Whats systole? Mijia Lai 2019.01.10 Definition The Systole of a - - PowerPoint PPT Presentation

What’s systole?

Mijia Lai 2019.01.10

Definition

The Systole of a compact metric space X is the least length of a non-contractible loop in X, denoted by sysπ1.

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Definition

The Systole of a compact metric space X is the least length of a non-contractible loop in X, denoted by sysπ1. Question Relation of systole with global geometric quantities, which are independent of curvature.

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Definition

d : X × X → R+ is called a metric if

• d(x, y) = d(y, x);
• d(x, y) ≥ 0, and = occurs if and only if x = y;
• d(x, z) + d(z, y) ≥ d(x, y), ∀x, y, z ∈ X.

(X, d) is called a metric space.

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Definition

d : X × X → R+ is called a metric if

• d(x, y) = d(y, x);
• d(x, y) ≥ 0, and = occurs if and only if x = y;
• d(x, z) + d(z, y) ≥ d(x, y), ∀x, y, z ∈ X.

(X, d) is called a metric space. It is called compact if every sequence in X has a convergent subsequence.

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A non-contractible loop

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History

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History

• 1947: Tutte studied the girth of a graph.

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History

• 1947: Tutte studied the girth of a graph.
• 1949: Loewner initiated the study on the torus.

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History

• 1947: Tutte studied the girth of a graph.
• 1949: Loewner initiated the study on the torus.
• 1950’s thesis: P.M. Pu on the projective plane and the

M¨

• bius band.

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History

• 1947: Tutte studied the girth of a graph.
• 1949: Loewner initiated the study on the torus.
• 1950’s thesis: P.M. Pu on the projective plane and the

M¨

• bius band.
• 1960-1970’s: Berger’s propaganda. (An impetus from Thom)

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History

• 1947: Tutte studied the girth of a graph.
• 1949: Loewner initiated the study on the torus.
• 1950’s thesis: P.M. Pu on the projective plane and the

M¨

• bius band.
• 1960-1970’s: Berger’s propaganda. (An impetus from Thom)
• 1983: Gromov’s spectacular inequality.

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History

• 1947: Tutte studied the girth of a graph.
• 1949: Loewner initiated the study on the torus.
• 1950’s thesis: P.M. Pu on the projective plane and the

M¨

• bius band.
• 1960-1970’s: Berger’s propaganda. (An impetus from Thom)
• 1983: Gromov’s spectacular inequality.
• Higher genus surface, k-systole, filling area, volume entropy,

Lusternik-Schnirelmann category, J-holomorphic curves...

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Loewner

Theorem sysπ1(T2)2 ≤ 2 √ 3 area(T2) holds for any metric on T2.

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Loewner

Theorem sysπ1(T2)2 ≤ 2 √ 3 area(T2) holds for any metric on T2.

• 2

√ 3 = γ2, which is the Hermite constant. (lattice) 6

Loewner

Theorem sysπ1(T2)2 ≤ 2 √ 3 area(T2) holds for any metric on T2.

• 2

√ 3 = γ2, which is the Hermite constant. (lattice)

• Equality holds if T2 = R2/L, where L is the lattice Z-spanned

by cubic roots. Hexagonal lattice.

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Uniformization of Torus

(T2, g) is conformal to a flat torus (T2, g0) i.e., g = ρ2g0, and any flat tours (T2, g0) is isometric to a parallelogram with two opposite sides identified. This means (T2, g0) is foliated by a family of closed geodesic ls of fixed length l for s ∈ [0, h]. By Fubini’s theorem, we have Area((T2, g)) =

• T2 ρ2dv0 =

h ds

• ls

ρ2dθ.

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H¨

• lder’s inequality implies:
• ls

ρ2dθ

• ls

1dθ ≥ (

• ls

ρdθ)2. Hence

• ls

ρ2dθ ≥ 1 l Lengthg2(ls). Scaling of flat is still flat, we can assume Area((T2, g)) = Area((T2, g0)) = h · l. Using mean value theorem, it follows that there exists s0 such that l ≥ Lengthg(ls0).

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Hermite constant

Hence the problem is reduced to flat tori, which in simple term, we ask for the biggest length of the shorter side among all parallelogram of area 1.

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P.M. Pu

Theorem sysπ1(RP2)2 ≤ π 2 area(RP2) holds for any metric on RP2.

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P.M. Pu

Theorem sysπ1(RP2)2 ≤ π 2 area(RP2) holds for any metric on RP2.

• Equality holds if RP2 is the antipodal quotient of round unit

sphere, for which the area is 2π and the length of a closed geodesic is π.

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Remarks

Isoperimetric inequality: 4πA ≤ l2.

• Systolic inequality is kind of a converse to isoperimetric

inequality.

• Isoperimetric inequality is with respect to a specified metric.
• Systolic inequality is a uniform inequality for all metrics.
• Higher genus surface: sysπ2

1 ≤ Carea, optimal C is not

• btained.

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Gromov

Theorem There exists a universal constant Cn, such that sysπ1(Tn) ≤ Cnvol(Tn, g)

1 n

holds for any metric g on Tn.

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One approach: topological dimension theory

Theorem (Brouwer 1909) There exists no homeomorphism from Rn to Rm if n = m.

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One approach: topological dimension theory

Theorem (Brouwer 1909) There exists no homeomorphism from Rn to Rm if n = m. If require the map is linear, we have two stronger facts from linear algebra.

• There exists no linear surjection from Rn to Rm if n < m.
• There exists no linear injection from Rn to Rm if n > m.

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Dropping the linear requirement, we have

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Dropping the linear requirement, we have

• there exists continuous surjection from Rn to Rm if n < m.

(Peano curves)

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Dropping the linear requirement, we have

• there exists continuous surjection from Rn to Rm if n < m.

(Peano curves)

• there exists no continuous injection from Rn to Rm if n > m.

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Dropping the linear requirement, we have

• there exists continuous surjection from Rn to Rm if n < m.

(Peano curves)

• there exists no continuous injection from Rn to Rm if n > m.

Smaller dimension can be stretched to cover a higher-dimensional space, but a higher-dimensional space may not be squeezed to fit into a lower-dimensional space.

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Open covering

{Ui} an open covering is called

• multiplicity of µ: if any point is contained in at most µ open

sets.

• diameter D: if diam(Ui) ≤ D.

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Lebesgue dimension theory

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Lebesgue dimension theory

• Lebesgue constructed an open covering of Rn with multiplicity

n + 1 and arbitrary small diameter.

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Lebesgue dimension theory

• Lebesgue constructed an open covering of Rn with multiplicity

n + 1 and arbitrary small diameter.

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Lebesgue dimension theory

• Lebesgue constructed an open covering of Rn with multiplicity

n + 1 and arbitrary small diameter.

• Lebesgue also proved the following: if Ui open sets cover the

unit cube of diameter 1, then some point of the n−cube must lie in at least n + 1 different Ui.

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Generalizations to Riemannian manifolds

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Generalizations to Riemannian manifolds

Theorem (Guth 2008) If (M, g) is an n dimensional Riemannian manifold with volume V , then there is an open cover with multiplicity n and diameter CnV

1 n .

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Generalizations to Riemannian manifolds

Theorem (Guth 2008) If (M, g) is an n dimensional Riemannian manifold with volume V , then there is an open cover with multiplicity n and diameter CnV

1 n .

Theorem (Gromov) If (Tn, g) has systole at least 1 and {Ui} an open cover of diameter

1 10, then some point lies in at least n + 1 different sets Ui. 17

Generalizations to Riemannian manifolds

Theorem (Guth 2008) If (M, g) is an n dimensional Riemannian manifold with volume V , then there is an open cover with multiplicity n and diameter CnV

1 n .

Theorem (Gromov) If (Tn, g) has systole at least 1 and {Ui} an open cover of diameter

1 10, then some point lies in at least n + 1 different sets Ui.

Hence sysπ1 ≤ CnV

1 n .

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