Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves

Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves

Abstract

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

\volume

1 \doi10.111

\authorheadline

M. Bruveris, P.W. Michor and D. Mumford

1]Martins Bruveris \cormark[1] \cortext[1]Corresponding author

2]Peter W. Michor

3]David Mumford

\MSC

58D15 (primary); 35G55, 53A04, 58B20 (secondary)

1 Introduction

Sobolev-type metrics on the space of plane immersed curves were independently introduced in [7, 17, 24]. They are used in computer vision, shape classification and tracking, mainly in the form of their induced metric on shape space, which is the orbit space under the action of the reparameterization group. See [14, 23] for applications of Sobolev-type metrics and [2, 18] for an overview of their mathematical properties. Sobolev-type metrics were also generalized to immersions of higher dimensional manifolds in [4, 5].

It was shown in [18] that the geodesic equation of a Sobolev-type metric of order is locally well-posed and this result was extended in [4] to a larger class of metrics and immersions of arbitrary dimension. The main result of this paper is to show global well-posedness of the geodesic equation for Sobolev-type metrics of order with constant coefficients. In particular we prove the following theorem:

Theorem 1.1.

Let and the metric on be given by

with and . Given initial conditions the solution of the geodesic equation

for the metric with initial values exists for all time.

Here denotes the space of all smooth, closed, plane curves with nowhere zero tangent vectors; this space is open in . We assume that and , are vector fields along , is the arc-length measure, is the derivative with respect to arc-length, is the unit length tangent vector to and is the Euclidean inner product on .

Thus if is a Sobolev-type metric of order at least 2, then the Riemannian manifold is geodesically complete. If the Sobolev-type metric is invariant under the reparameterization group , also the induced metric on shape space is geodesically complete. The latter space is an infinite dimensional orbifold; see [17, 2.5 and 2.10].

Theorem 1.1 seems to be the first result about geodesic completeness on manifolds of mappings outside the realm of diffeomorphism groups and manifolds of metrics. In the first paragraph of [9, p. 140] a proof is sketched that a right invariant -metric on the group of volume preserving diffeomorphisms on a compact manifold is geodesically complete, if . In [25] there is an implicit result that a topological group of diffeomorphisms constructed from a reproducing kernel Hilbert space of vector fields whose reproducing kernel is at least , is geodesically complete. For a certain metric on a group of diffeomorphisms on with kernel geodesic completeness is shown in [19, Thm. 2]. Metric completeness and existence of minimizing geodesics have also been studied on the diffeomorphism group in [6]. The manifold of all Riemannian metrics with fixed volume form is geodesically complete for the -metric (also called the Ebin metric).

Sobolev-type metrics of order 1 are not geodesically complete, since it is possible to shrink a circle to a point along a geodesic in finite time, see [18, Sect. 6.1]. Similarly a Sobolev metric of order 2 or higher with both is a geodesically incomplete metric on the space of plane curves modulo translations. In this case it is possible to blow up a circle along a geodesic to infinity in finite time; see Rem. 5.7.

In order to prove long-time existence of geodesics, we need to study properties of the geodesic distance. In particular we show the following theorem regarding continuity of curvature and its derivatives.

Theorem 1.2.

Let be a Sobolev-type metric of order with constant coefficients and the induced geodesic distance. If , then the functions

are continuous and Lipschitz continuous on every metric ball.

A similar statement can be derived for the -continuity of curvature and its derivatives; see Rem. 4.9.

The full proof of Thm. 1.1 is surprisingly complicated. One reason is that we have to work on the Sobolev completion (always with respect to the original parameter in ) of the space of immersions in order to apply results on ODEs on Banach spaces. Here the operators (and their inverses and adjoints) acquire non-smooth coefficients. Since we we want the Sobolev order as low as possible, the geodesic equation involves ; see Sect. 3.3. Eventually we use that the metric operator has constant coefficients. We have to use estimates with precise constants which are uniformly bounded on metric balls.

In [4] the authors studied Sobolev metrics on immersions of higher dimensional manifolds. One might hope that similar methods to those used in this article can be applied to show the geodesic completeness of the spaces with compact and a suitable Riemannian manifold. A crucial ingredient in the proof for plane curves are the Sobolev inequalities Lem. 2.14 and Lem. 2.15 with explicit constants, which only depend on the curve through the length. The lack of such inequalities for general will one of the factors complicating life in higher dimensions.

2 Background Material and Notation

2.1 The Space of Curves

The space

of immersions is an open set in the Fréchet space with respect to the -topology and thus itself a smooth Fréchet manifold. The tangent space of at the point consists of all vector fields along the curve . It can be described as the space of sections of the pullback bundle ,

In our case, since the tangent bundle is trivial, it can also be identified with the space of -valued functions on ,

For a curve we denote the parameter by and differentiation by , i.e., . Since is an immersion, the unit-length tangent vector is well-defined. Rotating by we obtain the unit-length normal vector , where is rotation by . We will denote by the derivative with respect to arc-length and by the integration with respect to arc-length. To summarize we have

The curvature can be defined as

and we have the Frenet-equations

The length of a curve will be denoted by . We define the turning angle of a curve by . Then curvature is given by .

2.2 Variational Formulae

We will need formulas that express, how the quantities , and change, if we vary the underlying curve . For a smooth map from to any convenient vector space (see [13]) we denote by

the variation in the direction .

The proof of the following formulas can be found for example in [18].

With these basic building blocks, one can use the following lemma to compute the variations of higher derivatives.

Lemma 2.3.

If is a smooth map , then the variation of the composition is given by

Proof.

The operator is linear and thus commutes with the derivative with respect to . Thus we have

2.4 Sobolev Norms

In this paper we will only consider Sobolev spaces of integer order. For the -norm on is given by

(1)

Given , we define the -norm on by

(2)

Note that in (2) integration and differentiation are performed with respect to the arc-length of , while in (1) the parameter is used. In particular the -norm depends on the curve . The norms and are equivalent, but the constants do depend on . We prove in Lem. 5.1, that if doesn’t vary too much, the constants can be chosen independently of .

The - and -norms are defined similarly,

and they are related via . Whenever we write or , we always endow them with the - and -norms.

For we shall denote by

the space of Sobolev immersions of order . Because of the Sobolev embedding theorem, see [1], we have and thus is well-defined. We will see in Sect. 3.2 that the -norm remains well-defined if .

The following result on point-wise multiplication will be used repeatedly. It can be found, among other places in [11, Lem. 2.3]. We will in particular use that can be negative.

Lemma 2.5.

Let and with Then multiplication is a bounded bilinear map

The last tool, that we will need is composition of Sobolev diffeomorphisms. For , define

the group of Sobolev diffeomorphisms. The following lemma can be found in [11, Thm. 1.2].

Lemma 2.6.

Let and . Then the composition map

is continuous.

Let and fix . Denote by the composition with . From Lem. 2.6 we see that is a bounded linear map . The following lemma tells us that the transpose of this map respects Sobolev orders.

Lemma 2.7.

Let , and . Then the restrictions of are bounded linear maps

On we have the identity .

Proof.

For , we obtain from Lem. 2.6 that is a map and by -duality we obtain that as required.

Now let , and . We replace by to simplify the formulas. By definition of the transpose

Thus we obtain and using Lem. 2.5 we see that for we also have . ∎

2.8 Notation

We will write

if there exists a constant , possibly depending on , such that the inequality holds.

2.9 Gronwall Inequalities

The following version of Gronwall’s inequality can be found in [22, Thm. 1.3.2] and [12].

Theorem 2.10.

Let , , be real continuous functions defined on and . We suppose that on we have the following inequality

Then

holds on .

We will repeatedly use the following corollary.

Corollary 2.11.

Let , be real, continuous functions on with and non-negative constants. We suppose that on we have the inequality

Then

holds in with .

Proof.

Apply the Gronwall inequality with , and , and note that implies . ∎

2.12 Poincaré Inequalities

In the later sections it will be necessary to estimate the -norm of a function by the -norm with , as well as the -norm by the -norm. In particular, we will need to know, how the curve enters into the estimates. The basic result is the following lemma, which is adapted from [15, Lem. 18].

Lemma 2.13.

Let and be absolutely continuous. Then

Proof.

Since , the following equality holds,

and hence after integration

Next we take the absolute value

Now we replace 0 by an arbitrary and repeat the above steps. ∎

This lemma permits us to prove the inequalities that we will use throughout the remainder of the paper.

Lemma 2.14.

Let and . Then

Proof.

From Lem. 2.13 we obtain the inequality

Next we use and Cauchy-Schwarz in

thus proving the first statement. To prove the second statement we note that and thus by Lem. 2.13

Hence

To prove the third statement we estimate

This completes the proof. ∎

The next lemma allows us to estimate the -norm using a combination of the - and the -norms, without introducing constants that depend on the curve.

Lemma 2.15.

Let , and . Then for ,

Proof.

Let us write and for and respectively to emphasize the dependence on the curve . Since , we can assume that has a constant speed parametrization, i.e. . The inequality we have to show is

Let . After a change of variables this becomes

(3)

Let and assume w.l.o.g. that is -valued. Define , which is an orthonormal basis of . Then and (3) becomes

Since for we have the inequality , the last inequality is satisfied, thus concluding the proof. ∎

An alternative way to estimate the -norm is given by the following lemma, which is the periodic version of the Gagliardo-Nirenberg inequalities (see [20]).

Lemma 2.16.

Let , and . Then for ,

If , the inequality also holds for .

2.17 The Geodesic Equation on Weak Riemannian Manifolds

Let be a convenient vector space, an open subset and a possibly weak Riemannian metric on . Denote by the canonical map defined by

with , and with denoting the canonical pairing between and . We also define via

with denoting the directional derivative at in direction . In fact is a smooth map

With these definitions we can state how to calculate the geodesic equation.

Lemma 2.18.

The geodesic equation – or equivalently the Levi-Civita covariant derivative – on exists if and only if is in the image of for all and the map

is smooth. In this case the geodesic equation can be written as

This lemma is an adaptation of the result given in [3, 2.4.1] and the same proof can be repeated; see also [16, Sect. 2.4].

3 Sobolev Metrics with Constant Coefficients

In this paper we will consider Sobolev-type metrics with constant coefficients. These are metrics of the form

with and . We call the order of the metric. The metric can be defined either on the space of (-)smooth immersions or for on the spaces of Sobolev -immersions.

3.1 The Space of Smooth Immersions

Let us first consider on the space of smooth immersions. The metric can be represented via the associated family of operators, , which are defined by

The operator for a Sobolev metric with constant coefficients can be calculated via integration by parts and is given by

The operator is self-adjoint, positive and hence injective. Since is elliptic, it is Fredholm with vanishing index and thus surjective. Furthermore its inverse is smooth as well. We want to distinguish between the operator and the canonical embedding from into , which we denote by . They are related via

Later we will simply write , especially when the order of multiplication and differentiation becomes important in Sobolev spaces.

3.2 The Space of Sobolev Immersions

Assume and let be a Sobolev metric of order . We want to extend from the space to a smooth metric on the Sobolev-completion . First we have to look at the action of the arc-length derivative and its transpose (with respect to ) on Sobolev spaces. Remember that we always use the -norm on Sobolev completions. We can write as the composition , where is interpreted as the multiplication operator . Its transpose is . These operators are smooth in the following sense.

Lemma 3.3.

Let and with . Then the maps

are smooth.

Proof.

For , the map is the composition of the following smooth maps,

Since , Lem. 2.5. concludes the proof. ∎

Using Lem. 3.3 we see that

is well-defined for . As the tangent bundle is isomorphic to , we can also write the metric as

Again we note that has to be interpreted as the multiplication operator on the spaces with . Thus the operator is given by

While it is tempting to “simplify” the expression for using the identity

one has to be careful, since the identity is only valid, when interpreted as an operator with . The left hand side however makes sense also for . Thus we have the operator

but the domain has to be at least for the operator