On open 3-manifolds School on Geometric Evolution Problems

G. Besson

Universit´eGrenoble Alpes

Centro De Giorgi June, 23rd 2014

INSTITUT i f FOURIER Outline Surfaces Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). Involves the structure at infinity of non compact surfaces.

The question

Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ? Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). Involves the structure at infinity of non compact surfaces.

The question

Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ?

Surfaces Involves the structure at infinity of non compact surfaces.

The question

Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ?

Surfaces Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). The question

Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ?

Surfaces Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). Involves the structure at infinity of non compact surfaces. I let X a simply connected closed 3-manifolds, X \ {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold.

Whitehead’s ”proof ” of the Poincar´econjecture

J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold.

Whitehead’s ”proof ” of the Poincar´econjecture

J. H. C. Whitehead suggested in 1934 a proof based on the following scheme :

I let X a simply connected closed 3-manifolds, X \ {∗} is contractible. 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold.

Whitehead’s ”proof ” of the Poincar´econjecture

J. H. C. Whitehead suggested in 1934 a proof based on the following scheme :

I let X a simply connected closed 3-manifolds, X \ {∗} is contractible. 3 I The only contractible 3-manifolds is R . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold.

Whitehead’s ”proof ” of the Poincar´econjecture

J. H. C. Whitehead suggested in 1934 a proof based on the following scheme :

I let X a simply connected closed 3-manifolds, X \ {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . , the so-called Whitehead manifold.

Whitehead’s ”proof ” of the Poincar´econjecture

J. H. C. Whitehead suggested in 1934 a proof based on the following scheme :

I let X a simply connected closed 3-manifolds, X \ {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3 Whitehead’s ”proof ” of the Poincar´econjecture

J. H. C. Whitehead suggested in 1934 a proof based on the following scheme :

I let X a simply connected closed 3-manifolds, X \ {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold. Outline 3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1.

On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1.

Whitehead manifolds

Take T1 ⊃ T2 ⊃ T3 ⊃ ... solid tori. On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1.

Whitehead manifolds

Take T1 ⊃ T2 ⊃ T3 ⊃ ... solid tori.

3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1. On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1.

Whitehead manifolds

Take T1 ⊃ T2 ⊃ T3 ⊃ ... solid tori.

3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1. Whitehead manifolds

Take T1 ⊃ T2 ⊃ T3 ⊃ ... solid tori.

3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1.

On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1. 3 3 I X = S \ W ⊂ S is a (the) whitehead manifold (genus one).

Theorem (J.H.C. Whitehead) X is contractible and not homeomorphic to R3.

Whitehead manifolds

I W = ∩Ti is the Whitehead continuum. Theorem (J.H.C. Whitehead) X is contractible and not homeomorphic to R3.

Whitehead manifolds

I W = ∩Ti is the Whitehead continuum. 3 3 I X = S \ W ⊂ S is a (the) whitehead manifold (genus one). Whitehead manifolds

I W = ∩Ti is the Whitehead continuum. 3 3 I X = S \ W ⊂ S is a (the) whitehead manifold (genus one).

Theorem (J.H.C. Whitehead) X is contractible and not homeomorphic to R3. Hausdorff dimension > 1.

Whitehead continuum Whitehead continuum

Hausdorff dimension > 1. 3 I Si = S \ Ti is diffeomorphic to a solid torus. I X is a limit of solid tori.

Whitehead manifolds : remarks

3 Ti is knotted in Ti−1 but not in S , hence I X is a limit of solid tori.

Whitehead manifolds : remarks

3 Ti is knotted in Ti−1 but not in S , hence 3 I Si = S \ Ti is diffeomorphic to a solid torus. Whitehead manifolds : remarks

3 Ti is knotted in Ti−1 but not in S , hence 3 I Si = S \ Ti is diffeomorphic to a solid torus. I X is a limit of solid tori. Whitehead link

The idea is that the core of Ti and the meridian of Ti−1 form the Whitehead link. Whitehead link

The idea is that the core of Ti and the meridian of Ti−1 form the Whitehead link. ; role of the two curves could be exchanged.

Whitehead link

Whitehead link is symmetric Whitehead link

Whitehead link is symmetric ; role of the two curves could be exchanged. Whitehead link

Whitehead link is symmetric ; role of the two curves could be exchanged. 3 I Define Si = S \ Ti .

I Si is a solid torus,

I Si ⊂ Si+1 knotted in the same way, S I symmetry ⇒ X = Si . I A more general construction.

Whitehead manifolds

Whitehead link is symmetric ; consequences. I Si is a solid torus,

I Si ⊂ Si+1 knotted in the same way, S I symmetry ⇒ X = Si . I A more general construction.

Whitehead manifolds

Whitehead link is symmetric ; consequences. 3 I Define Si = S \ Ti . I Si ⊂ Si+1 knotted in the same way, S I symmetry ⇒ X = Si . I A more general construction.

Whitehead manifolds

Whitehead link is symmetric ; consequences. 3 I Define Si = S \ Ti .

I Si is a solid torus, S I symmetry ⇒ X = Si . I A more general construction.

Whitehead manifolds

Whitehead link is symmetric ; consequences. 3 I Define Si = S \ Ti .

I Si is a solid torus,

I Si ⊂ Si+1 knotted in the same way, I A more general construction.

Whitehead manifolds

Whitehead link is symmetric ; consequences. 3 I Define Si = S \ Ti .

I Si is a solid torus,

I Si ⊂ Si+1 knotted in the same way, S I symmetry ⇒ X = Si . Whitehead manifolds

Whitehead link is symmetric ; consequences. 3 I Define Si = S \ Ti .

I Si is a solid torus,

I Si ⊂ Si+1 knotted in the same way, S I symmetry ⇒ X = Si . I A more general construction. I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , ∃i such that S ⊂ Si , I ∀k, πk (X ) = {1} ; X is contractible. Non-homeomorphy to R3 (less easy)

I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C.

Whitehead manifolds

Contractibility (easy) k k I For any k-sphere S in X , ∃i such that S ⊂ Si , I ∀k, πk (X ) = {1} ; X is contractible. Non-homeomorphy to R3 (less easy)

I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C.

Whitehead manifolds

Contractibility (easy)

I Si is null homotopic in Si+1 i.e. it is homotopic to a point. I ∀k, πk (X ) = {1} ; X is contractible. Non-homeomorphy to R3 (less easy)

I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C.

Whitehead manifolds

Contractibility (easy)

I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , ∃i such that S ⊂ Si , Non-homeomorphy to R3 (less easy)

I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C.

Whitehead manifolds

Contractibility (easy)

I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , ∃i such that S ⊂ Si , I ∀k, πk (X ) = {1} ; X is contractible. I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C.

Whitehead manifolds

Contractibility (easy)

I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , ∃i such that S ⊂ Si , I ∀k, πk (X ) = {1} ; X is contractible. Non-homeomorphy to R3 (less easy) Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C.

Whitehead manifolds

Contractibility (easy)

I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , ∃i such that S ⊂ Si , I ∀k, πk (X ) = {1} ; X is contractible. Non-homeomorphy to R3 (less easy)

I X is not simply connected at infinity. Whitehead manifolds

Contractibility (easy)

I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , ∃i such that S ⊂ Si , I ∀k, πk (X ) = {1} ; X is contractible. Non-homeomorphy to R3 (less easy)

I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if ∀C compact, ∃D compact, D ⊃ C s.t. any loop in X \ D is null homotopic in X \ C. 4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). 3 I S /W is not a manifold.

I Uncountably many examples (McMillan) (compare to countably many closed 3-manifolds). 3 I Uncountably many examples which do not embed in S (Kister-McMillan).

I Examples that cannot cover non-trivially any manifold (Myers).

Whitehead manifolds

What is known : 3 I S /W is not a manifold.

I Uncountably many examples (McMillan) (compare to countably many closed 3-manifolds). 3 I Uncountably many examples which do not embed in S (Kister-McMillan).

I Examples that cannot cover non-trivially any manifold (Myers).

Whitehead manifolds

What is known :

4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). I Uncountably many examples (McMillan) (compare to countably many closed 3-manifolds). 3 I Uncountably many examples which do not embed in S (Kister-McMillan).

I Examples that cannot cover non-trivially any manifold (Myers).

Whitehead manifolds

What is known :

4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). 3 I S /W is not a manifold. (compare to countably many closed 3-manifolds). 3 I Uncountably many examples which do not embed in S (Kister-McMillan).

I Examples that cannot cover non-trivially any manifold (Myers).

Whitehead manifolds

What is known :

4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). 3 I S /W is not a manifold.

I Uncountably many examples (McMillan) 3 I Uncountably many examples which do not embed in S (Kister-McMillan).

I Examples that cannot cover non-trivially any manifold (Myers).

Whitehead manifolds

What is known :

4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). 3 I S /W is not a manifold.

I Uncountably many examples (McMillan) (compare to countably many closed 3-manifolds). I Examples that cannot cover non-trivially any manifold (Myers).

Whitehead manifolds

What is known :

4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). 3 I S /W is not a manifold.

I Uncountably many examples (McMillan) (compare to countably many closed 3-manifolds). 3 I Uncountably many examples which do not embed in S (Kister-McMillan). Whitehead manifolds

What is known :

4 6 I X × R ' R (Glimm-Shapiro) and X × X ' R (Glimm). 3 I S /W is not a manifold.

I Uncountably many examples (McMillan) (compare to countably many closed 3-manifolds). 3 I Uncountably many examples which do not embed in S (Kister-McMillan).

I Examples that cannot cover non-trivially any manifold (Myers). I Question 1 : what is the ”best” metric on M ?

I Question 2 : what is the evolution of the Ricci flow (with surgery) ?

Strategy

The idea is to use geometry to understand these spaces and eventually what could be the right statement of geometrization. I Question 2 : what is the evolution of the Ricci flow (with surgery) ?

Strategy

The idea is to use geometry to understand these spaces and eventually what could be the right statement of geometrization.

I Question 1 : what is the ”best” metric on M ? Strategy

The idea is to use geometry to understand these spaces and eventually what could be the right statement of geometrization.

I Question 1 : what is the ”best” metric on M ?

I Question 2 : what is the evolution of the Ricci flow (with surgery) ? They can even not carry a CAT (0) distance.

Theorem (Gromov-Lawson, Chang-Weinberger-Yu) Whitehead manifolds cannot carry complete metrics of uniformly positive scalar curvature.

Theorem (Gang Liu) Whitehead manifolds cannot carry complete metrics of nonnegative Ricci curvature.

Whitehead manifolds : known results

Remark Whitehead manifolds cannot carry a complete metric of non-positive curvature. Theorem (Gromov-Lawson, Chang-Weinberger-Yu) Whitehead manifolds cannot carry complete metrics of uniformly positive scalar curvature.

Theorem (Gang Liu) Whitehead manifolds cannot carry complete metrics of nonnegative Ricci curvature.

Whitehead manifolds : known results

Remark Whitehead manifolds cannot carry a complete metric of non-positive curvature. They can even not carry a CAT (0) distance. Theorem (Gang Liu) Whitehead manifolds cannot carry complete metrics of nonnegative Ricci curvature.

Whitehead manifolds : known results

Remark Whitehead manifolds cannot carry a complete metric of non-positive curvature. They can even not carry a CAT (0) distance.

Theorem (Gromov-Lawson, Chang-Weinberger-Yu) Whitehead manifolds cannot carry complete metrics of uniformly positive scalar curvature. Whitehead manifolds : known results

Remark Whitehead manifolds cannot carry a complete metric of non-positive curvature. They can even not carry a CAT (0) distance.

Theorem (Gromov-Lawson, Chang-Weinberger-Yu) Whitehead manifolds cannot carry complete metrics of uniformly positive scalar curvature.

Theorem (Gang Liu) Whitehead manifolds cannot carry complete metrics of nonnegative Ricci curvature. Higher genus Whitehead manifolds Outline ∃ locally finite simplicial tree T and v 7→ Xv ∈ X defined on vertices of T

Xv1 v1

v2 v3 Xv 3 Xv2

v4

Xv4 v5 v6

Xv Xv5 6

2 such that removing 3-balls and gluing S × I to the Xv ’s according to the edges of T

Manifolds with infinite complexity X = a class of closed 3-manifolds. A manifold M is a connected sum of members of X if and v 7→ Xv ∈ X defined on vertices of T

Xv1 v1

v2 v3 Xv 3 Xv2

v4

Xv4 v5 v6

Xv Xv5 6

2 such that removing 3-balls and gluing S × I to the Xv ’s according to the edges of T

Manifolds with infinite complexity X = a class of closed 3-manifolds. A manifold M is a connected sum of members of X if

∃ locally finite simplicial tree T and v 7→ Xv ∈ X defined on vertices of T

Xv1

Xv 3 Xv2

Xv4

Xv Xv5 6

2 such that removing 3-balls and gluing S × I to the Xv ’s according to the edges of T

Manifolds with infinite complexity X = a class of closed 3-manifolds. A manifold M is a connected sum of members of X if

∃ locally finite simplicial tree T

v1

v2 v3

v4

v5 v6 2 such that removing 3-balls and gluing S × I to the Xv ’s according to the edges of T

Manifolds with infinite complexity X = a class of closed 3-manifolds. A manifold M is a connected sum of members of X if

∃ locally finite simplicial tree T and v 7→ Xv ∈ X defined on vertices of T

Xv1 v1

v2 v3 Xv 3 Xv2

v4

Xv4 v5 v6

Xv Xv5 6 Manifolds with infinite complexity X = a class of closed 3-manifolds. A manifold M is a connected sum of members of X if

∃ locally finite simplicial tree T and v 7→ Xv ∈ X defined on vertices of T

Xv1 v1

v2 v3 Xv 3 Xv2

v4

Xv4 v5 v6

Xv Xv5 6

2 such that removing 3-balls and gluing S × I to the Xv ’s according to the edges of T Manifolds with infinite complexity X = a class of closed 3-manifolds. A manifold M is a connected sum of members of X if

∃ locally finite simplicial tree T and v 7→ Xv ∈ X defined on vertices of T

Xv1 v1

v2 v3 Xv 3 Xv2

v4 ≈ M

Xv4 v5 v6

Xv 5 Xv6

2 such that removing 3-balls and gluing S × I to the Xv ’s according to the edges of T ; M. ≈

R3 3 2 I X = {S }, T = the line ; S × R

Examples

3 I X = {S }, T = the half-line ≈

R3 3 2 I X = {S }, T = the line ; S × R

Examples

3 I X = {S }, T = the half-line ≈

R3 3 2 I X = {S }, T = the line ; S × R

Examples

3 I X = {S }, T = the half-line ≈

R3 3 2 I X = {S }, T = the line ; S × R

Examples

3 I X = {S }, T = the half-line 3 2 I X = {S }, T = the line ; S × R

Examples

3 I X = {S }, T = the half-line

≈

R3 Examples

3 I X = {S }, T = the half-line

≈

R3 3 2 I X = {S }, T = the line ; S × R Graphs versus trees Graphs versus trees Theorem (Bessi`eres-B.-Maillot) M has a complete metric of bounded geometry and Scal > 1 iff there is a finite collection F of spherical manifolds such that M is a (maybe infinite) connected sum of copies of S2 × S1 and members of F.

One result

Definition (M, g) has bounded geometry if ∃Q, ρ > 0 such that | Sectg |6 Q and injg > ρ. One result

Definition (M, g) has bounded geometry if ∃Q, ρ > 0 such that | Sectg |6 Q and injg > ρ.

Theorem (Bessi`eres-B.-Maillot) M has a complete metric of bounded geometry and Scal > 1 iff there is a finite collection F of spherical manifolds such that M is a (maybe infinite) connected sum of copies of S2 × S1 and members of F. I Uses a version of the Ricci flow with surgery for non compact manifolds.

I What if we relax the assumptions ?

Remark

I Compact case due to Perelman + Schoen-Yau or Gromov-Lawson. I What if we relax the assumptions ?

Remark

I Compact case due to Perelman + Schoen-Yau or Gromov-Lawson.

I Uses a version of the Ricci flow with surgery for non compact manifolds. Remark

I Compact case due to Perelman + Schoen-Yau or Gromov-Lawson.

I Uses a version of the Ricci flow with surgery for non compact manifolds.

I What if we relax the assumptions ? R1 = space of complete metrics with scalar curvature ≥ 1 and bounded geometry.

Question If R1 6= ∅, is R1/Diff (M) path connected ?

The compact case is a result of F. Cod´aMarques.

Space of metrics with positive scalar curvature

Let M be an orientable open 3-manifold and define Question If R1 6= ∅, is R1/Diff (M) path connected ?

The compact case is a result of F. Cod´aMarques.

Space of metrics with positive scalar curvature

Let M be an orientable open 3-manifold and define

R1 = space of complete metrics with scalar curvature ≥ 1 and bounded geometry. The compact case is a result of F. Cod´aMarques.

Space of metrics with positive scalar curvature

Let M be an orientable open 3-manifold and define

R1 = space of complete metrics with scalar curvature ≥ 1 and bounded geometry.

Question If R1 6= ∅, is R1/Diff (M) path connected ? Space of metrics with positive scalar curvature

Let M be an orientable open 3-manifold and define

R1 = space of complete metrics with scalar curvature ≥ 1 and bounded geometry.

Question If R1 6= ∅, is R1/Diff (M) path connected ?

The compact case is a result of F. Cod´aMarques. Outline Question Let E ⊂ S3 (closed set), what is the ”best” complete metric on S3 \ E?

I If E is a link ; Thurston’s theory.

I What if E is a Cantor set or a fractal ?

An alternative question

Another way to ask the question is I If E is a link ; Thurston’s theory.

I What if E is a Cantor set or a fractal ?

An alternative question

Another way to ask the question is

Question Let E ⊂ S3 (closed set), what is the ”best” complete metric on S3 \ E? I What if E is a Cantor set or a fractal ?

An alternative question

Another way to ask the question is

Question Let E ⊂ S3 (closed set), what is the ”best” complete metric on S3 \ E?

I If E is a link ; Thurston’s theory. An alternative question

Another way to ask the question is

Question Let E ⊂ S3 (closed set), what is the ”best” complete metric on S3 \ E?

I If E is a link ; Thurston’s theory.

I What if E is a Cantor set or a fractal ? Facts

I All Cantor sets are homeomorphic. 3 3 I The homeomorphism may not extend to R or S .

Cantor sets in S3

Definition A Cantor set is a compact metrizable set which is totally discontinuous and has no isolated points. I All Cantor sets are homeomorphic. 3 3 I The homeomorphism may not extend to R or S .

Cantor sets in S3

Definition A Cantor set is a compact metrizable set which is totally discontinuous and has no isolated points. Facts 3 3 I The homeomorphism may not extend to R or S .

Cantor sets in S3

Definition A Cantor set is a compact metrizable set which is totally discontinuous and has no isolated points. Facts

I All Cantor sets are homeomorphic. Cantor sets in S3

Definition A Cantor set is a compact metrizable set which is totally discontinuous and has no isolated points. Facts

I All Cantor sets are homeomorphic. 3 3 I The homeomorphism may not extend to R or S . I The binary tree of spheres is a triadic Cantor. 3 I If E has ”holes” ; S \ E is not contractible.

I It is simply connected and has no metric of non positive curvature.

Theorem (Souto-Stover) There is a cantor set E ⊂ S3 such that the complement S3 \ E admits a complete hyperbolic metric.

Cantor sets in S3

Facts 3 I If E has ”holes” ; S \ E is not contractible.

I It is simply connected and has no metric of non positive curvature.

Theorem (Souto-Stover) There is a cantor set E ⊂ S3 such that the complement S3 \ E admits a complete hyperbolic metric.

Cantor sets in S3

Facts

I The binary tree of spheres is a triadic Cantor. I It is simply connected and has no metric of non positive curvature.

Theorem (Souto-Stover) There is a cantor set E ⊂ S3 such that the complement S3 \ E admits a complete hyperbolic metric.

Cantor sets in S3

Facts

I The binary tree of spheres is a triadic Cantor. 3 I If E has ”holes” ; S \ E is not contractible. Theorem (Souto-Stover) There is a cantor set E ⊂ S3 such that the complement S3 \ E admits a complete hyperbolic metric.

Cantor sets in S3

Facts

I The binary tree of spheres is a triadic Cantor. 3 I If E has ”holes” ; S \ E is not contractible.

I It is simply connected and has no metric of non positive curvature. Cantor sets in S3

Facts

I The binary tree of spheres is a triadic Cantor. 3 I If E has ”holes” ; S \ E is not contractible.

I It is simply connected and has no metric of non positive curvature.

Theorem (Souto-Stover) There is a cantor set E ⊂ S3 such that the complement S3 \ E admits a complete hyperbolic metric. I Loewner-Nirenberg (1974) : constant negative scalar curvature =⇒ H(n−2)/2(E) = +∞.

I Schoen-Yau (1988) : non-negative scalar curvature ; other results by ... Mazzeo-Pacard, Byde, Almir Silva Santos.

I D. Labutin (’05) : constant negative scalar curvature ⇐⇒ E is not thin.

Yamabe problem

Question : Complete metric of constant scalar curvature conformal to the standard one on Sn \ E (n ≥ 3) ? I Schoen-Yau (1988) : non-negative scalar curvature ; other results by ... Mazzeo-Pacard, Byde, Almir Silva Santos.

I D. Labutin (’05) : constant negative scalar curvature ⇐⇒ E is not thin.

Yamabe problem

Question : Complete metric of constant scalar curvature conformal to the standard one on Sn \ E (n ≥ 3) ?

I Loewner-Nirenberg (1974) : constant negative scalar curvature =⇒ H(n−2)/2(E) = +∞. I D. Labutin (’05) : constant negative scalar curvature ⇐⇒ E is not thin.

Yamabe problem

Question : Complete metric of constant scalar curvature conformal to the standard one on Sn \ E (n ≥ 3) ?

I Loewner-Nirenberg (1974) : constant negative scalar curvature =⇒ H(n−2)/2(E) = +∞.

I Schoen-Yau (1988) : non-negative scalar curvature ; other results by ... Mazzeo-Pacard, Byde, Almir Silva Santos. Yamabe problem

Question : Complete metric of constant scalar curvature conformal to the standard one on Sn \ E (n ≥ 3) ?

I Loewner-Nirenberg (1974) : constant negative scalar curvature =⇒ H(n−2)/2(E) = +∞.

I Schoen-Yau (1988) : non-negative scalar curvature ; other results by ... Mazzeo-Pacard, Byde, Almir Silva Santos.

I D. Labutin (’05) : constant negative scalar curvature ⇐⇒ E is not thin. Z 1/2 C(B(p, r) ∩ E)(2/n−2) dr ∀p ∈ E, = +∞. 0 C(B(p, r)) r

I C is a Bessel capacity.

I C(E1) ≥ C(E2) if E1 ⊃ E2. I Criterion satisfied if E submanifold of dim > (n − 2)/2.

I Satisfied for E = Whitehead continuum.

Labutin’s criterion

E ⊂ Sn is not thin means that I C is a Bessel capacity.

I C(E1) ≥ C(E2) if E1 ⊃ E2. I Criterion satisfied if E submanifold of dim > (n − 2)/2.

I Satisfied for E = Whitehead continuum.

Labutin’s criterion

E ⊂ Sn is not thin means that

Z 1/2 C(B(p, r) ∩ E)(2/n−2) dr ∀p ∈ E, = +∞. 0 C(B(p, r)) r I C(E1) ≥ C(E2) if E1 ⊃ E2. I Criterion satisfied if E submanifold of dim > (n − 2)/2.

I Satisfied for E = Whitehead continuum.

Labutin’s criterion

E ⊂ Sn is not thin means that

Z 1/2 C(B(p, r) ∩ E)(2/n−2) dr ∀p ∈ E, = +∞. 0 C(B(p, r)) r

I C is a Bessel capacity. I Criterion satisfied if E submanifold of dim > (n − 2)/2.

I Satisfied for E = Whitehead continuum.

Labutin’s criterion

E ⊂ Sn is not thin means that

Z 1/2 C(B(p, r) ∩ E)(2/n−2) dr ∀p ∈ E, = +∞. 0 C(B(p, r)) r

I C is a Bessel capacity.

I C(E1) ≥ C(E2) if E1 ⊃ E2. I Satisfied for E = Whitehead continuum.

Labutin’s criterion

E ⊂ Sn is not thin means that

Z 1/2 C(B(p, r) ∩ E)(2/n−2) dr ∀p ∈ E, = +∞. 0 C(B(p, r)) r

I C is a Bessel capacity.

I C(E1) ≥ C(E2) if E1 ⊃ E2. I Criterion satisfied if E submanifold of dim > (n − 2)/2. Labutin’s criterion

E ⊂ Sn is not thin means that

Z 1/2 C(B(p, r) ∩ E)(2/n−2) dr ∀p ∈ E, = +∞. 0 C(B(p, r)) r

I C is a Bessel capacity.

I C(E1) ≥ C(E2) if E1 ⊃ E2. I Criterion satisfied if E submanifold of dim > (n − 2)/2.

I Satisfied for E = Whitehead continuum. I What does that mean ?

I Start with n = 2.

Conical singularities

Question : Constant scalar curvature metrics with conical singularities on E ? I Start with n = 2.

Conical singularities

Question : Constant scalar curvature metrics with conical singularities on E ?

I What does that mean ? Conical singularities

Question : Constant scalar curvature metrics with conical singularities on E ?

I What does that mean ?

I Start with n = 2. Antoine’s necklace Antoine’s necklace birth Antoine’s necklace birth Antoine’s necklace birth