Constrained Willmore Surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint.


Introduction
In 1882 Felix Klein posed the question whether every abstract Riemann surface M can be realized in Euclidean 3-space. A positive answer to this question was given by A. Garsia [10] and R. Rüedy [15], who proved that every Riemann surface can be conformally embedded into R 3 . This immediately raises the question for the optimal geometric realizations in R 3 of a given abstract Riemann surface, or, slightly more general, for the optimal (especially beautiful, symmetric...) conformal immersions of a Riemann surface into space.
The same question may be raised about optimal realizations of topological types of surfaces, without the conformal constraint. In this context, there is a general agreement that an excellent way to make the notion of "optimal" precise is to look for surfaces that are critical points of the Willmore functional W(f ) = H 2 dσ.
This leads to the notion of constrained Willmore surfaces: an immersion f : M → R 3 of a Riemann surface M is a constrained Willmore surface if it is critical for W under compactly supported infinitesimal conformal variations of f . This generalizes the notion of Willmore surfaces, the critical points of W under all compactly supported variations. Constrained Willmore surfaces are a Möbius invariant class of surfaces with strong links to the theory of integrable systems.
The concept of constrained Willmore surfaces already appears in [13,21,14,7,16]. However, the Euler-Lagrange equation characterizing constrained Willmore surfaces is nowhere actually derived from the variational problem, except for the simplest cases of compact surfaces of genus g ≤ 1 in [7]. Examples of constrained Willmore surfaces are constant mean curvature surfaces in 3-dimensional space forms, see [13] for the Euclidean case and [14,7] or Section 4 below for general space forms.
In Section 2, we derive the Euler-Lagrange equation for conformal immersions of compact Riemann surfaces that are critical points of geometric functionals under infinitesimal conformal variations. In this Euler-Lagrange equation, a holomorphic quadratic differential appears as a Lagrange multiplier. In the case of immersions of non-compact surfaces, the same Euler-Lagrange equation is still a sufficient (but no longer necessary) condition for being critical under the conformal constraint.
In Section 3, we discuss examples of constrained minimal and constrained volume-critical surfaces, i.e., immersions which are critical for area or enclosed volume under the conformal constraint. This yields valuable insights into variational problems of the type studied in Section 2. In particular, Example 11 shows that the case of open surfaces is more delicate than the compact case: we construct a constrained minimal immersion that does not satisfy the Euler-Lagrange equation derived in Section 2.
In Section 4, we specialize the previous results to constrained Willmore surfaces and give a short proof of the fact that constant mean curvature surfaces are constrained Willmore. Moreover, we describe a phenomenon first observed in [3]: there are (globally smooth) immersions of compact surfaces that are constrained Willmore after removing a finite number of points, but not constrained Willmore as compact surfaces. This phenomenon seems to be typical for variational problems with conformal constraint: Example 8 in Section 3 shows that the same can be observed in the case of constrained minimal surfaces.
In the appendix we show that for non strongly isothermic immersions of compact surfaces all infinitesimal conformal variations may be realized as derivatives of genuine conformal variations, where strongly isothermic means that the immersion admits a non-trivial, holomorphic quadratic differential with the property that the null directions of its real part are principal curvature directions of the immersion. As a consequence, a non strongly isothermic immersion of a compact surface is a critical point of a geometric functional under infinitesimal conformal variations if and only if it is critical under genuine conformal variations.

Critical Points of Variational Problems with Conformal Constraint
In this section we derive the Euler-Lagrange equation characterizing immersions of a compact Riemann surface that are critical points of a geometric functional under the constraint that the conformal structure is infinitesimally preserved. In the case of non-compact surfaces, the equation thus obtained is still a sufficient condition for critical points, but (as shown in Section 3) it is not necessary.
Let M be a Riemann surface (without boundary throughout) and denote by F a functional (for non-compact surfaces it is sufficient that the functional F is well defined on compact subsets of M ) on the space of immersions of M into some 3-dimensional Riemannian manifold (M ,ḡ). We suppose that F is invariant under diffeomorphisms of M , i.e., that it depends only on the geometry of the immersion and not on the parametrization. Moreover, we suppose that F is differentiable in the following sense: for every immersion f : M →M there exists a 2-form grad(F ) ∈ Ω 2 (M ) such that the derivative of F in direction of a compactly supported infinitesimal variation with u ∈ C ∞ 0 (M ) a compactly supported function, X ∈ Γ 0 (T M ) a compactly supported vector field, and ξ the positive unit normal, is given bẏ Note thatḞ depends only on the normal variation, because F is invariant under diffeomorphisms.
In order to derive the Euler-Lagrange equation for variational problems with conformal constraint, we need to investigate the effect of compactly supported infinitesimal variations on the conformal structure. For this purpose we use the fact that on oriented surfaces a conformal structure is the same as a complex structure and can be represented by an endomorphism field J ∈ Γ(End(T M )) with J 2 = − Id. Lemma 1. The change of the conformal structure induced by a compactly whereÅ denotes the trace free part of the Weingarten operator A. For a compactly supported infinitesimal tangential variationḟ = df (X) ∈ Γ 0 (T f M ) the conformal structure changes according to For arbitrary vector fields Y ∈ Γ(T M ) we have g(JY, JY ) = g(Y, Y ) and g(JY, Y ) = 0. Infinitesimally, this becomeṡ Using the equation forġ andÅ = 1 2 (A + JAJ), this implies (2.2). A compactly supported infinitesimal variationḟ = uξ + df (X) is called conformal if the infinitesimal change of the conformal structure induced bẏ f vanishes. By Lemma 1 this is equivalent to (2.4) 2uÅJ + L X J = 0.
Clearly, compactly supported infinitesimal conformal variations can be obtained by taking the derivativeḟ = d dt |t=0 f t of a genuine compactly supported conformal variation f t , i.e., a family f t : M →M of immersions depending smoothly on t ∈ I ⊂ R such that on the complement of a compact subset of M all f t coincide with f and f = f 0 . It should be noted that, in contrast to the case of arbitrary compactly supported infinitesimal variations, it is not true that every compactly supported infinitesimal conformal variation is the derivative of a genuine conformal variation: an example of an infinitesimal conformal variation that does not admit an extension to a genuine conformal variation is the infinitesimal variation of a planar open annulus f given by a rotational symmetric function u ≥ 0 with compact support. However, in the appendix we prove that for immersions of compact surfaces that are not strongly isothermic, every infinitesimal conformal variation is obtained from a genuine conformal variation. Accordingly, we call an immersion a constrained Willmore surface, constrained minimal surface or constrained volume-critical surface if it is constrained F -critical for F the Willmore functional W, the area functional A, or the enclosed volume functional V.
For our derivation of the Euler-Lagrange equation we need the adjoint operators to the operators assigning to a normal or tangential variation the change of the conformal structure. We first define the adjoint of the operator δ : between the space of quadratic differentials Γ(K 2 ) and Γ 0 (End − (M )), where we use the convention that, for b a bilinear form, . With respect to these pairings the adjoint operator δ * : Γ(K 2 ) → Ω 2 (M ) of δ is given by The spaces involved in the definition of δ * are visualized by the diagram where the vertical arrows indicate the pairings defined in (2.6) and (2.7).
The operator X → L X J assigning to a tangential variation the induced change of the complex structure can be interpreted as follows: under the canonical isomorphisms T M ∼ = K −1 and End − (T M ) ∼ =KK −1 , with K the canonical bundle of the Riemann surface M , the operator X → L X J is the usual∂-operator on vector fields. We therefore denote it by Its dual operator is as well a∂-operator which we denote by It coincides with the usual∂-operator on quadratic differentials. As in the case of normal variations, we visualize this by the diagram As an application of this language we obtain the following sufficient condition for constrained F-critical immersions.
For the case of compact surfaces the following theorem shows that this sufficient condition is also necessary. The theorem follows essentially from the fact that, by elliptic theory of the∂-operator (or the so called Weyl-Lemma), for compact surfaces the space H 0 (K 2 ) = ker(∂ * ) of holomorphic quadratic differentials is finite dimensional and im(∂) = ker(∂ * ) ⊥ .
Theorem 3. Let f : M →M be a conformal immersion of a compact Riemann surface. Then f is constrained F-critical if and only if there is a holomorphic quadratic differential q ∈ H 0 (K 2 ) such that This proves the theorem, because ker(∂ * ) is finite dimensional and therefore (δ * (ker(∂ * ))) ⊥⊥ = δ * (ker(∂ * )).
Remark 4. We refer to q as the Lagrange multiplier of the constrained Fcritical immersion, because grad(F) = δ * (q) can be interpreted as the condition that grad(F ) is orthogonal to the space of conformal immersions f : M →M , see the appendix for details.
We conclude the section by briefly comparing three different notions of critical points of a geometric functional F on the space of immersions under the constraint that the conformal structure is fixed: i) The immersions satisfies the Euler-Lagrange equation ii) The immersion is constrained F-critical, i.e., it is critical under all infinitesimal conformal variations with compact support. iii) The immersion is F -critical under all genuine conformal variations with compact support.
Proposition 2 shows that i) always implies ii) and, by Theorem 3, if the underlying surface is compact i) and ii) are equivalent (we will see in Example 11 below that this is not the case for non-compact surfaces). By definition, ii) always implies iii). Corollary 21 in the appendix shows that ii) and iii) are equivalent for immersions of compact surfaces that are not strongly isothermic.

Constrained Minimal and Volume-Critical Surfaces
In this section we discuss examples of constrained minimal surfaces and constrained volume-critical surfaces, the critical points of area A and enclosed volume V under compactly supported infinitesimal conformal variations. These examples reveal several remarkable properties of solutions to variational problems under the constraint that the admissible variations preserve the conformal structure: • Example 9 shows that there are immersions of compact surfaces that are not constrained F-critical, but become constrained F-critical after removing a finite number of points. • Example 10 and 12 show that constrained minimal immersions are not necessarily analytic. This reflects the fact that the Lagrange multiplier appearing in the Euler-Lagrange equation for constrained minimal surfaces destroys its ellipticity. • Example 11 shows that, if the underlying Riemann surface is noncompact, constrained F-critical immersions do not necessarily admit a Lagrange multiplier q ∈ H 0 (K 2 ) such that grad(F) = δ * (q).
Lemma 5. Let f : M →M be an immersion of a Riemann surface M into a 3-dimensional Riemannian manifold (M ,ḡ). Under the compactly supported infinitesimal normal variationḟ = uξ, the area and volume functional change according tȯ where H denotes the mean curvature and dσ the area element of the immersion. Equivalently, the gradients of the functionals are the 2-forms A proof of this lemma is left to the reader. When considering the functionals A and V without the conformal constraint they behave rather differently: while V has no critical points at all, the critical points of the functional A are the minimal immersions into (M ,ḡ). In Euclidean space there exist no immersions of compact Riemann surfaces that are constrained minimal or constrained volume-critical, because homotheties do not change the conformal structure but scale A and V. In contrast to this, there are compact constrained minimal and constrained volume-critical surfaces in the 3-sphere.
Example 7 (Homogeneous Surfaces in Space Forms). Every non totally umbilic homogeneous surface in a space form of dimension 3 is constrained minimal and constrained volume-critical. This can be seen as follows. Because homogeneous surfaces have constant mean curvature, the Codazzi equation implies that the Hopf differential, i.e., the unique quadratic differential Q ∈ Γ(K 2 ) that satisfiesII = (Q +Q)ξ, whereII denotes the trace free second fundamental form, is holomorphic. From Re(Q) = 1 2 g(Å , ), A 2 = (H 2 − G) Id and dσ = 1 2 g(J ∧ ) follows that the Hopf differential satisfies where G denotes the Gaussian curvature, i.e., G = det(A). Since H 2 − G is a nonzero constant, there are λ 1 and λ 2 ∈ R such that grad(A) = λ 1 δ * (Q) and grad(V) = λ 2 δ * (Q), which, by Proposition 2, proves the statement. The non totally umbilic homogeneous surfaces in space forms are isometric to the following examples: the tori (u, v) → (r 1 e iu , r 2 e iv ) with r 2 1 + r 2 2 = 1 in S 3 ⊂ C 2 , the cylinders (u, v) → (r cos(u), r sin(u), v) in R 3 and the cones in the half space model of hyperbolic 3-space whose cusp lies on the infinity boundary and whose axis of rotation is perpendicular to the infinity boundary.
The exclusion of totally umbilic surfaces in the above statement is necessary: every piece of the plane is minimal but not constrained volume-critical, because every infinitesimal deformation with compact support is conformal. For the same reason, every piece of the round sphere is neither constrained minimal nor constrained volume-critical.
Example 8 (Discs of revolution). Let f : ∆ → R 3 be a conformal immersion of the open unit disc into R 3 that, on the punctured disc ∆\{0}, parametrizes a surface of revolution. Then every compactly supported variation of f that preserves the rotational symmetry is conformal.
The reason is that for immersions of the disc with rotational symmetry, the meridian curve has infinite length in the hyperbolic half plane whose boundary is the axis of rotation, and, moreover, such immersions are conformal if and only if, with respect to polar coordinates z = e x+iy , the meridian curve is parametrized by hyperbolic arc length. Thus, deforming the profile curve one can decrease A and V and therefore: An immersion of a disc with rotational symmetry is never constrained minimal nor constrained volume-critical, unless it is a planar disc and therefore minimal.
Example 9 (Spheres of revolution). Figure 2 shows four constrained minimal surfaces with rotational symmetry [17]. These surfaces can be smoothly extended to compact embedded spheres which are not constrained minimal. The preceding argument shows that only after removing both points on the axis of revolution one obtains constrained minimal surfaces. Interestingly, the round sphere with poles removed can be approximated by such punctured constrained minimal spheres with rotational symmetry although, by Example 8, a piece of the sphere itself is never constrained minimal.
Example 10 (Cylinders over plane curves). This example shows that, in contrast to minimal surfaces, constrained minimal surfaces are not necessarily analytic. Every cylinder over a plane curve is constrained minimal. To see this, assume that the plane curve γ(t) = (x(t), y(t)) is parametrized with respect to arc length. Then the cylinder f (u, v) = (x(u), y(u), v) is an isometric immersions of the plane with Weingarten operator A = −κ 0 0 0 and mean curvature H = − 1 2 κ where κ denotes the curvature of γ. For the holomorphic quadratic differential q = dz 2 with z = u + iv we have δ * (q) = −4κ dσ and therefore grad(A) = −2H dσ = κ dσ = − 1 4 δ * (q) which, by Proposition 2, proves the statement.
Example 11 (Constrained minimal surface without Lagrange multiplier q). We give now an example of a constrained minimal immersion that does not admit a holomorphic quadratic differential q with grad(A) = δ * (q). This shows that Theorem 3 does not hold in the non-compact case. The example is constructed as follows: take a planar domain as in Figure 3 and bend the hatched areas upwards such that all 3 fingers of the surface are cylinders over plane curves lying in three different planes perpendicular to the article.
As shown in Example 10 the surface thus obtained is locally constrained minimal and locally admits a holomorphic quadratic differential q such that the Euler-Lagrange equation grad(A) = δ * (q) is satisfied. It remains to be proven that the surface is globally constrained minimal but does not admit a global holomorphic quadratic differential q such that grad(A) = δ * (q).
To prove the first claim, we have to check that M grad(A)u = 0 for every compactly supported infinitesimal conformal variationḟ = uξ + df (X). On the white planar area grad(A) vanishes and δ(ũ) vanishes for all normal variationsũ. Hence, without loss of generality we may assume that u has support in a small neighborhood of the hatched areas: changing u on the white domain to achieve this would neither change the integral M grad(A)u nor would it destroy conformality of the variation. But this proves M grad(A)u = 0, because the 3 fingers of the surface are constrained minimal by Example 10.
The fact that there is no global holomorphic quadratic differential q such that grad(A) = δ * (q) can be seen as follows: as in Example 10, denote by q j , j = 1, ..., 3 the squares of the differentials of the cylinder coordinates on the three fingers. On each finger we then have grad(A) = − 1 4 δ * (q j ) for the respective j. Moreover, if there was a global holomorphic quadratic differential q with grad(A) = δ * (q), then on each finger there had to be t j ∈ R such that −4q = q j + t j iq j , which is impossible.
Example 12 (Hopf cylinders in S 3 ). All Hopf cylinders in S 3 , in particular all Hopf tori, are constrained minimal and constrained volume-critical. A surface in S 3 that is the preimage π −1 (γ) of an immersed curve γ in S 2 under the Hopf fibration π : S 3 → S 2 is called a Hopf cylinder or, if γ is closed, a Hopf torus. Taking arc length parameters of a horizontal lift of γ and of the Hopf fibers yields an isometric parametrization of the Hopf cylinder. With respect to the corresponding coordinates z = x + iy, the Weingarten operator takes the form A = −2κ −1 −1 0 , cf. [12], and the mean curvature H = −κ, where κ denotes the curvature of γ in S 2 . The holomorphic quadratic differential q = dz 2 satisfies δ * (q) = −8κdσ and δ * (iq) = 8dσ. By Lemma 5 and Proposition 2 this proves the claim. Using Theorem 15 of the next section one gets that a Hopf torus is constrained Willmore if and only if there are constants a, b ∈ R such that This equation means that the generating closed curve γ is a critical point of γ κ 2 ds for variations fixing the length and the enclosed area, i.e., γ is a generalized elastic curve, cf. [11] . This reflects the fact, see [12], that the Hopf torus generated by γ has Willmore energy W = π γ (κ 2 + 1)ds and that fixing the conformal type of a Hopf torus means fixing the length L and enclosed area A of γ, because the Hopf torus is isometric to R 2 /Γ with Γ generated by (2π, 0) and (A/2, L/2).

Constrained Willmore Surfaces
The Willmore energy or elastic bending energy was suggested as a global invariant of surfaces by T. Willmore [20] in 1965, but appears already in earlier work of S. Germain [9] and W. Blaschke [1]. Its applications range from the biophysics of membranes to string theory, where it was introduced by W. Helfrich and A. M. Polyakov, respectively. The Willmore energy of an immersion f : M →M into a 3-dimensional Riemannian manifold (M ,ḡ) is Remark 13. The Willmore functional W and the functionals M (κ 2 1 + κ 2 2 )dσ and M (κ 1 − κ 2 ) 2 dσ, with κ 1 , κ 2 denoting the principal curvatures, differ by multiples of M Kdσ only and therefore have the same critical points. Thus, the Willmore functional can be seen as a measure of the total amount of principal curvature and the defect from being totally umbilic.

Definition.
A conformal immersion f : M →M of a Riemann surface M into a 3-dimensional Riemannian manifoldM is constrained Willmore if it is a critical point of the Willmore functional W under compactly supported infinitesimal conformal variations.
For simplicity we restrict now to the case that (M ,ḡ) is a manifold of constant sectional curvature. Because both the functional and the constraint are invariant under conformal changes of the metricḡ, the notion of constrained Willmore surfaces for all three space forms coincides and could as well be considered from a purely Möbius geometric viewpoint, see e.g. [7].
A proof of the following theorem can be found in [19].

Theorem 14. A compactly supported infinitesimal variationḟ = uξ+df (X)
of an immersion f : M →M into a 3-dimensional space form changes the Willmore functional according to In particular, the gradient of the Willmore functional is the 2-form  For non-compact surfaces, the existence of a holomorphic quadratic differential q ∈ H 0 (K 2 ) that satisfies (4.5) implies that the surface is constrained Willmore.
It should be noted that Willmore surfaces correspond to the case that q = 0 in (4.5). In fact, even for immersions of non-compact surfaces the Euler-Lagrange equation of Willmore surfaces is a necessary condition: an immersion of a (possibly open) surface into a 3-dimensional space form is Willmore if and only if ∆H + 2H(H 2 − G) = 0.
In addition to being constrained Willmore, constant mean curvature surfaces in space forms are isothermic, which means that away from umbilics they admit conformal curvature line coordinates. Hence, constant mean curvature surfaces belong to two important surface classes of Möbius geometry: that of constrained Willmore surfaces and that of isothermic surfaces. For a constrained Willmore immersion of a torus into the conformal 3-sphere, J. Richter conversely proved (cf. [7]) that, if the immersion is also strongly isothermic as defined in the appendix, it has constant mean curvature with respect to some space form subgeometry. This generalizes a theorem of G. Thomsen [1]: a surface is Willmore and isothermic if and only if it is minimal in some space form.
In contrast to Thomsen's theorem, the global assumption in Richter's result is essential: an example, which is due to F. Burstall, of an isothermic, constrained Willmore surface that does not have constant mean curvature in some space form is the cylinder over the plane curve 1 in Figure 4. A fundamental property of constrained Willmore surfaces is the existence of an associated family depending on a spectral parameter, see e.g. [7]. This relation to the theory of integrable systems has important consequences. For example, in [4] the associated family of flat connections is used in order to show that to every constrained Willmore torus one can assign a compact Riemann surface, the so called spectral curve (a different approach to proving this is described in [16]). This fact allows to parameterize constrained Willmore tori explicitly in terms of holomorphic functions.
The case of constrained Willmore immersions of the sphere is the only case where a complete classification (in the sense of a reduction to a simpler algebraic geometric problem) is known: because there is only one conformal structure on the sphere, all constrained Willmore spheres are Willmore and one can apply R. Bryant's result [5,6] according to which all Willmore spheres in the conformal 3-sphere are complete Euclidean minimal surfaces of finite total curvature with planar ends (and therefore algebraic). Bryant proved that the Willmore energy of Willmore spheres is quantized and that the possible Willmore energies are W ∈ 4π(N * \{2, 3, 5, 7}). From the integrable systems point of view it is interesting that Willmore spheres are examples of so called soliton spheres, see [2].
Based on the idea of smooth ends [3] for constant mean curvature 1 surfaces in hyperbolic 3-space one can construct a family of globally smooth conformal immersions of the sphere that, by Corollary 16, are constrained Willmore after removing a finite number of points -the "ends" at which the immersion touches the ideal boundary of hyperbolic space. These so called Bryant spheres with smooth ends are constrained Willmore as punctured spheres only, but not as compact surfaces, because then they had to be Willmore which is impossible by Bryant's result on Willmore spheres. Nevertheless, they obey the same quantization of the Willmore energy as Willmore spheres, i.e., W ∈ 4π(N * \{2, 3, 5, 7}), and they also are soliton spheres. Figure 5 shows a Bryant sphere with two smooth ends. The phenomenon that an immersion of a surface becomes constrained Willmore only after removing a finite number of points does not occur for Willmore surfaces, i.e., for the critical points of the Willmore energy under all variations. In the example of Bryant spheres with smooth ends, the holomorphic quadratic differential in the Euler-Lagrange equation has poles at the ends, which have to be removed in order to obtain a constrained Willmore surface. Because the left hand side in (4.5) is globally smooth, the ends have to be umbilic points.

Appendix. Infinitesimal and genuine conformal transformations
Our definition of constrained F-critical immersions in Section 2 demands that the immersion is critical for all infinitesimal conformal variations with compact support. Constrained F-critical immersions according to this definition are in particular critical under all genuine conformal variations with compact support. In this appendix we show that for non strongly isothermic immersions of compact surfaces it is conversely true that being critical with respect to genuine variations implies being critical for all infinitesimal conformal variations.
Definition. We call a conformal immersion f : M →M of a Riemann surface M into a 3-dimensional Riemannian manifoldM strongly isothermic if there exists a non-trivial holomorphic quadratic differential q ∈ H 0 (K 2 ) with the property that the null directions of Re(q) are principal curvature directions of f .
An alternative characterization of strongly isothermic surfaces is provided by the following proposition.
Proposition 17. Let f be a conformal immersion of a Riemann surface and let q ∈ H 0 (K 2 ) be a non-trivial holomorphic quadratic differential. The null directions of Re(q) are principle curvature directions of f if and only if δ * (q) = 0 for δ * as in (2.8). The zeros of q are then umbilic points of f .
Proof. Away from the isolated zeros of q one can choose local coordinates z = x + iy : M ⊃ U → C such that dz 2 = iq on U . Then Re(q) = 2dxdy and we have δ * (q) = 8F dx ∧ dy, where F denotes the off-diagonal term of the Weingarten operator A with respect to the coordinates (x, y). Thus, δ * (q) = 0 holds on U if and only the (x, y)-parameter lines are curvature lines of f . Furthermore, it follows that the zeros of q are umbilics of f . Remark 18. The proof of Proposition 17 shows that a strongly isothermic immersion is isothermic in the sense that, away from umbilics, it admits conformal curvature line coordinates.
Examples of strongly isothermic immersions are surfaces of revolution and constant mean curvature surfaces in space forms. For the latter, a holomorphic quadratic differential satisfying δ * (q) = 0 is given by q = iQ, where Q denotes the Hopf differential. on M by the group of diffeomorphisms of M that are homotopic to the identity. It is well known, see e.g. [8,18], that T (M ) is a finite dimensional smooth manifold and that the projection Φ : C(M ) → T (M ) is differentiable. The kernel of its differential at J is the image of X ∈ Γ(T M ) → L X J =∂ X, i.e., the space that, with respect to the pairing (2.7), is perpendicular to the finite dimensional vector space of holomorphic quadratic differentials of the Riemann surface (M, J): ker(dΦ J ) =∂(Γ(T M )) = H 0 (K 2 ) ⊥ .
An immersion f : M →M induces a unique complex structure J ∈ C(M ) compatible with the orientation and the induced metric. Let τ denote the composition of the map f → J with Φ. By Lemma 1 and (2.5), for every infinitesimal variationḟ = uξ + df (X) of f , the differential dτ f of τ at f is dτ f (ḟ ) = dΦ J • δ(u).
Theorem 20. Let f : M →M be an immersion of a compact Riemann surface. If f is not strongly isothermic, then every infinitesimal conformal variation of f is the derivative of a genuine conformal variation.