Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves

Reparametrization-invariant Sobolev metrics on the space of immersed curves have been of central interest in recent years: from an application point of view, they take a central role in the area of mathematical shape analysis, see e.g. [41, 46, 6] and the references therein. These metrics also arise in higher-order gradient flows for various functionals [40, 37]. From a theoretical point of view, they are the natural generalization of right-invariant Sobolev metrics on the diffeomorphism group of $\mathbb{S}^{1}$; whose geodesic equations reduce to many important PDEs from hydrodynamics, including the Burgers, Camassa-Holm and Hunter-Saxton equations. For a comprehensive list of examples, see the book of Arnold and Khesin [1].

More recently, there has been an interest to extend the study of reparamerization-invariant Sobolev metrics to those of fractional order. This can be motivated, e.g., by applications in shape optimization in geometric knot theory [39, 28]; there, a main tool is using gradient-based approach for $H^{3/2}$ and $H^{3/2+\epsilon}$-type metrics, an exponent we show in this work to be critical for the completeness of the metric. Fractional order metrics have already been investigated in the context of the aforementioned geometric approach to hydrodynamics. Well-known PDEs, including the Surface Quasi-Geostrophic equations [44] and the modified Constantin-Lax-Majda equation [45], arise as reduced geodesic equations for right-invariant Sobolev metrics of fractional order on diffeomorphism groups.

The geometry of infinite dimensional Riemannian manifolds is subtle and susceptible to pathologies and many elementary facts from finite dimensional geometry do not necessarily carry over. Indeed a smooth exponential map may not exist [19], the geodesic distance can be degenerate or even vanish identically [22, 35, 32, 27], and almost all statements of the classical Hopf-Rinow theorem fail to hold [26, 34, 2].

In the context of reparametrization-invariant Sobolev metrics on spaces of immersions, generally speaking, the higher the order of the Sobolev metric, the better behaved the Riemannian structure. The goal of this current paper is to identify the exact thresholds in which transitions between “bad” and “good” behaviors occur, as described below, and are summarized in Table 1.

We will now describe the main results of the present article. Our central object of interest is the space $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ of smooth immersions of closed curves in $\mathbb{R}^{d}$, endowed with a reparametrization-invariant Sobolev metric of order $q\in[0,\infty)$, denoted by $G^{q}$. Each result is restated later in greater detail and generality, and includes also immersions of Sobolev (rather than smooth) regularity. Exact definitions of the spaces and the metrics considered here are given in Section 2.2.

Our first main result concerns the induced geodesic distance: Since the space of immersed curves $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ is an infinite dimensional manifold, the induced geodesic distance of a Riemannian metric is a-priori only guaranteed to be a semi-metric, i.e., distinct elements can be of zero distance [35, 22]. The following result characterizes precisely for which metrics this occurs:

A more detailed version of this Theorem is given in Theorem 3.1. The fact that for $q=0$ the geodesic distance collapses was obtained by Michor–Mumford [36], who also showed that the geodesic distance is not degenerate on the quotient shape space for $q\geq 1$. Using different methods we extend their results in both directions.

Our second main result concerns the well-posedness of the corresponding geodesic equation: Bauer–Bruveris–Kolev [5] showed that these equations are locally well-posed when the order of the metric is at least $1$. Here we determine the critical index for global existence, i.e, geodesic completeness of the metric:

A more detailed version of this theorem, including for results on geodesic convexity and metric completeness for curves of Sobolev regularity, is given in Theorem 4.1. Together with the local well-posedness result for the geodesic equation [5], our result implies that the corresponding geodesic equation is globally well-posed for $q>3/2$. This was previously known only for integer order metrics $q\geq 2$ [12].

The proof of this result extends a method previously used for proving completeness of integer-order metrics [12, 11]: first proving that for $q>3/2$ the space of $H^{q}$-Sobolev immersions, endowed with our reparametrization-invariant Sobolev metric of order $q$, is metrically-complete. Since this metric is a strong metric on this space, the geodesic completeness then follows by the only part of the Hopf-Rinow theorem that holds in infinite dimensions. The geodesic completeness in the smooth category is shown by an Ebin-Marsden-type no-loss-no-gain argument [21]. The main challenges here are the more complicated estimates that arise due to the fractional order norms. The proof for the geodesic incompleteness for $q<3/2$ follows from a simple example of shrinking circles.

The results of this paper are analogous to those on Sobolev metrics on the group of diffeomorphism of the circle $\operatorname{Diff}(\mathbb{S}^{1})$; in particular, the critical exponents for completeness and vanishing-distance turn out to be the same, see Table 1.

While the eventual results are mostly similar for $\operatorname{Diff}(\mathbb{S}^{1})$ and $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$, obtaining them for $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ is generally significantly harder. The reason for this is that $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ is a much richer space — one can think of $\operatorname{Diff}(\mathbb{S}^{1})$ as a subspace of $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ of curves with a fixed image. From another perspective, results on $\operatorname{Diff}(\mathbb{S}^{1})$ can be reduced to arguments over the Lie algebra $\mathfrak{X}(\mathbb{S}^{1})$ of vector fields, whereas there is no equivalent for this on $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$. This can be seen, for example, in the proof of non-vanishing distance of the geodesic distance for $q>1/2$ (Theorem 3.1): In the case of $\operatorname{Diff}(\mathbb{S}^{1})$, this is a simple application of the embedding $L^{\infty}\subset H^{q}$ [4]. However, using the same embedding for paths in $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$, we obtain weights, depending on the length of the curves in the path (whereas in the case of $\operatorname{Diff}(\mathbb{S}^{1})$ the length is fixed); these lengths are not controlled from below when $q\in(1/2,3/2]$, and thus a more convoluted argument is needed.

Similarly the global well-posedness for $q>\frac{3}{2}$ on $\operatorname{Diff}(\mathbb{S}^{1})$ follows directly by abstract arguments [16], using the fact that the $G^{q}$-metric extends to a strong, right invariant metric on the space of Sobolev diffeomorphisms of regularity $q$. Similar arguments can be used to show that the $G^{q}$-metric induces a strong, reparametrization-invariant metric on the space of curves of Sobolev regularity $q$. Due to the more intricate nature of $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ this result cannot be used to directly conclude global well-posedness. Instead one has to carefully bound the dependence of several geometric quantities on $G^{q}$-metric balls and use this to prove metric completeness by direct estimates.

The results of this paper also work for scale-invariant versions of the metrics $G^{q}$, and other length weights; it would be interesting to find optimal (or nearly optimal) conditions on the length-weights for which completeness holds for $q>3/2$, in the spirit of [15]. We expect our results to also extend to the case of manifold-valued curves, by combining the techniques of this paper with those of [11]. Regarding vanishing geodesic distance, it is still open whether, for $q<1/2$, the geodesic distance collapses completely, and whether it collapses also on shape space, which both hold in the case $q=0$ [36]. Finally, we note that the geodesic completeness for the critical index, i.e., for the $G_{c}^{3/2}$ metric on $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$, is still open. The completeness for the corresponding metric on $\operatorname{Diff}(\mathbb{S}^{1})$ was recently established [10].

In Section 2 we provide the necessary background, including definitions of the fractional Sobolev norms we use in this paper, the space of Sobolev immersions, and the reparametrization-invariant metrics. We also prove some useful inequalities, both on the Sobolev norms and on the Riemannian metrics. In Section 3 we restate and prove the results regarding the geodesic distance. In Section 4 we restate and prove the results regarding completeness properties. The appendix contains proofs of some results for fractional Sobolev spaces used throughout the paper.

CM and MB were partially funded by BSF grant #2022076. MB was partially funded by NSF grant DMS-1953244 and by the Austrian Science Fund grant P 35813-N. CM was partially funded by ISF grant 1269/19. PH was supported by the National University of Ireland’s Dr. Éamon de Valera Postdoctoral Fellowship.

Here we record some definitions and estimates pertaining to fractional Sobolev spaces. Our presentation follows closely that of Escher and Kolev [23].

Throughout this paper, we identify $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$, and let $\theta\in[0,1]$ be a parametrization of $\mathbb{S}^{1}$ (with $0\sim 1$). The fractional Sobolev space $H^{q}(\mathbb{S}^{1},\mathbb{R}^{d})$ for $q\in\mathbb{R}$ is acquired by completing the space of smooth functions $C^{\infty}(\mathbb{S}^{1},\mathbb{R}^{d})$ under the norm

where $\displaystyle\hat{f}(n)$ denotes the Fourier transform of $f$. We recall the Sobolev embedding theorem for fractional spaces:

Proofs of this statement can be found in many standard references, e.g., [43, Section 2.7.1]. We further define the space $H^{q}(\mathbb{S}^{1},\mathbb{S}^{1})$ to consist of all self-maps of the circle which, when composed with any chart, are in $H^{q}(\mathbb{S}^{1},\mathbb{R})$. If we require that $q>\frac{3}{2}$, it follows from the Sobolev Embedding Theorem that $H^{q}(\mathbb{S}^{1},\mathbb{S}^{1})\hookrightarrow C^{1}(\mathbb{S}^{1},\mathbb{S}^{1})$. Hence, by the Inverse Function Theorem, we may define the space of $H^{q}$-diffeomorphisms of the circle as

This space is an infinite-dimensional Hilbert manifold modeled on $H^{q}(\mathbb{S}^{1},\mathbb{R})$ c.f., [21]. It is in addition a half-Lie group [33, 8], i.e., a topological group under composition, where for any $\varphi\in\operatorname{\mathcal{D}}^{q}(\mathbb{S}^{1})$ right translation

is smooth, but left translation

is only continuous. It further acts on $H^{q}(\mathbb{S}^{1},\mathbb{R})$ where, again for fixed $\varphi\in\operatorname{\mathcal{D}}^{q}(\mathbb{S}^{1})$, the following map is smooth

Throughout our arguments we will require estimates on products and compositions with respect to the homogeneous Sobolev seminorm defined by

Finally, we note that this seminorm can be rewritten as

where $\Lambda:=H\partial_{\theta}$ is the pseudo-differential operator with symbol $\widehat{\Lambda}(m)=\left\lvert m\right\rvert$.

Our central estimates are as follows. From a notational standpoint, we will write $\left\lVert\cdot\right\rVert_{1}\simeq_{R}\left\lVert\cdot\right\rVert_{2}$, $\left\lVert\cdot\right\rVert_{1}\lesssim_{R}\left\lVert\cdot\right\rVert_{2}$, etc. to indicate an equivalence or inequality is valid up to a constant depending continuously on $R$.

Details of the proofs for these estimates can be found in Appendix A.

Here we introduce the setting for the results contained in this paper. Further details of the constructions can be found in Michor–Mumford [36].

We consider the space of smooth immersions of $\mathbb{S}^{1}$ into Euclidean space

It is an open subset of $C^{\infty}(\mathbb{S}^{1},\mathbb{R}^{d})$ and hence inherits the structure of an infinite-dimensional Frechét manifold with tangent space at the point $c\in\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ given by

Next, we define the space of smooth, orientation preserving diffeomorphisms of $\mathbb{S}^{1}$

This is an infinite-dimensional Frechét Lie group which acts on $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ on the right by composition

We are interested in equipping $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ with reparametrization-invariant Riemannian metrics $G$, i.e., for all $c\in\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d}),h,k\in T_{f}\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$ and $\varphi\in\operatorname{\operatorname{Diff}}(\mathbb{S}^{1})$

The importance of these metrics stems from the fact that they descend to Riemannian metrics on the shape space

which carries the structure of an infinite-dimensional orbifold.

A subclass of reparametrization-invariant metrics are Sobolev-type metrics of order $q\in\mathbb{R}$:

where $L^{q}_{c}$ is a self-adjoint, invertible, pseudo-differential operator of order $2q$ depending on $c$ in such a way as to keep $G$ reparametrization-invariant and $ds=\left\lvert c_{\theta}\right\rvert d\theta$ denotes integration with respect to arc length. It has been observed in [5], that the requirement that the metric (2.14) be invariant under $\operatorname{\operatorname{Diff}}(\mathbb{S}^{1})$ tells us that $\left\langle\cdot,\cdot\right\rangle_{G_{c}^{q}}$ is completely determined by its behavior on constant speed curves. In particular, if we define the constant speed reparametrization for $c$:

where $l_{c}$ denotes the length of $c$, we have $|(c\circ\psi^{-1}_{c})_{\theta}|=l_{c}$; which in turn gives us that $L^{q}_{c}=R_{\psi_{c}}L^{q}_{c\circ\psi_{c}^{-1}}R_{\psi_{c}^{-1}}$. We now assume $L^{q}_{c}$ has the form:

where, as before, $\Lambda:=H\partial_{\theta}$ is the pseudo-differential operator with symbol $\widehat{\Lambda}(m)=\left\lvert m\right\rvert$. In summary, our full and our homogeneous reparametrization-invariant metrics of interest are, respectively, given by:

and

We further denote differentiation with respect to arc length by $D_{s}=\frac{1}{\left\lvert c_{\theta}\right\rvert}\partial_{\theta}$. It is not difficult to show that $\left\lVert D_{s}h\right\rVert_{\dot{G}_{c}^{q}}=\left\lVert h\right\rVert_{\dot{G}_{c}^{q+1}}$. For constant speed curves this follows from integration by parts. The result for non-constant speed curves then follows from the reparametrization-invariance of $\dot{G}_{c}^{q}$.

We also consider alongside the above space $\operatorname{Imm}(\mathbb{S}^{1},\mathbb{R}^{d})$, its finite smoothness counterparts. For ${r}>\frac{3}{2}$ we have the Sobolev completions

The following result is due to Bauer et al. [5, Theorems 6.4 and 7.1].

We now express the induced homogeneous reparametrization-invariant norm in terms of the usual homogeneous Sobolev norm.

The next lemma, which establishes a Sobolev Embedding-type theorem for our reparametrisation-invariant norms, extends a result for first order metrics contained in [14, Lemma 2.14] to fractional orders. The important point here is the explicit dependence of the embedding constant on the length of the underlying curve.