Calibrations for the Volume of Unit Vector Fields in Dimension 2

We use the theory of calibrations to write an equation of a minimal volume vector field on a given Riemann surface.


Introduction
Ziller have used the theory of calibrations to prove that the minimal volume unit vector fields defined on the 3-dimensional sphere are the Hopf vector fields, [10]. Inspired by this celebrated result we try to parallel its main ideas on the setting of an oriented Riemannian 2-manifold.
In [10] an appropriate calibration 3-form ϕ is found on the total space of the unit tangent sphere bundle π : T 1 S 3 −→ S 3 . Sections of this bundle are the unit vector fields. Applying the theory of calibrations of Harvey and Lawson ( [11]), the corresponding embedded 3-dimensional submanifolds calibrated by ϕ are precisely the Hopf vector fields. They minimize volume globally in a unique homology class, namely the canonical class of the base S 3 which lies in The question of minimality in dimension 2 has been studied before and there are several important results eg. in [4][5][6][7]9,14]. A simple differential equation characterizing the 2-dimensional variational problem, ie. whose solutions are precisely the germs of minimal volume vector fields on a Riemannian surface, is partly missing. Such equation must of course have a space of solutions compatible with the base manifold isometries. Not to mention the topological obstructions to the existence of a unit vector field, such as Euler characteristic zero.
Let M denote a Riemann surface endowed with a unit norm class C 2 vector field X. By the original definition in [10], or [9], we have where g S is the Sasaki metric on T 1 M and e 0 , e 1 is any local orthonormal frame on M . Indeed, ∇ e0 X 2 + ∇ e1 X 2 is a frame invariant quantity. We denote by π : T 1 M −→ M the unit tangent sphere bundle of M , perhaps with boundary. T 1 M is a Riemannian submanifold of T M of metric contact type with contact 1-form e 0 , this is, a contact manifold with the metric g S and contact structure induced from the geodesic spray. N.B.: the present e 0 is a 1-form on the manifold T 1 M .
For M oriented, there exists a natural differential system of 1-forms e 0 , e 1 , e 2 globally defined on T 1 M -the well-known Cartan structural equations, which we like to see as the simplest case of a fundamental differential system introduced in [2].
It is clear how to find the global frame e 0 , e 1 , e 2 at each point u ∈ T 1 M such that π(u) = x ∈ M . The global vector field e 0 is the tautologial horizontal vector field, ie. the horizontal lift of u ∈ T x M to T u (T 1 M ), also known as geodesic spray vector field. Then e 1 is such that e 0 , e 1 is a well-defined direct orthonormal basis of horizontal vector fields. Finally e 2 is the vertical dual of e 1 , tangent to the S 1 fibres. Let us remark the dual of e 0 is the tautologial vertical vector field ξ, which gives T ( In this article we study the 2-forms ϕ = b 2 e 0 ∧e 1 +b 1 e 2 ∧e 0 +b 0 e 1 ∧e 2 on T 1 M which define a calibration, ie. a comass 1 and closed 2-form, seemingly appropriate for the study of unit vector fields on M .
Next, we endeavour to identify the existence of ϕ with that of a minimal vector field X. Since M is not required to satisfy any further condition, our main theorem becomes a local result; we deduce an equation of a minimal volume vector field in any bounded domain: letting A denote the C-valued function given essentially by the components of ∇ · X, we must have, on a conformal chart z of M , The function A is indeed globally defined.
In another article, [3], we have shown that the imaginary part of this Cauchy-Riemann equation is indeed the necessary condition for minimal volume, deduced by Gil-Medrano and Llinares-Fuster in [8] to coincide with the critical points of the functional (1). Our equation is a sufficient condition for minimality arising from a certain type of calibrations. A second main result is the solution of (2) on a manifold of constant negative sectional curvature K < 0.
Let us further remark that the existence of a parallel vector field, clearly an absolute minima for the volume functional, starts as a local metric issue. More precisely, the Riemann curvature tensor applied to X would imply flatness. On the other end, the theory of calibrations applies to manifolds with boundary and thus there is a path through geometry and topology to pursue.

Minimal Volume Over a Surface
We start by recalling some general ideas in any dimension.
Let (M, ) be an oriented Riemannian manifold of dimension n + 1. Recall the well-known metric and contact structure e 0 on the total space of π : T 1 M −→ M . As usual, we let e 0 denote the geodesic spray, i.e. the unit norm horizontal vector field such that dπ u (e 0 ) = u ∈ T π(u) M, ∀u ∈ T 1 M . Using duality of the Sasaki metric yields that, for any v ∈ T u (T 1 M ), we have e 0 (v) = e 0 , v = u, dπ(v) . It is thus easy to prove that e 0 is the restriction of the Liouville form pulled back from T * M to T M (we use musical isomorphism notation throughout; eg. e 0 = e 0 ).
Let ϕ be a degree n + 1 calibration defined on the manifold T 1 M . Let X ∈ X M be a class C 2 unit norm vector field on M and let us fix the H n+1 (T 1 M, R) homology class of X(M ). Since ϕ ≤ vol X , the minimal volume unit vector fields, within the same homology class, are those for which ϕ = vol X , ie. restricted to the Riemannian submanifold X(M ) the calibration coincides with the submanifold Riemannian volume. In other words, recalling vol X from [9,10], such unit vector fields are those for which X * ϕ = vol X ; corresponding to the so-called ϕ-submanifolds which are sections of π : T 1 M −→ M . Then the fundamental relation from [11] follows: for any unit X ∈ X M , The theory of calibrations holds for submanifolds-with-boundary of the calibrated manifold. So we may well focus on a fixed open subset, a domain Ω ⊂ M perhaps with non-empty boundary, and seek an immersion X : Ω → T 1 M giving a ϕ-submanifold. We remark that prescribing boundary values for X on a compact ∂Ω implies that certain moment conditions are satisfied, cf.
Recalling a useful notation π * , π for the horizontal, respectively vertical, canonical lift, cf. [1], we have the 'horizontal plus vertical' decomposition dX(Y ) = π * Y + π (∇ Y X) in T T M. Also, we may find local adapted frames e 0 , e 1 , . . . , e n , e 1+n , . . . , e 2n , indeed local oriented orthonormal moving frames on T 1 M with the e i+n the vertical mirror of the horizontal e i , i = 1, . . . , n.
π * X is the horizontal lift of X and thus π * X = e 0 restricted to the submanifold X(M ) ⊂ T 1 M . The horizontal e i project through dπ to a frame A ij e j+n (4) for i = 0, 1 . . . , n, where A ij = ∇ ei X, e j . Since X = 1, A i0 = 0. This implies that X * e 0 = X . We now suppose M is a Riemann surface and π : T 1 M −→ M is the unit circle tangent bundle. Let us search for the calibration ϕ.
As it is well-known, T 1 M is parallelizable: we have the global direct orthonormal frame e 0 , e 1 , e 2 , with e 2 the vertical mirror of e 1 . In particular π * vol M = e 0 ∧ e 1 .
The following formulas are well-known, cf. [2] and the references therein: where K = R(e 0 , e 1 )e 1 , e 0 is the Gauss curvature. Notice K is the pullback of a function on M and it is not necessarily a constant. Let us assume the abbreviation b = π * b for any given real function b on M ; this gives a function on T 1 M of course constant along the fibres.
This is a 2-calibration if it has comass 1 and dϕ = 0. Recall from [11] that comass 1 is defined by where ϕ * u = sup ϕ u , ζ : ζ is a unit simple 2-vector at u .

Proposition 1. The 2-form ϕ on T 1 M has comass 1 if and only if
The form ϕ is closed if and only if the function b 1 + √ −1b 0 is holomorphic.
Proof. For the first part, first, it is easy to deduce ϕ(u, v) The definition of comass 1 together with Cauchy inequality yields |(b 0 , b 1 , b 2 )| ≤ 1 and the requirement that the above supremum is 1. For the second part of the theorem, we note that db i (e 2 ) = 0, ∀i = 0, 1, 2, by construction. And therefore dϕ = 0 is equivalent to the condition db 1 (e 1 )+db 0 (e 0 ) = 0. As the frame varies along a single fibre we find Cauchy-Riemann equations. Hence the result.
Remark. There seems to be no advantage, later on, in considering general functions on T 1 M ; even if the equation db 2 (e 2 ) + db 1 (e 1 ) + db 0 (e 0 ) = 0 looks quite charmful. It is interesting to observe, by the way, that any two functions Vol. 78 (2023) Calibrations for the Volume of Unit Vector Fields Page 5 of 9 114 f, g on M , such that sup{f 2 + |∇g| 2 } = 1, define a calibration 2-form by f e 0 ∧ e 1 + dg ∧ e 2 .
Let us now seek for a calibration ϕ on T 1 M , intended for the new study on M .
Again let X ∈ X M have unit norm and be defined over (a domain contained in) M . We then have a unique vector field Y on the same domain such that X, Y is a direct orthonormal frame.
The differential of the map X is given by the identities dX(e 0 ) = e 0 + A 01 e 2 , dX(e 1 ) = e 1 + A 11 e 2 , with usual notation A ij = ∇ ei X, e j . In other words, abbreviating A i1 = A i , Recalling definition (1), we find On the other hand, Theorem 1. Suppose there exists a unit vector field X on M such that the C-valued function A = A 1 + √ −1A 0 satisfies the following equation, on a conformal chart z of M : corresponding to A/ 1 + |A| 2 being holomorphic. Then there exists a calibration ϕ on the total space of T 1 M for which X is a ϕ-submanifold. In particular, X is a unit vector field on M of minimal volume.
Proof. By Proposition 1, we search for a map b = (b 0 , b 1 , b 2 ) from M into the Euclidean ball of radius 1 and having a limit value in the S 2 boundary. Let us also denote A = (−A 0 , −A 1 , 1). Now, by (11) and (12), condition X * ϕ ≤ vol X is equivalent to b, A ≤ | A|.
Since we wish equality and since | b| ≤ 1 ≤ | A|, there is a unique solution: The corresponding ϕ is globally defined, with the same domain as X. Finally, one must have ϕ closed. Hence the function A/ 1 + |A| 2 must be holomorphic; and a straightforward computation leads to (13).