j.

. . . . . .

We fix X and assume that ||X|| = 1. Let V = gl(n,R). Definea linear map

T : V → V, Y 7→ [XT, [X,Y]],

where XT is the transpose of X. Then we have.Lemma..

.T is a semi-positive definite symmetric linear transformation ofV.

. . . . . .

The conjecture is equivalent to the statement that themaximum eigenvalue of T is not more than 2.Let α be the maximum eigenvalue of T. Then α > 0. Let Ybe an eigenvector of T with respect to α. Then we have

T(Y) = αY.

A straightforward computation gives

T([XT,YT]) = α[XT,YT].

Y and Y1 = [XT,YT] are linearly independent.Linear combine the two eigenfunctions, we get b11 = 0.

. . . . . .

T(Y) = αY.

. . . . . .

T(Y) = αY.

. . . . . .

Consider minimal submanifolds of sphere

...1 James Simons proved a gap theorem for the length of thesecond fundamental form of minimal submanifold insphere

...2 Generalizations of J. Simons result was obtained by Chern(陈省身)-do Carmo-Kobayashi, Yau (丘成桐), Shen(沈一兵), Wu (吴报强)-Song, Li-Li (李基民－李安民),Chen-Xu (陈卿－徐森林)

...3 Our results generalize all the above pinching theorems(for the first gap) with new method

. . . . . .

For the second gap......1 Chern (陈省身) Conjecture: if the scalar curvature of a

minimal hypersurface is constant, then the length of thesecond fundamental form is discrete.

...2 ��p Peng and Terng (彭家贵－滕楚莲) proved that, ifthe length of the second fundamental form is bigger thann, then it must be bigger than n + 1

9n ....3 Many generalizations...4 Evidence: for a complex hypersurface of CPn, if the scalar

curvature is a constant, then the length of the secondfundamental form can be represented by Chern classes;hence discrete.

. . . . . .

Basic Facts

Let M be a minimal submanifold of Sn+p....1 Let hα

ij be the second fundamental form (wrt orthonormalframes)

...2 Let Aα = (hαij )

...3 We have

∆Aα = nAα − ⟨Aα,Aβ⟩Aβ − [Aβ, [Aβ,Aα]].

. . . . . .

Basic Facts

...3 We have

. . . . . .

Basic Facts

...3 We have

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Basic Facts

...3 We have

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Simons-Bochner formula

2∆||σ||2 =

∑i,j,k,α

(hαijk)

2+n||σ||2−∑α,β

||[Aα,Aβ]||2−∑α,β

|⟨Aα,Aβ⟩|2.

The Simons (1969) pinching theorem

2∆||σ||2 ≥ n||σ||2 − 2

∑α̸=β

||Aα||2||Aβ||2 −∑α

||Aα||4

≥ n||σ||2 − (2− 1

p)||σ||4

. . . . . .

Simons-Bochner formula

2∆||σ||2 =

∑i,j,k,α

(hαijk)

2+n||σ||2−∑α,β

||[Aα,Aβ]||2−∑α,β

|⟨Aα,Aβ⟩|2.

The Simons (1969) pinching theorem

2∆||σ||2 ≥ n||σ||2 − 2

∑α̸=β

||Aα||2||Aβ||2 −∑α

||Aα||4

≥ n||σ||2 − (2− 1

p)||σ||4

. . . . . .

Some examples of minimal submanifolds:...1 Totally geodesic minimal submanifolds; ||σ||2 = 0

...2 Mr,n−r = Sr (√ rn)× Sn−r

(√n−r

); ||σ||2 = n

...3 The Veronese surface is defined as the immersion;||σ||2 = 4/3 = 2

u1 =1√3

yz, u2 =1√3

zx, u3 =1√3

xy, u4 =1

2√3(x2 − y2),

6(x2 + y2 − 2z2).

(x, y, z) ∈ S2.

. . . . . .

...2 Mr,n−r = Sr (√ rn)× Sn−r

(√n−r

); ||σ||2 = n

u1 =1√3

yz, u2 =1√3

zx, u3 =1√3

xy, u4 =1

2√3(x2 − y2),

6(x2 + y2 − 2z2).

(x, y, z) ∈ S2.

. . . . . .

...2 Mr,n−r = Sr (√ rn)× Sn−r

(√n−r

); ||σ||2 = n

u1 =1√3

yz, u2 =1√3

zx, u3 =1√3

xy, u4 =1

2√3(x2 − y2),

6(x2 + y2 − 2z2).

(x, y, z) ∈ S2.

. . . . . .

...2 Mr,n−r = Sr (√ rn)× Sn−r

(√n−r

); ||σ||2 = n

u1 =1√3

yz, u2 =1√3

zx, u3 =1√3

xy, u4 =1

2√3(x2 − y2),

6(x2 + y2 − 2z2).

(x, y, z) ∈ S2.

. . . . . .

It was found by J. Simons that there are rigidity theorems(pinching theorems) for minimal submanifolds of SN. Moreprecisely, the optimal results were

...1 For minimal hypersurface of dim n, if 0 ≤ ||σ||2 ≤ n, theneither ||σ|| = 0 or ||σ||2 = n (well known, Chern(陈省身)–do Carmo-Kobayashi);

...2 For high co-dimensional minimal submanifolds, if0 ≤ ||σ||2 ≤ 2

3n, then they are totally geodesic or

Veronese surface. (Li-Li (李基民－李安民), Chen-Xu(陈卿－徐森林))

. . . . . .

The new method

Taking ∆ on max ||Ar||2 instead of ||σ||2.New J. Simons formula:

2p∆fp =1

∑s+t=p−2

∑k,α,β

λsαλ

(D ∂

∂xkaαβ)2

+∑α

λp−1α

∑i,j,k

(hαijk)

+ nfp − fp+1 −

∑β ̸=α

||[Aβ,Aα]||2||Aα||p−1,

where fp = tr(Fp), F = (⟨Ai,Aj⟩).

lim f1pp = max ||Ar||.

. . . . . .

The new method

2p∆fp =1

∑s+t=p−2

∑k,α,β

λsαλ

(D ∂

∂xkaαβ)2

+∑α

λp−1α

∑i,j,k

(hαijk)

+ nfp − fp+1 −

∑β ̸=α

||[Aβ,Aα]||2||Aα||p−1,

. . . . . .

The new method

2p∆fp =1

∑s+t=p−2

∑k,α,β

λsαλ

(D ∂

∂xkaαβ)2

+∑α

λp−1α

∑i,j,k

(hαijk)

+ nfp − fp+1 −

∑β ̸=α

||[Aβ,Aα]||2||Aα||p−1,

. . . . . .

The technical heart of the estimate is the.Theorem..

∑||[A,Aj]||2 ≤ ||A||2(

∑||Aj||2 + max ||Aj||2)

where A1, · · · ,Am are orthogonal.

(Chern-do Carmo-Kobayashi gives 2||A||2(∑

||Aj||2).

. . . . . .

Let the matrix F = (⟨Ar,As⟩). Let λ2 be the second largesteigenvalue of the matrix..Theorem..

If0 ≤ ||σ||2 + λ2 ≤ n

then M is totally geodesic, one of the Mr,n−r, or a Veronesesurface.

. . . . . .

Let the matrix F = (⟨Ar,As⟩). Let λ2 be the second largesteigenvalue of the matrix..Theorem..

If0 ≤ ||σ||2 + λ2 ≤ n

then M is totally geodesic, one of the Mr,n−r, or a Veronesesurface.

. . . . . .

Generalization and Unification

||σ||2 is the trace of the fundamental matrix.

||σ||2 + λ2 ≤ ||σ||2 + 1

2||σ||2 ≤ n

||σ||2 ≤ 2

. . . . . .

||σ||2 + λ2 ≤ ||σ||2 + 1

2||σ||2 ≤ n

||σ||2 ≤ 2

. . . . . .

||σ||2 + λ2 ≤ ||σ||2 + 1

2||σ||2 ≤ n

||σ||2 ≤ 2

. . . . . .

.Corollary..

If the co-dimension is more than 1, and if

0 ≤ ||σ||2 + max ||Ar||2 ≤ n

then M is either totally geodesic, or a Veronese surface.

. . . . . .

When the equality is valid, that is, when

||σ||2 + λ2 ≡ n,

then the submanifolds must be of the three types mentionedabove. A typical result is in the next slide.

. . . . . .

.Theorem..

There is no minimal immersion Mn → S2n such that thesecond fundamental forms are

A1 = c1

n − 1

−1. . .

,A2 = c2

0 11 0

. . .0

A3 = c2

0 0 10 0 01 0 0

. . .0

,A4 = c2...

unless n = 2.Zhiqin Lu UC IrvinePinching Theorems

. . . . . .

Normal Scalar Curvature Conj.–1st application

...1 Consider a surface in R3. Principal curvature atP ∈ S : k1, k2

...2 Gaussian curvature: K(p) = k1 · k2

...3 Mean curvature: H(p) = k1+k22

...4 Then we have

H2(p) ≥ K(p) Extrinsic ≥ Intrinsic

. . . . . .

...4 Then we have

. . . . . .

...4 Then we have

. . . . . .

...4 Then we have

. . . . . .

Let Mn be an n-dimensional manifold isometrically immersedinto the space form Nn+m(c) of constant sectional curvature c.By the same reason,

|H|2 + c ≥ ρ,

where H is the mean curvature tensor. (B. Suceavă, etc)

. . . . . .

Let Mn be an n-dimensional manifold isometrically immersedinto the space form Nn+m(c) of constant sectional curvature c.By the same reason,

|H|2 + c ≥ ρ,

where H is the mean curvature tensor. (B. Suceavă, etc)

. . . . . .

The Conjecture

The (normalized) scalar curvature of the normal bundle isdefined as:

ρ⊥ =1

n(n − 1)|R⊥|,

where R⊥ is the curvature tensor of the normal bundle. Moreprecisely, let ξ1, · · · , ξm be a local orthonormal frame of thenormal bundle. Then

ρ⊥ =2

n(n − 1)

( n∑1=i<j

m∑1=r<s

⟨R⊥(ei, ej)ξr, ξs⟩2) 1

. . . . . .

The Conjecture

ρ⊥ =1

n(n − 1)|R⊥|,

ρ⊥ =2

n(n − 1)

( n∑1=i<j

m∑1=r<s

. . . . . .

The Conjecture

ρ⊥ =1

n(n − 1)|R⊥|,

ρ⊥ =2

n(n − 1)

( n∑1=i<j

m∑1=r<s

. . . . . .

ρ⊥ is always nonnegative. In the study of submanifold theory,De Smet, Dillen, Verstraelen, and Vrancken proposed thefollowing Normal Scalar Curvature Conjecture.Conjecture..

Let h be the second fundamental form, and let H = 1n trace h

be the mean curvature tensor. Then

ρ+ ρ⊥ ≤ |H|2 + c.

. . . . . .

ρ+ ρ⊥ ≤ |H|2 + c.

. . . . . .

ρ+ ρ⊥ ≤ |H|2 + c.

. . . . . .

...1 Partial results were obtained by Chern (陈省身)-doCarmo-Kobayashi, Rouxel, Guadalupe-Rodriguez, DeSmet-Dillen-Verstraelen-Vrancken, Choi-Lu (陆志勤),Dillen-Fastenakels-Veken, etc

...2 I proved the conjecture in 2007

...3 Ge-Tang (唐梓州) gave an independent proof slightlylater.

. . . . . .

The algebraic version of the Normal Scalar CurvatureConjecture can be stated as follows:.Theorem..

Let A1, · · · ,Am be symmetric matrices. For n,m ≥ 2, we have

||Ar||2)2 ≥ 2(∑r<s

||[Ar,As]||2).

. . . . . .

The algebraic version of the Normal Scalar CurvatureConjecture can be stated as follows:.Theorem..

Let A1, · · · ,Am be symmetric matrices. For n,m ≥ 2, we have

||Ar||2)2 ≥ 2(∑r<s

||[Ar,As]||2).

. . . . . .

Erdös-Mordell inequality (1935)–2nd application

PA + PB + PC ≥ 2(PPA + PPB + PPC).

. . . . . .

The observation

Erdös-Mordell inequality is the dual version of the NormalScalar Curvature Conjecture!

...1 Is a2 + b2 + c2 ≥ ab + bc + ca a generalization ofa2 + b2 ≥ 2ab?

...2 Not really!

...3 The correct generalization is: if α + β + γ = π, then

a2 + b2 + c2 ≥ 2ab cos γ + 2bc cosα + 2ac cos β,

which implies the Erdös-Mordell inequality.

. . . . . .

The observation

...2 Not really!

. . . . . .

The observation

...2 Not really!

. . . . . .

The observation

...2 Not really!

. . . . . .

...1 We consider vectors x⃗, y⃗, z⃗ in R2 such that the anglesbetween them are π/2 + γ/2, π/2 + β/2, π/2 + α/2,respectively. Moreover, we assume that that norm ofx⃗, y⃗, z⃗ are a, b, c.

...2 Let X,Y,Z be the corresponding 3× 3 skew-symmetricmatrices.

...3 Then the EM inequality is equivalent to

||[X,Y]||2+||[Y,Z]||2+||[Z,X]||2 ≤ 1

8(||X||2+||Y||2+||Z||2)2.

. . . . . .

||[X,Y]||2+||[Y,Z]||2+||[Z,X]||2 ≤ 1

8(||X||2+||Y||2+||Z||2)2.

. . . . . .

||[X,Y]||2+||[Y,Z]||2+||[Z,X]||2 ≤ 1

8(||X||2+||Y||2+||Z||2)2.

. . . . . .

The conjecture

We expect the following generalization of the EM inequality istrue..Conjecture..

Let A1, · · · ,Am be skew-symmetric matrices. Then

||Ar||2)2 ≥ c(m, n)(∑r<s

||[Ar,As]||2) n ≥ 4

||Ar||2)2 ≥ 8c(m, n)(∑r<s

||[Ar,As]||2) n = 3

. . . . . .

.Theorem..

We have

(||X||2 + ||Y||2)2 ≥ ||[X,Y]||2 n ≥ 4

(||X||2 + ||Y||2)2 ≥ 8||[X,Y]||2 n = 3 (This implies EM)

for skew-symmetric matrices.

For proof of the above results we need the Böttcher-WenzelTheorem!

. . . . . .

Proof of the key inequality

First, we prove the following elementary lemma.Lemma..

Suppose η1, · · · , ηn are real numbers and

η1 + · · ·+ ηn = 0, η21 + · · ·+ η2n = 1.

Let rij ≥ 0 be nonnegative numbers for i < j. Then we have∑i<j

(ηi − ηj)2rij ≤

∑i<j

rij + Max(rij). (1)

. . . . . .

If η1 ≥ · · · ≥ ηn, and rij are not simultaneously zero, then theequality in the above holds in one of the following three cases:

...1 rij = 0 unless (i, j) = (1, n),(η1, · · · , ηn) = (1/

√2, 0, · · · , 0,−1/

√2);

...2 rij = 0 if 2 ≤ i < j, r12 = · · · = r1n ̸= 0, and(η1, · · · , ηn) =(√

(n − 1)/n,−1/√

n(n − 1), · · · ,−1/√

n(n − 1));...3 rij = 0 if i < j < n, r1n = · · · = r(n−1)n ̸= 0, and(η1, · · · , ηn) =(1/√

n(n − 1), · · · , 1/√

n(n − 1),−√

(n − 1)/n);

. . . . . .

√2, 0, · · · , 0,−1/

√2);

(n − 1)/n,−1/√

n(n − 1), · · · ,−1/√

n(n − 1), · · · , 1/√

n(n − 1),−√

(n − 1)/n);

. . . . . .

√2, 0, · · · , 0,−1/

√2);

(n − 1)/n,−1/√

n(n − 1), · · · ,−1/√

n(n − 1), · · · , 1/√

n(n − 1),−√

(n − 1)/n);

. . . . . .

The key lemma

.Lemma..

Let A be an n × n diagonal matrix of norm 1. Let A2, · · · ,Ambe symmetric matrices such that1.⟨Aα,Aβ⟩ = 0 if α ̸= β;2. ||A2|| ≥ · · · ≥ ||Am||.Then we have

m∑α=2

||[A,Aα]||2 ≤m∑

||Aα||2 + ||A2||2. (2)

. . . . . .

The equality holds iff, after an orthonormal base change andup to a sign, we have

A3 = · · · = Am = 0, and

0 − 1√2

0. . .

A2 = c

0 1√

21√2

0. . .

. . . . . .

The equality holds iff, after an orthonormal base change andup to a sign, we have

A3 = · · · = Am = 0, and

0 − 1√2

0. . .

A2 = c

0 1√

21√2

0. . .

. . . . . .

OrFor two real numbers λ = 1/

√n(n − 1) and µ, we have

A1 = λ

n − 1

−1. . .

and Aα is µ times the matrix whose only nonzero entriesare 1 at the (1, α) and (α, 1) places, where α = 2, · · · , n.

. . . . . .

We assume that each Aα is not zero. Let Aα = ((aα)ij), where(aα)ij are the entries for α = 2, · · · ,m. Let

δ = Maxi̸=j

m∑α=2

(aα)2ij.

η1. . .

Then by the previous lemma, we havem∑

||[A,Aα]||2 ≤m∑

||Aα||2 + 2δ.

. . . . . .

Thus it remains to prove that

2δ ≤ ||A2||2.

To see this, we identify each Aα with the (column) vector A⃗α

in R 12

n(n+1) as follows:

Aα 7→ (a12, · · · , a1n, a23, · · · , a2n, · · · , a(n−1)n,1√2

a11, · · · ,1√2

ann)T.

Let µα be the norm of the vector Aα. Then we have

µ2α =

2||Aα||2

for α = 2, · · · ,m.

. . . . . .

Extending the set of vectors {A⃗α/µα}2≤α≤m into anorthonormal basis of R 1

2n(n+1)

A⃗2/µ2, · · · , A⃗m/µm, A⃗m+1, · · · , A⃗ 12

n(n+1)+1,

we get an orthogonal matrix. Apparently, each row vector ofthe matrix is a unit vector. Thus we have

m∑α=2

(µα)−2(aα)2ij ≤ 1

for fixed i < j. Since µ2 ≥ · · · ≥ µm, we getm∑

(aα)2ij ≤ µ22 ≤

2||A2||2.

This completes the proof.Zhiqin Lu UC IrvinePinching Theorems

. . . . . .

In conclusion...

...1 many more generalizations in linear algebra

...2 the maximum commutator on simple (real or complex)Lie algebras

...3 the “correct” quantity for the optimal pinching theorem?

...4 the Peng-Terng (彭家贵－滕楚莲)-type gap theorem

...5 comass problem in calibrated geometry

...6 the parabolic version of the result, high co-dimensionalmean curvature flow

...7 the linear algebra applicable to Ricci flow

. . . . . .

In conclusion...

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In conclusion...

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In conclusion...

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In conclusion...

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In conclusion...

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In conclusion...

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Thank you!

pinching theorems on minimal submanifolds of …zlu/talks/2011-uci-geometry/...prelude introduction...

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