Constant curvature conical metricsxuwenzhu/materials/Conic2_slides.pdfLocally near a cone point with angle 2ˇ , there are coordinates (geodesic polar coordinates) such that the metric

Constant curvature conical metrics

Xuwen Zhu (UC Berkeley)

Joint with Rafe Mazzeo and Bin Xu

Xuwen Zhu (UC Berkeley) Constant curvature conical metrics 1 / 28

Outline

1 Constant curvature conical metrics

2 Spherical metrics with large cone angles

3 Compactification of configuration family

4 Generalized deformation rigidity


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


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


Some examples of constant curvature conical metrics

Translationsurfaces

4π

4π

Branched coversof constantcurvaturesurfaces

β

β

Sphericalfootballs

αβ

4π

α + β

“Heart”: footballsglued alonggeodesics



Translationsurfaces

4π

4π


β

β

Sphericalfootballs

αβ

4π

α + β




Translationsurfaces

4π

4π


β

β

Sphericalfootballs

αβ

4π

α + β




Translationsurfaces

4π

4π


β

β

Sphericalfootballs

αβ

4π

α + β



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


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.


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


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.


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.


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.


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)


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)


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


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.


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].


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.


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


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:


When two points collide

Scale back the distance between two cone points (“blow up”)


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


Iterative structure

When there are several levels of distance: scale iteratively


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.


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


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


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


A tricotomy theorem


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.


A tricotomy theorem








A tricotomy theorem








A tricotomy theorem








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


Thank you for your attention!


Documents

Constant curvature conical metricsxuwenzhu/materials/Conic2_slides.pdfLocally near a cone point with angle 2ˇ , there are coordinates (geodesic polar coordinates) such that the metric