Curvatures and anisometry of maps

Curvatures and anisometry of maps

Benoît R. Kloeckner Université de Grenoble I, Institut Fourier
CNRS UMR 5582
BP 74
38402 Saint Martin d’Hères cedex
France
benoit.kloeckner@ujf-grenoble.fr
Abstract

We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The proofs relate to a conjectural isoperimetric inequality for manifolds whose curvature is bounded above, and to a higher-dimensional generalization of the Schwarz-Ahlfors lemma.

1 Introduction

One of the basic facts of Riemannian geometry is that curvatures are isometry invariants: this explains for example why one cannot design a perfect map of a region on the earth. In this article, we shall be interested in quantifying this fact: how far from being an isometry a map from a region of a manifold to another manifold must be, when the source and target manifolds satisfy incompatible curvature bounds?

When the source manifold is the round -sphere and the target manifold is the Euclidean plane, this question is a cartography problem: a round sphere is a relatively good approximation of the shape of the Earth. It has been considered by Milnor [Milnor] who described the best map when the source region is a spherical cap. Surprisingly, it seems like no other cases of the general question above have been considered.

1.1 Distortion and anisometry

To fill this gap, one has first to ask how we should measure the isometric default of a map from a domain in a manifold to a manifold , assumed to be a diffeomorphism on its image. Milnor uses the distortion, defined as follows. Let and be the Lipschitz constants of , i.e.

and , are respectively the greatest and least numbers satisfying such an inequality. Then the distortion of is the number .

However, when the target manifold is not Euclidean, the distortion is ill-suited: it is zero for maps that are not isometries, but mere homotheties. More disturbing is the case when is positively curved and is negatively curved: to minimize distortion, one is inclined to take with a very small image, so that the curvature of barely matters. To make this case more interesting, we propose the following definition of anisometry:

This quantity generalizes distortion in the sense that when ,

1.2 Azimuthal maps

To describe our results we will need to introduce a specific family of maps between model spaces. All considered manifolds will be of the same fixed dimension ; we set for the simply connected manifold of constant curvature (thus a sphere, the Euclidean space or a hyperbolic space).

Given a point , we have polar coordinates ( a positive real, a unit tangent vector at ) given by the exponential map:

where is less than the conjugate radius and may be any point but the antipodal point to (when ).

Definition.

An azimuthal map is a map where is a geodesic ball, which reads in polar coordinates centered at and as

where is a linear isometry from to and is a differentiable function. In other words, we have

The function is then called the distance function of .

As we consider only model spaces, is irrelevant and the function defines a unique azimuthal map up to isometries. The azimuthal map associated to each of the following distance functions bears a special name:

  • : equidistant azimuthal map,

  • with : -contracting azimuthal map

Moreover, given and there exists exactly one family of conformal azimuthal maps and a unique volume-preserving map (see below for details).

1.3 Description of the results

We shall not state our results in the greatest generality in this introduction, please see below for details.

Our main results have the following form: we assume satisfies some kind of lower curvature bound associated with a parameter , that satisfies some kind of upper curvature bound (or more general geometric assumption) associated with a parameter , and that is a map (possibly satisfying extra assumptions) from a geodesic ball of center and radius in to .

Our methods provide half-local results, and we shall always assume that is bounded above by some number. This bound shall be explicit most of the time and depends only on synthetic geometrical properties of and . In some cases (e.g. when the target is a Hadamard manifold) this bound will be completely harmless.

We then conclude that there is an azimuthal map (where is any geodesic closed ball of radius in ) such that

with equality if and only if is conjugated to by isometries.

For simplicity, we shall write to mean that the Ricci tensor and the metric tensor of satisfy the usual bound

Similarly, means that the sectional curvature of is not greater than at any tangent -plane.

We shall always assume implicitly that (or more generally ) and are complete; recall that they have the same dimension .

{theo}

[General maps] Assume , where , and where is an explicit positive constant.

Then any map satisfies

where is:

  • the equidistant azimuthal map when ,

  • the -contracting azimuthal map when , where is such that the boundaries of and have equal volume.

Moreover in case of equality and are conjugated by isometries (in particular, the source and image of have constant curvature and ).

One can write explicitly, see below. This theorem is proved using a rather direct generalization of Milnor’s argument who considers the constant curvature, -dimensional case.

Remark.
  1. It is interesting to see that the sign of has such an influence on the optimal map: when the best map is isometric along rays issued from the center, and increases distances in the orthogonal directions, while when the best map induces an isometry on the boundaries but contracts the radial rays. Of course, when all -contracting azimuthal maps are equivalent up to an homothety, and as long as their anisometries are equal.

  2. The hypothesis on can be relaxed thanks to the generalized Günther inequality proved with Greg Kuperberg [KK:Gunther]. In particular, can be replaced by mixed curvatures bounds like

    see Section 2.3 and Theorem 3 for the most general hypothesis and the above reference for various classical assumptions that imply this general hypothesis.

  3. The precise expression of is given page 3. In many cases one can adapt the result and its proof to larger but we favored clarity over exhaustivity. For example, what happens for close to is that the boundary of becomes very small, and one can improve the equidistant azimuthal map by making it dilating along the rays.

We shall then consider maps satisfying special conditions. Two prominent examples are volume-preserving maps and conformal maps. In cartography, both make sense: area is obviously a relevant geographic information, and for many historical uses (e.g. navigation) measurement of angles on the map have been needed. Moreover, asking a map to be conformal means that zooming into the map will decrease arbitrarily the distortion of a smaller and smaller region. We therefore ask whether in general, asking to be volume-preserving or conformal increases the anisotropy lower bound by much.

In the theorems below, we shall make the assumption that satisfies the best isoperimetric inequality holding on , meaning that for all smooth ,

where is the isoperimetric profile of defined by

This assumption can be replaced by in some cases.

One says that is a Hadamard manifold if and is simply connected; it is conjectured that all Hadamard manifolds satisfy the isoperimetric inequality of whenever , but this conjecture has only been proved in a handful of cases: when [Weil, Aubin], [Kleiner], [Croke] and for small enough domains [KK:petitprince]. Moreover, the similar conjecture when holds in dimension for uniquely geodesic domains [KK:petitprince]. When the curvature assumption can generally be relaxed as for Theorem 1.3, see Section 2.3 below and [KK:petitprince].

This means that in most dimensions, our results below hold under a curvature assumption only conditionally to a strong conjecture; but note that even in the case when these results are new.

{theo}

[Volume-preserving maps] Assume , satisfies the best isoperimetric inequality holding on for some , and .

Then any volume-preserving map satisfies

where is the unique volume-preserving azimuthal map .

Assume further that the only domains in satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in . Then whenever , the domain of has constant curvature and its range is isometric to a constant curvature ball . However, there are uncountably many different maps achieving equality.

Remark.

Here we have put little restriction on (we only restrict it below the injectivity radius at for simplicity), but in fact stronger restriction can appear when one wants to apply the result. Indeed, if one is only able to show that small enough domains of satisfy the desired isoperimetric inequality, then one can still use Theorem 1.3 for small enough : then a map either has a small image, or a large .

{theo}

[Conformal maps] Assume , satisfies the best isoperimetric inequality holding on for some , and where is an explicit positive constant.

Then any conformal map satisfies

where is:

  • the conformal azimuthal map with when ,

  • the conformal azimuthal map that induces an isometry on the boundaries when .

Assume further that the only domains in satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in . Then whenever , the maps and are conjugated by isometries (in particular, the domain and range of have constant curvature and ), except that when one can compose with any homothety such that we still have , and still get an optimal map.

Remark.

We shall see that can in fact be chosen independently of (but depending on ). Moreover, when we can take .

Conformal maps are rare in higher dimension, so we also tackle quasi-conformal maps, whose angular distortion is controlled. Recall that a smooth map is said to be -quasiconformal if at each point in its domain, we have , i.e. its infinitesimal distortion is uniformly bounded; conformal maps are precisely the -quasiconformal maps. {theo} Assume , satisfies the best isoperimetric inequality holding on for some , let be a number greater than and assume where is some positive constant.

Then any -quasiconformal map satisfies

where is an explicit -conformal azimuthal map, which is but not .

Assume further that the only domains in satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in . Then whenever , the maps and are conjugated by isometries (in particular, the domain and range of have constant curvature and ), except that when one can compose with any homothety such that we still have , and still get an optimal map.

Remark.

Here the constant is less explicit than in the other result, but it is still perfectly constructive. Moreover we shall see that when , we can take .

It is also interesting to compare what we obtain from the above inequalities when is small. {coro} If and , any map satisfies

If is conformal, then

If is volume-preserving, then

Remark.
  1. In this Corollary, one can easily replace the curvature assumptions by scalar curvature bounds, since only small balls are considered. Note that the isoperimetric inequality needed in Theorems 1.3 and 1.3 has been proved to be true for small enough domains under the curvature assumption (or even in some cases) by Johnson and Morgan [Johnson-Morgan] and under by Druet [Druet]. To obtain a Taylor series, these strict assumptions are sufficient (but then the remainder term cannot be made explicit).

  2. In all our results, one consider maps from the higher-curvature manifold to the lower-curvature one. These results imply similar estimates for maps , because either such a map contracts some distances by much (hence has large anisometry), or its image contains a ball of radius bounded below, allowing us to apply the results above to . However, the estimates one gets that way are certainly not sharp, and we do not know whether is optimal in any of the situation treated above; it seems that even the case of a map from a ball in the plane to a round -sphere is open. One might want to perturb the equidistant azimuthal map to enlarge the boundary of its image, so as to limit the distortion along the boundary. It is not clear whether this can be achieved without increasing distortion too much anywhere else.

Organization of the paper

Next section gives notations and some background. We prove our main results in the following three sections (general maps, then volume-preserving maps, then conformal and quasi-conformal maps). The technique we use in the conformal and quasiconformal cases turns out to have been used by Gromov to generalize the Schwarz-Pick-Ahlfors lemma. In the final Section 6, we shall state and prove a result of this flavor that seems not to be in the literature (but certainly is in its topological closure).

Acknowledgments

It is a pleasure to thank Charles Frances, Étienne Ghys and Pierre Pansu for interesting discussions related to the content of the present article.

2 Toolbox

2.1 Notations

Let be the model space of curvature and dimension , i.e. a round sphere when , the Euclidean space when , and a hyperbolic space when .

We denote by (respectively ) the geodesic closed ball (respectively sphere) of radius and center in . When there is no ambiguity, we let and . To simplify notation, we set (respectively ) for any geodesic closed ball (respectively sphere) of radius in .

The volumes of manifolds, submanifolds and domains shall be denoted either by or . We let be the -dimensional volume of the unit sphere in .

When there is no ambiguity, shall denote .

When is a point in a manifold and a tangent vector at , we let be the time of the geodesic issued from with velocity .

We shall denote by the unit tangent bundle of a Riemannian manifold , by the injectivity radius at and by the injectivity radius of .

2.2 Geometry of model spaces

The model spaces are well understood, let us recall a few facts about them.

2.2.1 Trigonometric functions

It will be convenient to use the functions defined by

We then set

and

We shall also use occasionally

and we have the derivatives and .

A trigonometric formula that will prove useful is

We shall need the following Taylor series:

2.2.2 Volumes

Let be a point on , be a positive real and be a unit tangent vector at ; then setting we can express the volume measure on by the formula

where is Lebesgue measure on and is the volume measure on the unit tangent sphere naturally identified with the unit round sphere .

In this volume formula, one can decompose the density into factors (in the direction of the ray from the pole) and (in the orthogonal directions). This shows that up to isometry there exists exactly one azimuthal, conformal map from the ball such that is a homothety of ratio (i.e. ), whose distance function is driven by the following differential equation:

Moreover the -dimensional volume of a geodesic sphere of radius is

where is the volume of ; when we only consider below the conjugate radius . We also name the volume of geodesic balls of :

Given and , there is exactly one volume-preserving azimuthal map, defined by the distance function

That is as above is clearly necessary for an azimuthal map to be volume-preserving, but the local volume formula shows that it is also sufficient.

It is known that in the least perimeter domains of given domains are balls, so that the isoperimetric profile of is given by

Note that the lesser is , the greater is and the more stringent is the corresponding isoperimetric inequality.

Using the above Taylor series, we get:

2.3 Candle functions and comparison

To study anisometry of maps under curvature bounds of the domain and range, we will need some tools of comparison geometry, relating the geometry of and to the geometry of and . We will notably rely on Bishop and Günther’s inequality, which in their common phrasing compare volume of balls. It will be useful to discuss their more general form, which is about comparing Jacobians of exponential maps.

Given a point , a vector and a real number , let and define the candle function as a normalized Jacobian of the exponential map by

where denotes the Riemannian volume and is the spherical measure on .

In the case of , this function does not depend on nor on and is equal to .

{defi}

The manifold is said to satisfy the candle condition if for all , and all it holds

The manifold is said to satisfy the logarithmic candle derivative condition if for all , and all it holds

(1)

where and .

The name “candle condition” is motivated by the fact that describes the fade of the light of a candle (or of the gravitational fields generated by a punctual mass) in .

By integration, implies that spheres and ball of radius at most have volume at least as large as the volume of the spheres and balls of equal radius in .

The candle condition is an integrated version of the logarithmic candle derivative condition, which itself follows for from the sectional curvature condition : this is known as Günther’s theorem, see [GHL]. With Greg Kuperberg, we proved in [KK:Gunther] that it also follows from a weaker curvature bound, involving the “root-Ricci curvature”. In particular, we proved that manifolds satisfying a relaxed bound on and a suitably strengthened bound on still satisfy a LCD condition, and therefore a Candle one.

The strong form of Bishop’s Theorem is that the reversed inequality in (1) holds under the curvature lower bound (for the conjugate time of ). The corresponding comparison on the volumes of spheres and balls follow and are also referred to as Bishop’s inequality.

We shall establish Theorem 1.3 using the comparison of spheres; the assumption can therefore be relaxed to where can be taken to be e.g. when is a Hadamard manifold or chosen suitably otherwise, see the proof below.

2.4 Volume of ellipsoids and hyperplanes

A couple of our arguments will rely on a simple and classical lemma, which we state and prove for the sake of completeness.

Let be a scalar product in Euclidean space of dimension , endowed with the standard inner product . We shall denote by and the largest, respectively smallest numbers such that

and say that is at most -distorted if . We shall also denote by the determinant of , that is the ratio of the volume of its unit ball to the volume of Euclidean unit ball (both volumes computed with respect to the Lebesgue measure associated to ).

{lemm}

Let be the restriction of to any hyperplane. Then we have

and

There is equality in this second inequality if and only if has eigenvalues and , with respective multiplicity and , and the hyperplane defining is the eigenspace of .

Proof.

Let be the eigenvalues of and be the eigenvalues of . In particular, . Then Rayleigh quotients show that for all . It follows

But by the distortion bound, we have

so that and

and the desired inequality follows. The equality case is straightforward. ∎

3 General maps

Assume that and that satisfies for some (on which we shall put some restriction latter on).

Let be a diffeomorphism on its image, where , the injectivity radius of at .

In what follows, we shall assume bounds involving : our convention is that this number is whenever . {lemm} If , we have

The proof is a mere generalization of Milnor’s argument in [Milnor].

Proof.

Denote by the geodesic sphere of center and radius . Bishop’s inequality ensures that . On the other hand, encloses the ball of of radius centered at . Given a unit vector , let be the first time at which hits , and be the angle between and the outward normal to . We have and obviously . Moreover when , we have

so that the comparison candle function is increasing on .

Then, letting be the candle function of at and be the usual measure on the unit Riemannian sphere, we get

There is at least one point on at which the Jacobian of the restriction of to is at least

and the lemma follows. ∎

Let us now define the -bound .

Definition.

Let be the greatest number such that for all we have , if :

(2)

and if

(3)
Remark.
  1. depends on and only through their curvature/candle bounds and and their injectivity radii,

  2. if we do not insist on a uniform bound over possible centers, we can replace by ,

  3. if is a Hadamard manifold, then (or ),

  4. if and are large enough, when we have and when ,

  5. in some cases (e.g. when one can apply Klingenberg’s Theorems, see [Cheeger-Ebin] Theorems 5.9 and 5.10), the curvature bound on is sufficient to get an estimate on , and therefore to get a bound that does not depend on the injectivity radius of the range.

{theo}

Assume , satisfies for some (e.g. ) and defined above. Let be any smooth map.

If then

and there is equality if and only if is conjugated via isometries to the equidistant azimuthal map from to (in particular, and its image must have constant curvatures and ).

If then, letting be the number in such that , we have

and there is equality if and only if is the -contracting azimuthal map from to (in particular, and its image must have constant curvatures and ).

Notice that is the dilation coefficient that makes the volumes of the spheres and coincide; it makes the -contracting azimuthal map a non-dilating map, i.e. when .

Proof.

We can assume is small enough to apply Lemma 3, otherwise the way we designed ensures that is so large that is a least the claimed lower bound.

Let us start with the case. From Lemma 3 we have

(4)

The derivative of the right-hand side with respect to is

when and

when . This shows that the right-hand side of (4) achieves its minimum when , so that

In case of equality, one must have and , therefore there is equality in Lemma 3. This forces and its image to have constant curvatures and and must be mapped to the geodesic sphere of radius and center in . Since , must then map to for all . Each ray must be mapped to a curve of length at its that connects to the boundary of the image of , therefore unit rays are mapped to unit rays. The whole map then depends only on its derivative, which must preserve the norms. It follows that is azimuthal equidistant, up to isometries.

In the case, (4) also holds but is not optimal anymore. Indeed, the derivative of its right-hand side is positive both when and when since . But when , the lower bound on given by Lemma 3 is less than . It follows

Which is achieved by . The case of equality is treated as above. ∎

{coro}

In the above setting,

where the implied constant in the remainder term only depends on the curvature bounds.

4 Area-preserving maps

Let us now prove Theorem 1.3 in the following form.

{theo}

Assume , satisfies the best isoperimetric inequality holding on , and . Then any volume-preserving map satisfies

and equality is achieved by the unique volume-preserving azimuthal map .

Assume further that the only domains in satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in . Then whenever , the domain of has constant curvature and its range is isometric to a constant curvature ball . However, there are uncountably many different maps achieving equality.

Proof.

The key point is the following Lemma, which is a direct adaptation of Theorem 3.5 in [Johnson-Morgan]. {lemm} Under the assumption and for all , we have

If there is equality, then is isometric to .

Proof of Lemma.

Setting , the strong form of Bishop’s inequality yields for all

By integration, it comes

Since is concave, we have ; therefore

In case of equality, we must have , which implies . The equality case in Bishop’s inequality then implies that is isometric to . ∎

Now, since is volume-preserving and satisfies an isoperimetric inequality, we have

Then, there must be a point on such that the Jacobian of is at least so that

but also, since , Lemma 2.4 shows that a direction transverse to the boundary must be contracted by and

These two bounds combined imply the desired inequality on .

It is straightforward to see that the unique volume-preserving azimuthal map realizes equality. Moreover, if there is equality then there must be equality in the Lemma, so that is isometric to , and there must be equality in the isoperimetric inequality on .

However, is far from being the only optimal map: both and are realized on the boundary, and for all ,

If we compose with any diffeomorphism of close to identity and supported on some , we get another optimal map. ∎

5 Conformal and quasiconformal maps

The following result is the heart of our results for quasiconformal maps; it’s formulation has been chosen to avoid repetition of arguments while keeping as much flexibility as we shall need, and it is therefore rather technical. {theo}[Main quasiconformal inequality] Assume is a -quasiconformal maps from to , where and satisfies the isoperimetric inequality of .

Let be the function defined by

If , assume further that the volume of the image of is not greater than the volume of an hemisphere of curvature .

Then, for all we have

where is the radius of a ball in that has the same volume as the image of .

The proof of this inequality follows a simple idea: at each time , the isoperimetric inequality forces the image of the sphere of radius to have large volume, and the quasiconformality then translate this into a large increase in the volume of the image of the ball. These two effects therefore amplify one another. At , we get a lower bound on , and using the isoperimetric inequality again we bound from below the perimeter of the image of the -ball. Comparing with the perimeter of the ball, we get a lower bound on .

Proof of the main quasiconformal inequality.

For convenience, for all set . In particular, .

Using Hölder’s inequality we get

where is the Jacobian of at . Let be the restriction of along : using Lemma 2.4, Bishop’s inequality and the isoperimetric inequality on it comes

(5)

Let be defined by

and let us compute :