The exponential map on the area-preserving diffeomorphism group for - - PowerPoint PPT Presentation

The exponential map on the area-preserving diffeomorphism group for a bounded surface

Stephen C. Preston University of Colorado Stephen.Preston@colorado.edu math.colorado.edu/˜prestos (joint work with Gerard Misio lek) January 13, 2015

Q: What is the goal of differential geometry? A: To prove global (nonlinear) results using local (linearized) computations.

For example,

◮ Sphere theorem: positive pinched curvature implies

homeomorphism to sphere

◮ Hadamard-Cartan: nonpositive curvature implies exponential

map is covering

◮ Local flow: geodesic field nonzero implies local unique

minimizers

◮ Toponogov: curvature bounds imply distance bounds ◮ Hopf-Rinow: metric completeness implies geodesic

completeness

Many of these results fail in infinite dimensional geometry.

◮ Lack of local compactness (unavoidable, though Palais-Smale

can replace)

◮ Topology not generated by geometry (weak metrics) ◮ Metrics and connections are not necessarily smooth ◮ Fundamental theorem of linear algebra fails

Most of the work in the field has been in four areas:

• 1. Discover applications in PDE
• 2. Compute local quantities like curvature
• 3. Explore paradoxes and counterexamples
• 4. Establish criteria for “good behavior”

Strong metrics

M is locally diffeomorphic to a Hilbert space, and the Hilbert inner product gives the Riemannian metric. Typical examples:

◮ Hilbert ellipsoid {(x1, x2, . . .) ∈ ℓ2 | ∞ k=1 akx2 k = 1} for some

positive numbers ak. (Good source of counterexamples.)

◮ H1(S1, M), the space of loops γ in Riem. mfd. M with

• S1|γ′(t)|2 dt < ∞. Tangent bundle looks like H1(S1, TM)

and coordinate charts come from the Riemannian exponential map on M. (Used for finding closed geodesics a la Klingenberg.)

Smooth (strong) Riemannian metric implies:

• 1. smooth Levi-Civita connection
• 2. smooth geodesic vector field on TM
• 3. smooth exponential map taking v to expp(v), defined by the

geodesic γ with γ(0) = p, γ′(0) = v, and expp(v) = γ(1)

• 4. existence and uniqueness in small neighborhoods of

minimizing geodesics (inverse function theorem)

• 5. Hadamard-Cartan theorem (nonpositive curvature implies

exponential map is globally defined and a covering) Global theorems (e.g., Hopf-Rinow) typically fail due to loss of local compactness. “Semiglobal” theorems involving transversality depend on Sard’s theorem, which requires Fredholmness.

An operator K is compact if whenever xn is a bounded sequence, K(xn) has a convergent subsequence. For example integral

• perators are compact (basically by the Ascoli theorem).

Theorem (Fredholm alternative)

Suppose K is a compact operator from a Hilbert space E to itself and that λ ∈ C. Then either λI − K has a nontrivial nullspace, or λI − K is invertible. If λ = 0, we say that λI − K is a Fredholm operator. Its range is closed, and its kernel and cokernel are both finite-dimensional. More generally, an operator T : E → F is Fredholm iff it is of the form T = Ω + K, where K is compact and Ω is invertible.

Suppose M and N are Hilbert manifolds. A smooth map F : M → N is called a Fredholm map if dFp : TpM → TF(p)N is a Fredholm operator. By a result of Smale, Fredholm maps satisfy Sard’s theorem (the set of critical values is a meager set). To study degree theory and transversality in the infinite-dimensional context, we want maps to be Fredholm. In our context, we want the Riemannian exponential map to be

• Fredholm. Its differential solves the Jacobi equation: if

γ(t) = expp(tv) then J(t) = (d expp)tv(tw) where J(0) = 0, J′(0) = w, and D2J dt2 + R

• J(t), ˙

γ(t)

• ˙

γ(t) = 0.

On the loop space, the curvature operator J → R(J, ˙ γ)˙ γ is compact. Thus J(t) satisfies the integral equation P−1

γ(t)J(t) = tw −

t (t − τ)P−1

γ(τ)R(J(τ), ˙

γ(τ))˙ γ(τ) dτ. where Pγ is parallel transport along γ. The equation for J expresses (d expp)tv as “invertible plus compact.” Here we used the facts that

◮ the limit of a compact operator is compact ◮ the composition of a compact and a bounded operator is

compact

On the Hilbert ellipsoid the exponential map is not Fredholm. Singularities may be

◮ monoconjugate (d exp is not injective) ◮ or epiconjugate (d exp is not surjective)

We may construct examples (Grossmann 1965) where:

• 1. there is a monoconjugate point of infinite order
• 2. there is a sequence of monoconjugate points converging to an

epiconjugate point Biliotti, Exel, Piccione, Tausk, 2006:

• 1. every monoconjugate point is epiconjugate
• 2. if M is separable, there are countably many monoconjugate

points

• 3. the closure of the set of monoconjugate points is the set of

epiconjugate points

• 4. at any accumulation point of the monoconjugate points, the

range of d exp is not closed

Weak metrics

In most applications, the natural Riemannian metric topology does not make M a manifold. (Geodesic equation is a physical PDE.) Examples:

◮ volumorphism group Diffµ(M) (fluids) ◮ symplectomorphism and contactomorphism groups ◮ shape space Emb(S1, M)/Diff(S1) ◮ Bott-Virasoro group Diff(S1) ⋉ R (KdV equation) ◮ Diff(M) with H1 metric (EPDiff) ◮ A(S1, R2), arclength-parametrized curves

First problem: the geodesic equation may not be smooth, even if the metric is. Has to be checked in each case. Typically either the exponential map is C ∞ or fails to be C 1.

Smooth exponential map:

◮ Diffµ(M) with L2 metric (Ebin-Marsden 1970) ◮ Diff(S1) with H1 metric (Misio

lek 2002, Constantin-Kolev 2002)

◮ symplectomorphism and contactomorphism groups with L2

metric (Ebin 2012, Ebin-Preston 2015) Nonsmooth exponential map:

◮ Bott-Virasoro group (L2 or H1), KdV and CH equations

(Constantin-Kolev 2002)

◮ A(S1, R2) (L2), whip equation

Typically for a given Hilbert manifold, there is a “critical” Sobolev index r0 such that Sobolev metrics of order Hr give smooth geodesics for r ≥ r0 and not for r < r0 (Misio lek, P. 2010). Failure of C 1 smoothness in a Hilbert space is frequently shown by finding conjugate points arbitrarily close to the identity (if exp were C 1, inverse function theorem would prevent this).

Assume M is a Hilbert manifold with weak Riemannian metric and smooth exponential map. When is this map Fredholm? Applications:

◮ Morse Index Theorem (Misio

lek, P., 2010)

◮ Surjectivity of exponential map (Misio

lek, P., 2010, Shnirelman 2014)

◮ Morse-Littauer Theorem (Misio

lek, 2015)

◮ (Conjecture) Global existence of exponential map ◮ (Conjecture) Bounds on growth of gradients

Two known techniques for proving Fredholmness:

• 1. Compactness of curvature operator
• 2. Compactness of coadjoint operator, if M is a group

If M is a group with right-invariant metric, then the geodesic γ satisfies dγ dt = DRγu, du dt + ad⋆

u u = 0,

with γ(0) = id and u(0) = u0. This is the Euler-Arnold equation. The system is not an ODE, though it often becomes one for γ when u is eliminated. We then have γ(t) = expid(tu0). The equation may be written in the form d

dt (Ad⋆ γ u) = 0, which

implies conservation of vorticity Ad⋆

γ(t) u(t) = u0 and thus that γ

satisfies dγ dt = Ad⋆

γ(t)−1 u0,

which is also often an ODE on M.

Write J(t) = (d expid)tu0(tw0), so that J(0) = 0 and J′(0) = w0. If J(t) = dRγ(t)y(t) for y(t) ∈ TidM, then dy dt − adu y = z, dz dt + ad⋆

u z + ad⋆ z u = 0.

Let y = Adγ v and z = Adγ w. Then (Ebin, Misio lek, P. 2006; Misio lek, P. 2010) dv dt = w, d dt

• Ad⋆

γ Adγ w

• + Ad⋆

γ ad⋆ Adη w Ad⋆ γ−1 u0 = 0.

It turns out that the last term is just Ku0(w) := ad⋆

w u0.

Integrating, we get Λ(t)dv dt + Ku0v(t) = w0 where Λ(t) = Ad⋆

γ(t) Adγ(t), so that

v(t) = t Λ(τ)−1w0 dτ + t Λ(τ)−1Ku0v(τ) dτ, with Ω(t) = t

0 Λ(τ)−1 dτ.

The solution operator Φ(t) := w0 → v(t) thus satisfies Φ(t) = Ω(t) + Γ(t), where Ω(t) = t Λ(τ)−1 dτ and Γ(t) = t Λ(τ)−1Ku0Φ(τ) dτ. Important notes:

◮ The Jacobi equation is an ODE on TM, but this splitting is

usually not an ODE. It loses derivatives. We approximate u0 by smooth ˜ u0 so that ˜ γ(t) = expid(t˜ u0) is smooth, to make Ω and Γ continuous.

◮ Λ = Ad⋆ γ Adγ is positive-definite in the weak topology since

Adγ is invertible: Ad−1

γ

= Adγ−1. Thus so are Λ−1 and Ω =

• Λ−1 dτ.

◮ If Ku0 is a compact operator, then so is Γ(t). Hence

compactness of Ku0 gives “weak Fredholmness” immediately (that is, Fredholmness of the completion of (d expid)tu0 in the weak topology).

◮ “Strong Fredholmness,” which is what we really need, is

harder and relies on commutator estimates.

When Ku is compact (for all u):

• 1. Volumorphism group Diffµ(M) when dim M = 2 in L2 metric

(Ebin, Misio lek, P. 2006)

• 2. Symplectomorphisms and contactomorphisms (Benn,

Chhay-P.)

• 3. Diff(M) with H1 metric (Misio

lek, P. 2010) When Ku is not compact:

• 1. Diffµ(M) when M is three-dimensional (Ebin, Misio

lek, P. 2006) The 3D ideal Euler equation for an incompressible fluid is the only known example with integer-order Sobolev metric where the exponential map is smooth but not Fredholm. Pathological conjugate points are typical. They were classified in (P. 2006, P. 2008) and used to study the blowup problem in (P. 2010). Note: for any r > 0 the exponential map on Diffµ(M) is Fredholm in any dimension.

To obtain strong Fredholmness, we want to know that Ω(t) = t

0 Λ(τ)−1 dτ is invertible in the strong topology. Assume

M is a group of Hs diffeomorphisms of manifold M (or a subgroup, or a central extension) and the weak metric is L2.

• 1. It is sufficient to show that
• w, Ω(t)w

Hs ≥ C(t) w, w Hs − w, Υ(t)w Hs where Υ(t) is some compact operator and C(t) > 0.

• 2. For this, it is sufficient to show that
• w, Λ(t)w

Hs ≥ c(t)w2

Hs − b(t)w2 Hs′ where s′ < s. This

implies a similar estimate for w, Λ(t)−1w Hs but is easier to prove.

• 3. Working in charts on M, we just need to estimate the

difference between

• k≤s,|α|=k
• M

Dαw, DαΛw dµ and

• k≤s,|α|=k
• M

Dαw, ΛDαw dµ.

• 4. Typically the commutator [Dα, Λ] is a lower-order differential
• perator and thus compact.

Explicitly if M = Diffs

µ(M) then Λ = P(Dγ†Dγ) where P is the

• rthogonal projection onto TidDiffs

µ(M), the space of vector fields

divergence-free and tangent to ∂M.

• 1. Dγ†Dγ is just some matrix multiplier with smooth
• coefficients. Commutator is fine.
• 2. If M has no boundary, then Dαw is divergence-free whenever

w is.

• 3. But if M has boundary, DNw will not be tangent to the

boundary if DN denotes the normal derivative. In fact [DN, P] is a differential operator of order one! Example: Let w = sin nx sin ny ∂x + cos nx cos ny ∂y on the annulus [0, 2π] × S1. Then P(w) = w and [∂x, P]w = Q(∂xw), where Q = 1 − P. Now Q(∂xw) = ∇q where ∆q = 0 and qx(0, y) = qx(1, y) = n sin ny. Thus q =

• − tanh nπ cosh nx + sinh nx
• sin ny

and ∇q is of order n while w is of order 1.

In fact we can prove that Λ(t): TidDiffs

µ(M) → TidDiffs µ(M) is not

• f the form “positive-definite plus compact” if M has a boundary.

On the periodic upper half-plane M = S1 × R+, let γ(t, x, y) = (x + te−y, y) and f (x, y) = ye−2ny cos nx. Write sgrad g = P(Dγ†Dγ sgrad f ). Then the dominant term in

• sgrad f , Λ(t) sgrad f

H3 is I(t) =

• M
• i+j=3

(∂i

x∂j yf )(∂i x∂j yg) dx dy

and we have I(t) −n6 for t > 7 and all n. If Λ(t) were positive-definite up to a compact operator, this could

• nly be true for finitely many n.

Of course, the fact that the proof fails does not mean the result is false! Λ(t) is still invertible, but we do not yet know if Ω(t) = t

0 Λ(τ)−1 dτ is invertible.

There is one situation where the proof of strong Fredholmness works even when M is a surface with boundary.

Theorem (Misio lek, P.)

Let M be a surface with totally geodesic boundary and u0 a smooth velocity field on M which extends to a smooth ˜ u0 on the double ˜

• M. Then for any t > 0, (d expid)tu0 is (strongly) Fredholm
• n TidDiffs

µ(M) for any s ≥ 3.

In this case we can use the Fredholmness result on Diffs

µ( ˜

M). Furthermore the operator Λ(t) is “positive-definite plus compact.” We conjecture that strong Fredholmness is false for Diffs

µ(M2) for

s ≥ 3 in all other situations.

Why this matters:

• 1. Kiselev-Sverak (2014) proved that 2D vorticity gradients can

have double-exponential growth. ∇ω(t)L∞ ≥ aebect. Their example relies crucially on M having boundary.

• 2. Shnirelman (1994) proved that there are area-preserving

diffeomorphisms that cannot be reached by finite-energy

• paths. This also relies crucially on M having boundary. On

the other hand Shnirelman recently announced that the exponential map on Diffµ(T2) is surjective (using Fredholmness and quasiruled structure). Failure of strong Fredholmness in the boundary case would be a strong clue about how such pathologies relate to global geometric properties.