Conservation laws for CMC surfaces
Daniel Fox & Joe S. Wang
jswang12@gmail.com
Estes Park, July, 2013
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
1 / 23
Symmetries and conservation laws
Symmetries and conservation laws
1
Symmetry - predictability, stability, applicability
- new solutions from the old ones.
Conservation law - cohomological representation of symmetry.
2
Three types of symmetry for CMC surfaces in a three dimensional
space form
- classical symmetry
- higher order symmetry
- spectral symmetry.
3
Historically, conservation law was introduced as a fundamental
principle of natural science.
• Mayer - mechanics of heat (thermodynamics), heat = work.
• Helmhöltz - mechanical and chemical foundation of physiology,
unifying principle underlying nature (heat, light, electricity,
magnetism).
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
2 / 23
Symmetries and conservation laws
Noether’s theorem
3
Noether’s theorem in calculus of variation (working hypothesis)
symmetry 99K conservation law.
”symmetry” - in a narrower sense, certain algebra of vector fields on
the associated infinite jet space.
4
Burgers’ equation
ut = uux + uxx
- (finitely generated) infinite dimensional algebra of symmetry vector
fields involving higher and higher order derivatives
- bound on the jet order of a conservation law (to be defined below)
for an even order scalar evolution equation.
• Symmetry vector fields and conservation laws - innate symmetry of
a differential equation, may behave differently.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
3 / 23
Differential equation for CMC surfaces
Elliptic Monge-Ampere system
M: three dimensional space form of constant curvature ,
X = Gr+ (2, TM): Grassmann bundle of oriented 2-planes.
Definition
Differential equation for CMC-δ surfaces is an exterior differential system
I defined on X such that an integral surface projects to a possibly singular
CMC-δ surface in M. It is an elliptic Monge-Ampere system.
Problem. Determine the space of global conservation laws for CMC
surface system (with a view to application to the global geometry of CMC
surfaces).
Remark.
Assume γ 2 = + δ 2 6= 0.
Depending on the sign of γ 2 , a CMC surface is locally described by
elliptic sinh, or cosh-Gordon equation.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
4 / 23
Differential equation for CMC surfaces
Bonnet theorem
1
The geometric data for a CMC surface consist of the triple

 Riemann surface Σ
conformal metric g
(?)

Hopf differential II ∈ H 0 (Σ, K 2 )
that satisfy the compatibility equation
Rg = γ 2 − |II|2g .
2
Let π : Σ̃ → Σ be the universal cover. (?) induces a CMC-δ
immersion
x̃ : Σ̃ ,→ M
which realizes (π ∗ g , π ∗ II). x̃ is unique up to motion by isometries of
the ambient space M.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
5 / 23
Classical conservation laws
Classical symmetries
1
Let g be the six dimensional algebra of Killing vector fields on M.
For Y ∈ g,
∇ · Y = 0 −→ Y = ∇ × Z Y .
2
Let Σ ,→ M be a CMC-δ surface.
∇Σ · Y T = 2δhe3 , Y i, e3 unit normal to Σ.
3
Let Γ ,→ Σ be an oriented curve, representing [Γ] ∈ H1 (Σ, R).
e1 tangent to Γ, e2 = e3 × e1 along Γ.
Let D ,→ M be a surface in M with boundary ∂D = Γ.
n unit normal to D.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
6 / 23
Classical conservation laws
Momentum class
Definition (Kusner)
Momentum class µ ∈ H 1 (Σ, R) ⊗ g∗ is defined by
Z
Z
µ([Γ], Y ) = he2 , Y i + 2δ hn, Y i
ZΓ
ZD
= he2 , Y i + 2δ he1 , Z Y i
Γ
ZΓ
Y
Y
= ϕ .
(dϕ = 0 on Σ)
Γ
Example
For a curve Γ ,→ M with a prescribed tangent 2-plane, the momentum
class gives an obstruction for Γ to bound a CMC-δ surface Σ that satisfies
the given overdetermined boundary condition and for which the boundary
class [Γ] ∈ H1 (Σ, R) is trivial.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
7 / 23
Classical conservation laws
Square root of Hopf differential
1
Let II ∈ H 0 (Σ, K 2 ) be the Hopf differential (holomorphic second
fundamental form).
√
Consider ω = II. It is a holomorphic 1-form and hence
dω = 0.
• But, [ω] is not a classical conservation law.
2
It is necessary to take into consideration the higher order derivatives.
Remark. Noether’s theorem for classical conservation laws.
g ' {classical conservation laws}.
The momentum class accounts for all the conservation laws defined on X .
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
8 / 23
Higher order conservation laws
Prolongation
Recall (X , I) = (X (0) , I(0) ) is an elliptic Monge-Ampere system.
Definition
The infinite prolongation is defined as the limit
(X (∞) , I(∞) ) = lim (X (k) , I(k) ).
The sequence of Pfaffian systems satisfies the inductive closure condition
dI(k) ≡ 0
mod I(k+1) , k ≥ 1.
The ideal I(∞) is Frobenius, and (Ω∗ (X (∞) )/I(∞) , d) is a complex.
Definition
A conservation law is an element in the characteristic cohomology
C = H 1 (Ω∗ (X (∞) )/I(∞) , d).
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
9 / 23
Higher order conservation laws
Spectral sequence
Due to Vinogradov, Tsujishita, Bryant & Griffiths, ...
1
Consider the filtration
F p Ω∗ (X (∞) ) = Image{I(∞) ∧ I(∞) ... ∧ : Ω∗ (X (∞) ) → Ω∗ (X (∞) )}.
From the associated graded F p Ω∗ (X (∞) )/F p+1 Ω∗ (X (∞) ), a standard
construction yields the spectral sequence
(Erp,q , dr ),
2
dr has bidegree (r , 1 − r ).
General fact: the following sub-complex is exact.
0 → E10,1 ,→ E11,1 → E12,1 .
• E10,1 = C: space of conservation laws.
C = {ϕ ∈ Ω1 (X (∞) )|dϕ ≡ 0 mod I(∞) }/{dΩ0 (X (∞) ) + Ω1 (I(∞) )}
= H 1 (Ω∗ (X (∞) )/I(∞) , d).
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
10 / 23
Higher order conservation laws
Jacobi fields
• E11,1 : the space of cosymmetries (symbols, or generating functions)
- the algebra of higher order symmetry vector fields (derivations) of
I(∞) (self-adjoint linearization).
- a symmetry vector field is uniquely generated by a Jacobi field. A
Jacobi field is a scalar function in the kernel of the Jacobi operator
E := ∆ − 2(γ 2 + |II|2g ).
• The derivative d1 : E10,1 ,→ E11,1 maps a conservation law to its
generating Jacobi field.
Example
Higher order Noether’s theorem claims d1 : E10,1 ,→ E11,1 is surjective.
C ' {Jacobi fields}.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
11 / 23
Higher order conservation laws
Computation
1
From the exact sequence
0 → I(∞) → Ω∗ (X (∞) ) → Ω∗ (X (∞) )/I(∞) → 0,
locally
C ' H 2 (I(∞) , d) ' ker{d1 : E11,1 → E12,1 }.
H 2 (I(∞) , d) is easier to analyze than C.
• An analysis shows that there exist at most two conservation laws
at each odd order (weight).
2
Apply a differential algebraic recursion to generate the desired infinite
sequence of conservation laws.
The recursion formula is originally due to Pinkall & Sterling,
Dorfmann, ..., in slightly different forms.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
12 / 23
Higher order conservation laws
Enhanced prolongation
1
Direct prolongation (X (∞) , I(∞) ) is a general, mechanical process.
• On the basis of (X (∞) , I(∞) ), build an enhanced prolongation which
incorporates the geometric properties of the differential system.
2
Enhanced prolongation is derived from the loop Lie algebra valued
formal Killing field equation
• dX(λ) + [ψ(λ), X(λ)] = 0.
Here λ is the spectral parameter, and
ψ(λ) : so(4, C) ⊗ [λ, λ−1 ] − valued extended Maurer-Cartan form,
X(λ) : so(4, C) ⊗ [[λ, λ−1 ]] − valued formal Killing field.
Recall conformal metric g = ξ ◦ ξ,
Hopf differential II = h2 ξ 2 .
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
13 / 23
Higher order conservation laws
Enhanced prolongation
Structure equation.
da2n−1 ≡ (iγc 2n + ih2 b 2n )ξ + (iγb 2n−2 + ih̄2 c 2n−2 )ξ,
iγ
i
db 2n − ib 2n ρ ≡ a2n+1 ξ + h̄2 a2n−1 ξ,
2
2
i
iγ
dc 2n + ic 2n ρ ≡ h2 a2n+1 ξ + a2n−1 ξ,
2
2
da2n+1 ≡ (iγc 2n+2 + ih2 b 2n+2 )ξ + (iγb 2n + ih̄2 c 2n )ξ,
iγ
i
db 2n+2 − ib 2n+2 ρ ≡ a2n+3 ξ + h̄2 a2n+1 ξ,
2
2
i
iγ
dc 2n+2 + ic 2n+2 ρ ≡ h2 a2n+3 ξ + a2n+1 ξ, mod I(∞) .
2
2
• b, c’s determine a’s, but a’s do not determine b, c’s.
• Repeats a single three term relation, which reflects the decomposition
so(4, C) = sl(2, C) ⊕ sl(2, C).
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
14 / 23
Higher order conservation laws
Enhanced prolongation
It can be summarized by the following diagram.
2
∂
ih̄2 ξ
b 2n
2
∂
iγ ξ
y
#
... a2n−1
e
2
∂
iγ ξ
2
∂
ih̄2 ξ
a2n+1
d
;
c 2n
2
∂
ih2 ξ
b 2n+2
2
∂
iγ ξ
z
2
∂
iγ ξ
%
a2n+3
9
c 2n+2
...
2
∂
ih2 ξ
• Sequence of coefficients a2j+1 , b 2j+2 , c 2j+2 can be solved in terms of an
inductive differential algebraic formula.
−j
- define ∂ξ hj = hj+1 , j ≥ 2. Set zj = h2 2 hj .
- the coefficients a, b, c’s are (up to scale) weighted homogeneous
polynomials in zj ’s (weight zj = j − 2).
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
15 / 23
Higher order conservation laws
Recursion
Set the initial circuit
−1
1
−3
a1 = 0, b 2 = −iγh2 2 , c 2 = ih22 , a3 = h2 2 ∂ξ h2 .
1
Define
âij = a2i+1 a2j+3 − 2b 2i+2 c 2j+2 − 2b 2j+2 c 2i+2 , 1 ≤ i ≤ j.
Consider the partial sum and its modification
X
ŝn =
âij , n ≥ 1,
i+j=n
1≤i≤j
m̂n =
Daniel Fox & Joe S. Wang ()
ŝn + 21 (−â n2 , n2 + an+1 an+3 )
ŝn + 12 (an+2 )2
Conservation laws for CMC surfaces
when n is even,
when n is odd.
Estes Park, July, 2013
16 / 23
Higher order conservation laws
Formulae
2
Formula for a2n+3 :
a
2n+3
1
=
γh2
1
1
2 2n+1
2
−∂ξ a
+ ∂ξ (h2 m̂n ) .
2
• The function a2n+3 is a Jacobi field.
3
Formula for b 2n+2 , c 2n+2 :
b
2n+2
c 2n+2
i
1 21
2n+1
=
−∂ξ a
− h2 m̂n ,
2h2
2
1
1
i
2n+1
2
−∂ξ a
+ h2 m̂n .
=
2γ
2
• Set
ϕn = c 2n+2 ξ + b 2n ξ.
The 1-form ϕn represents a conservation law of order (weight)
2n + 3.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
17 / 23
Higher order conservation laws
Recursion
2
∂
ih̄2 ξ
b 2n
2
∂
iγ ξ
y
#
... a2n−1
e
2
∂
iγ ξ
2
∂
ih̄2 ξ
a2n+1
;
d
c 2n
2
∂
ih2 ξ
b 2n+2
2
∂
iγ ξ
z
2
∂
iγ ξ
%
a2n+3
9
c 2n+2
...
2
∂
ih2 ξ
Theorem (Pinkall & Sterling, Fox & Wang)
a) The sequence of coefficients a2n+1 in the enhanced prolongation are
Jacobi fields.
b) The sequence of 1-forms ϕn = c 2n+2 ξ + b 2n ξ represent nontrivial
conservation laws.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
18 / 23
Higher order conservation laws
Classification
Results.
a) C = C classical ⊕ C vertical .
b) For each n ≥ 0 there exist two conservation laws ϕn , ϕn of order 2n + 3.
C vertical = hϕn , ϕn i∞
n=0 .
c)∗ A higher-order Noether’s theorem holds.
C ' {Jacobi field}.
Remark. ϕ0 , ϕ0 are smooth, but for n ≥ 1 the 1-forms ϕn , ϕn have pole
type singularities at the umbilics.
The residues provide a sequence of characteristic numbers for
umbilics.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
19 / 23
Spectral symmetry and conservation law
Spectral symmetry
Recall the compatibility equation for the pair (g , II)
Rg = γ 2 − |II|2g .
1
Consider the spectral deformation
(g , II) −→ (g , λII)
for a unit complex number λ (associate surfaces).
• The corresponding spectral Jacobi field, spectral symmetry
(shadow), and conservation law are neither classical nor vertical.
2
They are defined on an integrable extension (Wahlquist-Estabrook
prolongation) of (X (∞) , I(∞) ) - nonlocal objects.
Spectral symmetry S plays an important role in the analysis of the
(local) higher-order conservation laws.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
20 / 23
Spectral symmetry and conservation law
Integrable extension
3
Affine bundle Z modeled on g with I(∞) ⊂ J , dJ ≡ 0 mod J .
(Z , J )
g
(X (∞) , I(∞) )
Spectral symmetry S - a horizontal vector field on Z such that
LS I(∞) ⊂ J .
• S is an intermediate object (not a symmetry of J ).
4
The corresponding nonlocal spectral conservation law [ϕS ] exists.
• [ϕS ] - a secondary characteristic cohomology class
[ϕS ] ∈ H 1 (Σ, R)/hC classical i.
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
21 / 23
Remarks
Extension of symmetries
Classical Killing field: classical cv laws
enhanced prolongation
Loop algebra Killing field: higher order cv laws
integrable extension
Affine Kac-Moody algebra Killing field: secondary cv laws
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
22 / 23
Remarks
Application to high genus CMC surfaces
For a CMC surface Σ of genus ≥ 2.
√
1 Double cover for
II
so that
√
Σ̂ → Σ
√
II ∈ H 0 (Σ̂, K 1 ). Let U = ( II)0 .
ĝ = genus(Σ̂)
√
r = number of zero points of II (even)
Set m = ĝ + 2r .
2
For a choice of conservation laws φ = (φ1 , φ2 , ... φm )t , Abel-Jacobi
map is defined as (Lφ is the integral lattice of residues and periods)
uφ : Σ̂ \ U → Cm /Lφ
(' comlex m-torus, generically).
Can one employ such uφ to linearize the CMC surface in any way?
Daniel Fox & Joe S. Wang ()
Conservation laws for CMC surfaces
Estes Park, July, 2013
23 / 23