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 { ( x 1 , x 2 , . . . ) ∈ ℓ 2 | � ∞ k =1 a k x 2 k = 1 } for some positive numbers a k . (Good source of counterexamples.) ◮ H 1 ( S 1 , M ), the space of loops γ in Riem. mfd. M with S 1 | γ ′ ( t ) | 2 dt < ∞ . Tangent bundle looks like H 1 ( S 1 , 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 T M 3. smooth exponential map taking v to exp p ( v ), defined by the geodesic γ with γ (0) = p , γ ′ (0) = v , and exp p ( 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 x n is a bounded sequence, K ( x n ) has a convergent subsequence. For example integral operators 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 λ I − K has a nontrivial nullspace, or λ I − K is invertible. If λ � = 0, we say that λ I − 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 dF p : T p M → T F ( 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 ) = exp p ( tv ) then J ( t ) = ( d exp p ) tv ( tw ) where J (0) = 0, J ′ (0) = w , and D 2 J � � dt 2 + 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 � t P − 1 ( t − τ ) P − 1 γ ( t ) J ( t ) = tw − γ ( τ ) R ( J ( τ ) , ˙ γ ( τ ))˙ γ ( τ ) d τ. 0 where P γ is parallel transport along γ . The equation for J expresses ( d exp p ) 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 Diff µ ( M ) (fluids) ◮ symplectomorphism and contactomorphism groups ◮ shape space Emb ( S 1 , M ) / Diff ( S 1 ) ◮ Bott-Virasoro group Diff ( S 1 ) ⋉ R (KdV equation) ◮ Diff ( M ) with H 1 metric (EPDiff) ◮ A ( S 1 , R 2 ), 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: ◮ Diff µ ( M ) with L 2 metric (Ebin-Marsden 1970) ◮ Diff ( S 1 ) with H 1 metric (Misio� lek 2002, Constantin-Kolev 2002) ◮ symplectomorphism and contactomorphism groups with L 2 metric (Ebin 2012, Ebin-Preston 2015) Nonsmooth exponential map: ◮ Bott-Virasoro group ( L 2 or H 1 ), KdV and CH equations (Constantin-Kolev 2002) ◮ A ( S 1 , R 2 ) ( L 2 ), whip equation Typically for a given Hilbert manifold, there is a “critical” Sobolev index r 0 such that Sobolev metrics of order H r give smooth geodesics for r ≥ r 0 and not for r < r 0 (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 d γ du dt = DR γ u , dt + ad ⋆ u u = 0 , with γ (0) = id and u (0) = u 0 . 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 ) = exp id ( tu 0 ). The equation may be written in the form d dt (Ad ⋆ γ u ) = 0 , which implies conservation of vorticity Ad ⋆ γ ( t ) u ( t ) = u 0 and thus that γ satisfies d γ dt = Ad ⋆ γ ( t ) − 1 u 0 , which is also often an ODE on M .

Write J ( t ) = ( d exp id ) tu 0 ( tw 0 ), so that J (0) = 0 and J ′ (0) = w 0 . If J ( t ) = dR γ ( t ) y ( t ) for y ( t ) ∈ T id M , then dy dz dt − ad u y = z , dt + ad ⋆ u z + ad ⋆ z u = 0 . Let y = Ad γ v and z = Ad γ w . Then (Ebin, Misio� lek, P. 2006; Misio� lek, P. 2010) dv d � � Ad ⋆ + Ad ⋆ γ ad ⋆ Ad η w Ad ⋆ dt = w , γ Ad γ w γ − 1 u 0 = 0 . dt It turns out that the last term is just K u 0 ( w ) := ad ⋆ w u 0 . Integrating, we get Λ( t ) dv dt + K u 0 v ( t ) = w 0 where Λ( t ) = Ad ⋆ γ ( t ) Ad γ ( t ) , so that � t � t Λ( τ ) − 1 w 0 d τ + Λ( τ ) − 1 K u 0 v ( τ ) d τ, v ( t ) = 0 0 � t 0 Λ( τ ) − 1 d τ. with Ω( t ) =

The solution operator Φ( t ) := w 0 �→ v ( t ) thus satisfies Φ( t ) = Ω( t ) + Γ( t ), where � t � t Λ( τ ) − 1 d τ Λ( τ ) − 1 K u 0 Φ( τ ) d τ. Ω( t ) = and Γ( t ) = 0 0 Important notes: ◮ The Jacobi equation is an ODE on T M , but this splitting is usually not an ODE. It loses derivatives. We approximate u 0 by smooth ˜ u 0 so that ˜ γ ( t ) = exp id ( t ˜ u 0 ) is smooth, to make Ω and Γ continuous. ◮ Λ = Ad ⋆ γ Ad γ is positive-definite in the weak topology since = Ad γ − 1 . Thus so are Λ − 1 and Ad γ is invertible: Ad − 1 γ Λ − 1 d τ . � Ω = ◮ If K u 0 is a compact operator, then so is Γ( t ). Hence compactness of K u 0 gives “weak Fredholmness” immediately (that is, Fredholmness of the completion of ( d exp id ) tu 0 in the weak topology). ◮ “Strong Fredholmness,” which is what we really need, is harder and relies on commutator estimates.