IsometricSubmersions of

Teich Spcs

Mark Greenfield

Teichmuller theory

Isometricsubmersions

Main theorem andproof ideas

Conclusion

Isometric Submersions of Teichmuller Spaces

Mark GreenfieldUniversity of Michigan

Joint with Dmitri Gekhtman

April 29, 2020(Virtual) Hyperbolic Lunch at University of Toronto

IsometricSubmersions of

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Mark Greenfield

Teichmuller theory

Isometricsubmersions

Main theorem andproof ideas

Conclusion

Outline

1. Quick review of necessary facts on Teichmuller theory

2. Definition and example of Isometric submersion

3. Main theorem and overview of proof method

IsometricSubmersions of

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Conclusion

Teichmuller spaces

Denote Sg ,n genus g surface with n puncturesTeichmuller space of Sg ,n:

Tg ,n = {(X , h) | h : Sg ,n → X homeo, X Riemann surface}/ ∼

where (X , h) ∼ (Y , j) if

j ◦ h−1 : X → Y

is isotopic (same marking) to a biholomorphism (sameRiemann surface).

Teichmuller metric: dTeich

((X , h), (Y , j)

)= inff∼j◦h−1 K (f )

Smallest quasiconformal distortion respecting the marking.

IsometricSubmersions of

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Some facts and notation

1. Mod(S) = Homeo+(S)/Homeo0(S) mapping class(“change-of-marking”) group

2. Tg ,n is a complex manifold with dimC Tg ,n = 3g − 3 + n

3. Teichmuller metric intrinsic to C-structure:dTeich = dKob

4. We often write just X instead of (X , h) (locally okay).

5. Write X := X ∪ {filled-in punctures} for X punctured

IsometricSubmersions of

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Conclusion

Local structure of Teichmuller space

I Holomorphic quadratic differential q on a Riemannsurface X is locally q(z)dz2 with q(z) holomorphic

I q is integrable if the 1-norm is finite:

||q|| :=

∫X|q| <∞

I Define Q(X ) = {integrable holomorphic q.d.’s on X};dimCQ(X ) = 3g − 3 + n.

I For X punctured:Q(X ) = Q(X ) ∪ {q.d.’s w/ simple poles at punctures}

I Fact: at (X , h) ∈ Tg ,n:

T ∗(X ,h)Tg ,n = Q(X )

IsometricSubmersions of

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Conclusion

Using the space Q(X )

Let φ0, . . . , φl be a basis of C-VS Q(X ). Thenfj := φj/φ0 : X → C is a meromorphic function. The map

F := (f1, . . . , fl) : X → Cl

is also meromorphic. Fact: if 2g + n ≥ 5, then z 7→ [1,F (z)]defines a holomorphic embedding Φ : X → Pl . This is calledthe bi-canonical embedding.

RemarkThe 2g + n condition comes from Riemann-Roch and a littlebit of algebra, which we will not discuss here.

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Conclusion

Maps between Teichmuller spaces

Theme: given F : Tg ,n → Tk,m preserving analytic/geometricstructure, show F is induced by some map f : Sg ,n ↔ Sk,mpreserving topological structure.

Example of f inducing F : if ϕ ∈ Mod(S) then(X , h) 7→ (X , h ◦ ϕ−1) is a map Tg ,n → Tg ,n

Beautiful central idea: geometry (metric, C-structure, etc.)of Tg ,n reflects topology of Sg ,n

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Maps between Teichmuller spaces: past results

Theorem (Royden, ’71)

If F : Tg → Tg is a biholomorphism then F is induced bysome ϕ ∈ Mod(Sg )

Generalized several times (Earle, Kra, Gardiner, Lakic ’70s -’90s), finally by Markovic (2003) to include all ∞-typesurfaces.

IsometricSubmersions of

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Maps between Teichmuller spaces: recentquestions

What about different Teichmuller spaces?

Conjecture (Antonakoudis)

Every isometric embedding Tg ,n ↪→ Tk,m is the pull-backfrom a covering map Sk,m → Sg ,n.

I If f : Sk,m → Sg ,n is a cover, then h : Sg ,n → X lifts toh′ : Sk,m → f ∗X giving a slice of Tk,m

What about the dual question?

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Conclusion

Finsler manifolds

DefinitionA Finsler manifold is a manifold M together with acontinuous norm on the tangent bundle TM.

I More general than Riemannian (norm vs. inner product)

I Allows distances but not angles

I Easy examples: L1 and L∞ on Rn

I Hard example: Teichmuller space with dTeich

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Teichmuller space as a Finsler manifold

Some classical facts:

The infinitesimal dTeich gives Tg ,n a Finsler structure.

There is a dual pairing between the holomorphic tangentspace and the space of quadratic differentials for eachX ∈ Tg ,n:

TXTg ,n ↔ Q∗(X )

The dual norm for the (Finsler) infinitesimal Teichmullermetric is the 1-norm on the space of quadratic differentials:

supv∈TXTg,n

||φ · v ||T||v ||T

= ||φ|| =

∫X|φ| for φ ∈ Q(X )

IsometricSubmersions of

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Isometric submersions between Finsler manifolds

DefinitionAn isometric submersion F : M → N isa C 1 submersion such that for all x ∈ M,

dFx(unit ball in TxM) = unit ball in TF (x)N

“Metric form” of projection

Equivalent: the co-derivative

dF ∗x : T ∗F (x)N → T ∗xM

is an isometric embedding with respect todual Finsler norms.

Figure: Infinitesimalisometric submersion

IsometricSubmersions of

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Teichmuller theory

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Conclusion

Key example of isometric submersion: theforgetful map

F : Tg ,n → Tg ,m, where m < n

Coderivative is inclusion:

dF ∗X : QD(X ) ↪→ QD(X − {p}){q.d.’s holo on X

}↪→{

q.d.’s holo on X − {p}}

Figure: Forgetful map for T2,3 → T2,1.

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Our question

What are the possible holomorphic isometricsubmersions between Teichmuller spaces?

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The result

Theorem ((Dmitri) Gekhtman - G, ’19)

Let F : Tg ,n → Tk,m be a holomorphic isometric submersionwith k ≥ 1 and 2k + m > 4.Then g = k , n ≥ m, and F is a forgetful map.

Remark

1. We do not assume same genus.

2. Conjecture: For T2,0, T1,2, the only exceptions involveT2,0∼= T0,6 and T1,2

∼= T0,5.

IsometricSubmersions of

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Conclusion

The three main steps in the proof

Given F : Tg ,n → Tk,m, pick X ∈ Tg ,n, thendF ∗F (X ) : QD(F (X ))→ QD(X ) is an isometric embedding.

1. Use methods of Markovic and Earle-Markovic (2003) tofind a holomorphic map h : X → F (X ) inducing dF ∗F (X )

2. Riemann-Hurwitz and dim’n count ⇒ h is forgetful

3. Show h varies continuously across Tg ,n, so the sameforgetful map works everywhere

Topological answer, to a geometric question... via analyticmethods!

IsometricSubmersions of

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Teichmuller theory

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Conclusion

Finding the map h : X → F (X ): adaptingMarkovic/Earle-Markovic

Pick X ∈ Tg ,n and write T := dF ∗X : T ∗F (X )Tk,m → T ∗XTg ,nLet φ0, . . . , φl be a basis of Q(F (X )), write ψi = Tφi(ψi ’s are linearly independent in Q(X )).Define the bicanonical map Φ and a related map Ψ:

Φ = [1, φ1/φ0, . . . , φl/φ0] and Ψ = [1, ψ1/ψ0, . . . , ψl/ψ0].

Then we have F (X )Φ↪−→ Pl Ψ←− X

LemmaFortunately, Φ

(F (X )

)= Ψ(X )!

IsometricSubmersions of

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Conclusion

Finding the map h: analytic tools

For F (X )Φ↪−→ Pl Ψ←− X , why is Φ(F (X )) = Ψ(X )?

Define measures µ(K ) =∫K |φ0| and ν(L) =

∫L |ψ0| for

Borel K ⊆ F (X ) and L ⊆ X .

Then for any (λ1, . . . , λl) ∈ Cl :∫F (X )

∣∣∣∣1 +λ1f1 + · · ·+λl fl

∣∣∣∣dµ =

∫X

∣∣∣∣1 +λ1g1 + · · ·+λlgl

∣∣∣∣dνby ch. of vars. and linearity (fj = φj/φ0, gj = ψj/ψ0).

By an auspicious theorem of W. Rudin (1976), under theseconditions, Φ∗(µ) = Ψ∗(ν). The above claim follows.

Define h : X → F (X ) by h = Φ−1 ◦Ψ. It is holomorphic andnonconstant!

IsometricSubmersions of

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Showing h is forgetful: some classical ideas

So far: Given any X in the “big” Teich space Tg ,n, obtain

holomorphic h : X → F (X ).I This is a nonconstant holo map

between closed Rmn. sfc., thuscovering map!

I χ(X ) = 2− 2g , dim Tg = 3g − 3,

χ(F (X )) = 2− 2k , dim Tk = 3k − 3

I By Riemann-Hurwitz:2− 2g = d(2− 2k) + b

I Dim’n of space of X ∈ Tg which cancover sfc in Tk : dim Tk + b

Contradiction unless d = 1, b = 0. So the surfaces are thesame genus g = k , but one has fewer of punctures, m ≤ n.Thus h is forgetful (up to mapping classes).

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Local to global: very quick overview

Must ensure h has same topological type for all X ∈ Tg ,n.

I Cg ,n = universal curve (all surfaces in Tg ,n together)

I Package together all h’s for H : Cg ,n → Cg ,mI F∗ = P(dF )

I Since H = Φ−1 ◦ F∗ ◦Ψ, H must be holo (⇒ cts.)

Conclude that h has same topo. type (i.e. induced by samef : Sg ,n → Sg ,m) for all X ∈ Tg ,n.

IsometricSubmersions of

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Conclusion

Completing the proof

There must exist a single f : Sg ,n → Sg ,m such that:

commutes up to isotopy for all Y ∈ Tg ,n. So, f must be aninclusion map, filling in n −m punctures, up to Mod(Sg ,n).

The map F : Tg ,n → Tk,m is induced by f : Sg ,n → Sg ,m, andg = k .

IsometricSubmersions of

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Conclusion

Some open questions

I The remaining cases of T1,2∼= T0,5 and T2,0

∼= T0,6 withsurfaces of exceptional type

I Holomorphic isometric submersions between ∞-typeTeichmuller spaces

I Can we get rid of “holomorphic” or “isometric” fromthe conditions “holomorphic isometric submersion”?

I Not both, since a topological submersion would onlyremember that Tg ,n ∼= R6g−6+2n

I Not “submersion” since then we are back to Royden’stheorem

I More generally, other kinds of rigidity, or isometricsubmersions/embeddings with other metrics onTeichmuller space

IsometricSubmersions of

Teich Spcs

Mark Greenfield

Teichmuller theory

Isometricsubmersions

Main theorem andproof ideas

Conclusion

Thank you!

Main References

I C. Earle and V. Markovic. Isometries between the spaces of L1

holomorphic quadratic differentials on Riemann surfaces of finite type.Duke Math J., 120(2):433-440, 2003.

I Gekhtman, D., Greenfield, M. Isometric submersions of Teichmullerspaces are forgetful. To appear in Israel Journal of Mathematics.Available at arXiv:1901.02586. See additional citations therein.

I V. Markovic. Biholomorphic maps between Teichmuller spaces. DukeMath J., 120(2):405-431, 2003.