Constant curvature conical metricsxuwenzhu/materials/Conic2_slides.pdfLocally near a cone point with...
Transcript of Constant curvature conical metricsxuwenzhu/materials/Conic2_slides.pdfLocally near a cone point with...
Constant curvature conical metrics
Xuwen Zhu (UC Berkeley)
Joint with Rafe Mazzeo and Bin Xu
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Outline
1 Constant curvature conical metrics
2 Spherical metrics with large cone angles
3 Compactification of configuration family
4 Generalized deformation rigidity
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Constant curvature metric with conical singularities
Consider a compact Riemann surface M, with the following data:k distinct points p = (p1, . . . ,pk )
Angle data ~β = (β1, . . . , βk ) ∈ (0,∞)k
Curvature constant K ∈ {−1,0,1}Area AConformal structure c given by M
A constant curvature metric with prescribed conical singularities is asmooth metric with constant curvature, except near pj the metric isasymptotic to a cone with angle 2πβj .
(Gauss–Bonnet) χ(M, ~β) := χ(M) +k∑
j=1
(βj − 1) =1
2πKA
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Local structure near a cone point
Locally near a cone point with angle 2πβ, there are coordinates(geodesic polar coordinates) such that the metric is given by
g =
dr2 + β2r2dθ2 K = 0;
dr2 + β2 sin2 rdθ2 K = 1;
dr2 + β2 sinh2 rdθ2 K = −1
In the flat case, relating to the conformal structure
r =1β|z|β, then g = |z|2(β−1)(d |z|2 + |z|2dθ2) = |z|2(β−1)|dz|2
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Some examples of constant curvature conical metrics
Translationsurfaces
4π
4π
Branched coversof constantcurvaturesurfaces
β
β
Sphericalfootballs
αβ
4π
α + β
“Heart”: footballsglued alonggeodesics
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Some examples of constant curvature conical metrics
Translationsurfaces
4π
4π
Branched coversof constantcurvaturesurfaces
β
β
Sphericalfootballs
αβ
4π
α + β
“Heart”: footballsglued alonggeodesics
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Some examples of constant curvature conical metrics
Translationsurfaces
4π
4π
Branched coversof constantcurvaturesurfaces
β
β
Sphericalfootballs
αβ
4π
α + β
“Heart”: footballsglued alonggeodesics
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Some examples of constant curvature conical metrics
Translationsurfaces
4π
4π
Branched coversof constantcurvaturesurfaces
β
β
Sphericalfootballs
αβ
4π
α + β
“Heart”: footballsglued alonggeodesics
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An object with many aspectsThe study of constant curvature conical metrics is related to a lot ofproblems:
Magnetic vortices: solitons of gauged sigma-models on aRiemann surfaceMean Field EquationsToda system: multi-dimensional versionHigher dimensional analogue: Kähler-Einstein metrics with conicalsingularitiesBridge between the (pointed) Riemann moduli spaces: cone anglefrom 0 to 2π
This object can be studied in many approaches:PDE: singular Liouville equationComplex analysis: developing maps and Schwarzian derivativesSynthetic geometry: cut-and-glue method
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The simple cases
Theorem (’88 McOwen, ’91 Troyanov, ’92 Luo–Tian)
For any compact surface M and conical data (p, ~β) withχ(M, ~β) ≤ 0; or
χ(M, ~β) > 0, ~β ∈ (0,1)k
I k = 2, β1 = β2; orI k ≥ 3, βj + k − χ(M) >
∑i 6=j βi ,∀j ,
there is a unique constant curvature metric with the prescribedsingularities.
Theorem (’15 Mazzeo–Weiss)There is a well-defined (6γ − 6 + 3k)-dimensional manifold which isthe moduli space of the constant curvature metrics when all coneangles are less than 2π.
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Positive curvature with big cone angles
There has been a lot of development:Calculus of variations and min-max method:
I ’91 TroyanovI ’02 Bartolucci–TarantelloI ’11-’19 Bartolucci, Carlotto, De Marchis, Malchiodi, et. al.
Integrable systems:I ’02-’19 Chen, Ku, Lin, Wang, et. al.
Complex analysis:I ’00 Umehara–Yamada, ’00 EremenkoI ’01-’19 Eremenko, Gabrielov, Tarasov, et. al.I ’14 Chen–Wang–Wu–Xu
Synthetic geometry:I ’16-’17, Mondello–PanovI ’17 Dey
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A very bad moduli space structure
Existence?
For a given angle data ~β and conformal class (c, p), does there exist aspherical conical metric?
Answer: Not always.
[’16 Mondello–Panov] There is a “holonomy condition” for ~β whenM = S2.[’17 Chen–Lin] The existence of a metric with one 4π singularityon a flat tori depends on the conformal class c.[’18 Chen–Kuo–Lin–Wang] There is a nonempty open interior inthe space of impossible p for ~β = (π, π, π,3π) on the sphere.
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A very bad moduli space structure
Uniqueness?
For given admissible (~β, p, c), is the metric unique?
Answer: Usually no.
[’14 Chen–Wang–Wu–Xu, ’17 Eremenko] There is a (noncompact)one-parameter family of reducible metrics associated to the samedata (~β, p, c).[’11 Bartolucci–De Marchis–Malchiodi] For a generic large coneangles ~β on a surface with positive genus, there exist multiplesolutions.
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A very bad moduli space
Connected?Is the solution space connected?
Answer: No.
[’17 Mondello–Panov] On S2 there exists ~β such that the space ofadmissible p has an arbitrarily large number of connectedcomponents.
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Obstruction in deformationDeformation?For a given spherical conical metric g, is there a smooth neighborhoodin the moduli space parametrized by (c, ~β, p)?
Answer: Not always.
Theorem (Z, ’19)
Fixing angles (α, β,4π, α + β) on S2, the only local deformation for the“heart” is a one-parameter family which belongs to the same markedconformal class.
A = αB = β
C = 4π
D = α + β
A B
C2 C3
C1 C4
D1 D2
α β
β1 β2(a) (b)
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Obstruction in deformationDeformation?For a given spherical conical metric g, is there a smooth neighborhoodin the moduli space parametrized by (c, ~β, p)?
Answer: Not always.
Theorem (Z, ’19)
Fixing angles (α, β,4π, α + β) on S2, the only local deformation for the“heart” is a one-parameter family which belongs to the same markedconformal class.
A = αB = β
C = 4π
D = α + β
A B
C2 C3
C1 C4
D1 D2
α β
β1 β2(a) (b)
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The obstruction in analysisThe analytic problem of solving the singular Liouville equation
∆g0u − Ke2u + Kg0 = 0
has obstructions!The linearized operator is given by ∆g − 2K for a constantcurvature metric gThe Friedrichs extension of the Laplacian ∆g is self-adjoint andhas discrete spectrumWhen K = −1, ∆g − 2K is invertible; K = 0, the only kernel is theconstantWhen K = 1 and ~β ∈ (0,1)k : the first nonzero eigenvalue of ∆gsatisfies λ1 ≥ 2KWhen at least one βi > 1: the argument would not work any moreEigenfunctions become too singular when cone angle increases,so the Lichnerowicz type argument would not work
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Eigenvalue 2: obstruction to operator invertibility
Intuition: when angles increase, eigenvalues of the LaplaciandecreaseExample: the heart, and any numbers of footballs glued alongmeridians
β1
β1
β2
β2
4π
4π
Strata with eigenvalue 2 appear in the interior, and extend toinfinity.
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Spectrum and Monodromy
The metric g can be seen as a pull back g = f ∗gS2
Developing map f : M → S2 is a multivalued functionThe monodromy of f is contained in PSU(2)
When the monodromy is U(1), the conical metric is calledreducible.
Theorem (Xu–Z, in progress)If a spherical conical metric g has reducible monodromy, then2 ∈ spec (∆g).
And there are a lot of reducible metrics [’17 Eremenko].
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Deligne–Mumford compactification of moduli space
To understand the analytic problem, there is an analogue in theRiemann moduli space theory:
1
1
0
x1 x4 x2x3
x2 x4 x1x3
x3 x4 x2x1
The moduli space of spheres withfour puncturesM0,4 is identified withCP1 \ {0,1,∞}.The Deligne–Mumford–KnudsencompactificationM0,4 is given byadding three points, eachrepresenting a singular fiber of anodal surface.The “bubble” of the nodal surface isproduced when two marked pointscollide.
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Degenerating fiber metrics on the compactification
t ∈ D0
The “bubbles” are surfaces with cuspsCan be viewed as a Lefschetzfibration where fibers degenerate intosurfaces with cusps
Theorem (Melrose–Z, ’16, ’17)The fiber metrics and Weil–Petersson metrics are polyhomogeneouson a resolved space with new coordinates.
Polyhomogeneous: complete expansion with possibly non-integerexponent and logarithmic powers under the new coordinates
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Degenerating fiber metrics on the compactification
t ∈ D0
The “bubbles” are surfaces with cuspsCan be viewed as a Lefschetzfibration where fibers degenerate intosurfaces with cusps
Theorem (Melrose–Z, ’16, ’17)The fiber metrics and Weil–Petersson metrics are polyhomogeneouson a resolved space with new coordinates.
We developed an C∞ analogue for conical metrics:
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When two points collide
Scale back the distance between two cone points (“blow up”)
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When two points collideScale back the distance between two cone points (“blow up”)Half sphere at the collision point, with two cone points over thehalf sphere:
Flat metric on the half sphere, and curvature K metric on theoriginal surface
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Iterative structure
When there are several levels of distance: scale iteratively
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Iterative structure“bubble over bubble” structureHigher codimensional faces from deeper scalingFlat conical metrics on all the new faces
C123
C12
p1
p2
p3
Iterative singular structures: Albin–Leichtnam–Mazzeo–Piazza,Albin–Gell-Redman, Degeratu–Mazzeo, Kottke–Singer, etc.
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Resolution of the configuration space
This “bubbling” process can be expressed in terms of blow-up ofproduct Mk ×M → Mk
E2
C2
π2
p1
p2
(p1; p2)
p2
p1
Figure: “Centered” projection of C2 → E2
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Results about fiber metrics
Theorem (Mazzeo–Z, ’17)
For any* given ~β, the family of constant curvature metrics with conicalsingularities is polyhomogeneous on this resolved space.
*The metric family can be hyperbolic / flat (with any cone angles),or spherical (with angles less than 2π, except footballs)Solving the curvature equation uniformly
∆g0u − Ke2u + Kg0 = 0
The bubbles with flat conical metrics represent the asymptoticproperties of merging cones
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Spherical metrics with large cone angles
To resolve obstruction: allowing cone points to splitThe splitting creates extra dimensions, which fills up the cokernelof the linearized operator ∆g − 2The direction of splitting is determined by the expansion of theeigenfunctionsAn eigenfunction gives a 2K -tuple ~bThe tangent of splitting directions are given by vectors ~A that areorthogonal to all such ~bThe bigger dimension of eigenspace, the more constraint on thedirection of splittingHow to get the splitting direction from ~A: “almost” factorizingpolynomial equations
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A tricotomy theorem
Theorem (Mazzeo–Z, ’19)
Let (M,g0) be a spherical conic metric. Denote K =∑k
j=1 max{[βj ],1}.Let ` be the multiplicity of the eigenspace of ∆g0 with eigenvalue 2.There are three cases:
1 (Local freeness) If 2 /∈ spec (∆g0), then g0 has a smoothneighborhood parametrized by conformal structure, cone positionsand angles.
2 (Partial rigidity) If 1 ≤ ` < 2K , then for any nearby admissibleangles and conformal structures, there exists a 2K − ` dimensionalp-submanifold X that parametrizes nearby cone metrics.
3 (Complete rigidity) If ` = 2K , then there is no nearby sphericalcone metric obtained by moving or splitting the conic points of g0.
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A tricotomy theorem
Theorem (Mazzeo–Z, ’19)
Let (M,g0) be a spherical conic metric. Denote K =∑k
j=1 max{[βj ],1}.Let ` be the multiplicity of the eigenspace of ∆g0 with eigenvalue 2.There are three cases:
1 (Local freeness) If 2 /∈ spec (∆g0), then g0 has a smoothneighborhood parametrized by conformal structure, cone positionsand angles.
2 (Partial rigidity) If 1 ≤ ` < 2K , then for any nearby admissibleangles and conformal structures, there exists a 2K − ` dimensionalp-submanifold X that parametrizes nearby cone metrics.
3 (Complete rigidity) If ` = 2K , then there is no nearby sphericalcone metric obtained by moving or splitting the conic points of g0.
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A tricotomy theorem
Theorem (Mazzeo–Z, ’19)
Let (M,g0) be a spherical conic metric. Denote K =∑k
j=1 max{[βj ],1}.Let ` be the multiplicity of the eigenspace of ∆g0 with eigenvalue 2.There are three cases:
1 (Local freeness) If 2 /∈ spec (∆g0), then g0 has a smoothneighborhood parametrized by conformal structure, cone positionsand angles.
2 (Partial rigidity) If 1 ≤ ` < 2K , then for any nearby admissibleangles and conformal structures, there exists a 2K − ` dimensionalp-submanifold X that parametrizes nearby cone metrics.
3 (Complete rigidity) If ` = 2K , then there is no nearby sphericalcone metric obtained by moving or splitting the conic points of g0.
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A tricotomy theorem
Theorem (Mazzeo–Z, ’19)
Let (M,g0) be a spherical conic metric. Denote K =∑k
j=1 max{[βj ],1}.Let ` be the multiplicity of the eigenspace of ∆g0 with eigenvalue 2.There are three cases:
1 (Local freeness) If 2 /∈ spec (∆g0), then g0 has a smoothneighborhood parametrized by conformal structure, cone positionsand angles.
2 (Partial rigidity) If 1 ≤ ` < 2K , then for any nearby admissibleangles and conformal structures, there exists a 2K − ` dimensionalp-submanifold X that parametrizes nearby cone metrics.
3 (Complete rigidity) If ` = 2K , then there is no nearby sphericalcone metric obtained by moving or splitting the conic points of g0.
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An example: open-heart surgery
We obtain a deformation rigidity for the “heart”The cone point with angle 4π is split into two separate pointsIn the equal splitting case: 4π → (3π,3π)
Only one direction of admissible splitting:
αβ
3π 3π
α + β
Yes
αβ
4π
α + β
αβ
3π
3π
α + β
No
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Thank you for your attention!
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