A new approach to the Fraser-Li conjecture with the Weierstrass representation formula
aa r X i v : . [ m a t h . DG ] S e p THE FRASER-LI CONJECTURE AND THE LIOUVILLE TYPEBOUNDARY VALUE PROBLEM
JAEHOON LEE AND EUNGBEOM YEON
Abstract.
In this paper, we provide a sufficient condition for a curve on a surfacein R to be given by an orthogonal intersection with a sphere. This result makesit possible to express the boundary condition entirely in terms of the Weierstrassdata without integration when dealing with free boundary minimal surfaces in aball. Moreover, we show that solving the Fraser-Li conjecture in the case of realanalytic boundaries is equivalent to proving that a solution of the Liouville typeboundary value problem in an annulus has radial symmetry. This suggests a newPDE theoretic approach to the Fraser-Li conjecture. Introduction
A topic of free boundary minimal surfaces has been a very active field of research.One of the most widely accepted conjectures is the following.
Conjecture 1.1 (Fraser and Li, [3]) . The critical catenoid is the only embedded freeboundary minimal annulus in B , up to rigid motions. The above open question deals with the free boundary analog of the Lawson con-jecture in which the Clifford torus is the only embedded minimal torus in S . Law-son’s conjecture was proved in [1] by applying a maximum principle to a two-pointfunction obtained from the geometric observation of the inner and outer spheres ofa surface in S . It is tempting to check whether a similar method holds, but twoboundary components of the surface make it difficult to apply a maximum principletype method to the Fraser-Li conjecture.Instead, another useful tool, called the Weierstrass representation formula , is usedin the classical minimal surface theory:Re
Z (cid:20)
12 (1 − g ) ω, i g ) ω, gω (cid:21) , where g is a meromorphic function and ω is a holomorphic one-form on a Riemannsurface. Since the Weierstrass representation formula is presented in the integralform, it is difficult to translate all the information related to free boundary into thedata g and ω .Meanwhile, two facts on boundary curves can be obtained from the free boundarycondition. More generally, let Σ be a surface in R that meets a sphere orthogonally Mathematics Subject Classification.
Key words and phrases.
Free boundary minimal surfaces, the Weierstrass representation formula,the Fraser-Li conjecture, the Liouville equation. along a curve Γ. As implied by the Terquem-Joachimsthal theorem [6], Γ is acurvature line on Σ. Moreover, as the conormal vector of Σ along the curve coincideswith the unit normal vector to the sphere, geodesic curvature along Γ computed onΣ is identical with normal curvature on the sphere equal to 1. It turns out that theconverse is also true. Indeed, we gained the below observation as the two conditionsare so powerful:
Proposition 1.2 (Proposition 3.3 in Section 3) . Let Γ be a compact real analyticcurve on a surface Σ in R . Suppose that Γ contains only finitely many umbilicpoints of Σ . If Γ is a line of curvature and has constant geodesic curvature c on Σ , then there exists a sphere S of radius | c | (if c = 0 , it means a plane) where Σ intersects S orthogonally along Γ . When Γ is a piecewise real analytic curve, thesame result can be obtained once a sphere is replaced by a union of spheres. This proposition generalizes the well-known fact that if a surface contains a prin-cipal geodesic, then it meets the plane containing the geodesic perpendicularly.Since curvature lines on a nonplanar minimal surface are real analytic and containpossibly a finite number of umbilic points, the proposition is applied to both a curveon the interior of a minimal surface and real analytic boundaries. It should be notedthat the conditions in the proposition can be expressed in terms of the Weierstrassdata ( g, ω ) without undergoing a process of integration. In this way, the propositionsolves difficulties associated with using the representation formula for free boundaryminimal surfaces in a ball.After applying the above proposition and the Weierstrass representation formula,we found a close relationship between the Fraser-Li conjecture and the Liouvilleequation. More specifically, we demonstrated that solving the Fraser-Li conjecturein real analytic boundaries is equivalent to proving that the solution of the followingLiouville equation, whenever it exists, shows a radial symmetry for arbitrary
R > < ǫ <
1, and a nonzero real constant C : ∆ v + 2 C e v = 0 in A (1 − ǫ, R + ǫ ) , ∂v∂n = 2 e − v − | z | = 1 , ∂v∂n = R e − v + R if | z | = R, R π R R r e − v d r d θ = R π e − v (1 ,θ ) d θ + R π Re − v ( R,θ ) d θ. (E[ R , ǫ , C ])The main idea is to construct a free boundary minimal annulus in a ball from thesolution of the equation above: Theorem 1.3 (Theorem 4.1 in Section 4) . Let v be a solution of E[ R , ǫ , C ] . Thereexist a minimal immersion X : A (1 − ǫ, R + ǫ ) → R and a unit sphere where theyintersect orthogonally along level curves | z | = 1 and | z | = R . Moreover, the solution v satisfies v = log | z | λ , where λ is a metric factor induced by X . This equivalence enables us to approach the Fraser-Li conjecture by using PDEtheoretic methods. It can be remarked that Jim´enez [5] solved the Liouville equationin an annulus to classify constant curvature annuli. Although it was successfulto deal with the boundary in [5] by considering the Schwarzian derivative of a
HE FRASER-LI CONJECTURE AND THE LIOUVILLE EQUATION 3 meromorphic function (for instance, h in (4.1)), the free boundary condition doesnot work well with this method. Moreover, if we translate the conjecture into aproblem of the Gauss map by using the Weierstrass data, it is easy to see that theboundary condition depends not only on a geometric property, but on a specificparametrization. In this regard, only using the holomorphic function theory is adifficult approach. The equivalence suggests a new research direction to the Fraser-Li conjecture.The paper is organized as follows. In Section 2, the Weierstrass representationand the well-known fact on a characterization of spherical space curves are reviewed.In Section 3, we prove that necessary conditions obtained from the orthogonal in-tersection with a sphere actually become the sufficient condition for the existenceof a sphere that meets a surface orthogonally. In the final section, the equivalencebetween the Fraser-Li conjecture and the Liouville equation is addressed. Acknowledgements
The authors would like to express their gratitude to Jaigyoung Choe for helpfulcomments concerning the application of the main theorem. This work was supportedin part by NRF-2018R1A2B6004262.2.
Preliminaries
Weierstrass representation.
We recall the Weierstrass representation of aminimal surface in R . Since coordinate functions are harmonic, it can be expressedas a conformal harmonic immersion from a Riemann surface C into R :Re Z (cid:20)
12 (1 − g ) ω, i g ) ω, gω (cid:21) , where g is a meromorphic function and ω is a holomorphic one-form on C such that g has order n pole at p ∈ C if and only if ω has order 2 n zero at p ∈ C . Note that g corresponds to the Gauss map and ( g, ω ) is called a Weierstrass data .The induced metric is d s = 14 (1 + | g | ) | ω | and the second fundamental form is given byRe { d g · ω } . Note that to obtain a well-defined immersion from C , it should satisfy the periodcondition: Re Z δ (cid:20)
12 (1 − g ) ω, i g ) ω, gω (cid:21) = 0for every closed curves δ on C . Otherwise, the Weierstrass data gives rise to aperiodic minimal surface. J. LEE AND E. YEON
Characterization of spherical space curves.
We may recall an elementaryfact on spherical curves, which is one of the main ingredients of the paper. Since acurve in R is determined (up to a rigid motion) by the curvature and torsion, it ispossible to characterize spherical space curves in terms of them as follows.Let α = α ( t ) : I → R be an arclength parametrized curve such that τ ( t ) = 0and κ ′ ( t ) = 0 for all t ∈ I . Here, κ is the curvature and τ is the torsion. Then α ( I ) ⊂ S ( R ) if and only if (cid:18) κ (cid:19) + (cid:18)(cid:18) κ (cid:19) ′ (cid:19) (cid:18) τ (cid:19) = R . (2.1)Moreover, if α ( I ) ⊂ S ( R ), then the unit normal vector of the sphere can be ex-pressed as − Rκ n + κ ′ Rκ τ b, (2.2)where n and b are normal and binormal vector of α , respectively.3. Sufficient condition to meet a sphere orthogonally
In this section, we discuss a sufficient condition for a surface to meet a sphereorthogonally. We first prove two lemmas that describe the local nature of orthogonalintersections. Then by combining two lemmas and some global arguments, we provethe main result.
Lemma 3.1.
Let Γ be a line of curvature on a surface Σ with the constant geodesiccurvature c = 0 . Moreover, assume that Γ has a non-vanishing torsion as a curvein R and does not contain any umbilic points. Then there exists a sphere of radius | c | such that it intersects Σ orthogonally along Γ .Proof. Let t be the unit tangent vector of Γ. Let us denote the unit conormal vectoralong Γ as ν and the unit normal vector of Σ as N such that { N, t, ν } is positivelyoriented in R . Also we write the unit normal and binormal vectors of Γ by n and b , respectively. Here, b is given by t ∧ n .As { N, ν } and { b, n } form oriented orthonormal bases for the normal plane of thecurve, we may write ( b = cos θN + sin θνn = − sin θN + cos θν (3.1)for some function θ defined on Γ.Let κ and τ be the curvature and torsion of Γ as a space curve. Since Γ is a lineof curvature on Σ, h∇ t N, ν i = −h N, ∇ t ν i = 0 (3.2)and we obtain from (3.1) that h∇ t N, n i = cos θ h∇ t N, ν i = 0 , h∇ t ν, n i = − sin θ h∇ t ν, N i = 0 . HE FRASER-LI CONJECTURE AND THE LIOUVILLE EQUATION 5
Here, ∇ means the Riemannian connection in R . Then it follows from the Frenet-Serret formulas and (3.1) that − τ = h∇ t b, n i = h∇ t (cos θN + sin θν ) , n i = ( D t θ ) h− sin θN + cos θν, n i + cos θ h∇ t N, n i + sin θ h∇ t ν, n i = D t θ, where D t denotes the directional derivative. Hence τ = − D t θ .Again by (3.1) and the Frenet-Serret formulas, the geodesic curvature of Γ com-puted on Σ is given by c = h∇ t t, ν i = h κn, ν i = κ cos θ. (3.3)Since Γ does not contain umbilic points and c = 0, we have κ = 0 , cos θ = 0 , sin θ = 0 . The constancy of the geodesic curvature along Γ implies that0 = D t ( κ cos θ ) = ( D t κ ) cos θ − κ sin θ ( D t θ ) = ( D t κ ) cos θ + τ κ sin θ, where we used τ = − D t θ in the last step. Therefore we obtain D t κτ κ = − tan θ = 0 . (3.4)Now we compute 1 κ + (cid:18) D t (cid:18) κ (cid:19)(cid:19) τ = 1 κ (cid:18) D t κ ) τ κ (cid:19) = 1 κ (1 + tan θ )= 1 c so that the characterization result in Section 2 shows that Γ lies on a sphere ofradius | c | . Moreover, we deduce from (cid:28) − κ n + D t κκ τ b, N (cid:29) = − κ h n, N i + 1 κ · D t κτ κ h b, N i = 1 κ sin θ − κ tan θ · cos θ = 0that the sphere intersects Σ orthogonally. (cid:3) Next, we have the following lemma for a vanishing torsion case:
Lemma 3.2.
Let Γ be a line of curvature on a surface Σ with the constant geodesiccurvature c = 0 , which does not contain any umbilic points. If the torsion of Γ isidentically zero, then Γ is a part of a circle. Therefore Σ is orthogonal to a sphereof radius | c | along Γ . J. LEE AND E. YEON
Proof.
We may use the same notation as in the proof of Lemma 3.1. Since the torsionis identically zero, Γ lies on a plane orthogonal to b . Moreover, τ = − D t θ = 0 impliesthat Σ has a constant contact angle θ with the plane along Γ. Now (3.3) shows thatΓ has a constant curvature on the plane and the result follows. (cid:3) Combining Lemma 3.1 and 3.2, we obtain the proposition:
Proposition 3.3.
Let Γ be a compact real analytic curve on a surface Σ in R .Suppose that Γ contains only finitely many umbilic points of Σ . If Γ is a line ofcurvature and has constant geodesic curvature c on Σ , then there exists a sphere S of radius | c | (if c = 0 , it means a plane) where Σ intersects S orthogonally along Γ .When Γ is a piecewise real analytic curve, the same result can be obtained once asphere is replaced by a union of spheres.Proof. It is well-known that if there exists a principal geodesic on Σ, then it is aplane curve and Σ intersects that plane orthogonally. So we may assume that thegeodesic curvature is nonzero, i.e., c = 0.We will consider the real analytic case first. Since Γ contains finitely many umbilicpoints of Σ, it divides into finite pieces by umbilic points: Γ = Γ ∪ Γ ∪ · · · ∪ Γ k .Moreover, by the analyticity, the torsion of Γ satisfies either one of the following: itis identically zero, or it vanishes only at finitely many points.In the first case, we may apply Lemma 3.2 to each Γ i and obtain a sphere S i ofradius | c | for each Γ i . Then we can conclude that all S i ’s must be the same by thecontinuity of the curvature of Γ.For the second case, each Γ i divides into finite pieces by torsion-vanishing points.Then Lemma 3.1 implies that we can find a sphere of radius | c | for each piece, andagain by the continuity, we complete the proof.If Γ is a piecewise real analytic curvature line, then its singular points can occurat umbilic points. In this case, it is not possible to use the continuity argumentat singular points to obtain only one sphere as in the above. Instead, a similarargument shows that there exists a union of spheres that intersects Σ orthogonallyalong Γ. (cid:3) The Fraser-Li conjecture and the Liouville equation
In this section, the equivalence between the Fraser-Li conjecture and the Liouvilleequation will be studied. This equivalence not only suggests a possible new approachto the Fraser-Li conjecture but also provides an interesting boundary value problemof partial differential equations to study.Let F : A (1 , R ) → R be a free boundary minimal annulus in a ball B with realanalytic boundaries, where A (1 , R ) := { z ∈ C | < | z | < R } . The real analyticityimplies that F can be extended as a minimal immersion defined in a slightly largerannulus A (1 − ǫ, R + ǫ ) for some ǫ >
0. By abuse of notation, we may also denoteit by F : A (1 − ǫ, R + ǫ ) → R . HE FRASER-LI CONJECTURE AND THE LIOUVILLE EQUATION 7
By using the Hopf differential, one can show that the second fundamental formis given by σ = Re (cid:8) C z d z (cid:9) for some real constant C . From this we have | σ | = 2 C r λ . Here, λ is the metric factor defined by ds = λ | d z | and r = | z | . Then Simons’identity ∆ log | σ | = − | σ | (cf. [2], p. 71) gives1 λ ∆ log 2 C r λ = − C r λ , where we used ∆ = 4 ∂∂z ∂∂ ¯ z .Now let v := log r λ . Since log r is a harmonic function, the Simons identityimplies that v satisfies the Liouville equation in A (1 − ǫ, R + ǫ ):∆ v + 2 C e v = 0 . Moreover, it follows from the free boundary condition that the geodesic curvatureof level curves r = 1 and r = R are equal to 1, which can be expressed as − rλ (cid:18) rλ ∂λ∂r (cid:19) = 1 if r = 1 , − rλ (cid:18) rλ ∂λ∂r (cid:19) = − r = R. Hence we obtain ∂v∂n = 2 e − v − r = 1 ,∂v∂n = 2 R e − v + 2 R if r = R, where n is the inner unit normal vector to A (1 , R ).Combining all things above, we observe that v gives rise to the solution of thefollowing Liouville type boundary value problem: ∆ v + 2 C e v = 0 in A (1 − ǫ, R + ǫ ) , ∂v∂n = 2 e − v − | z | = 1 , ∂v∂n = R e − v + R if | z | = R, R π R R r e − v d r d θ = R π e − v (1 ,θ ) d θ + R π Re − v ( R,θ ) d θ. (E[ R , ǫ , C ])The last equation comes from 2 | F ( A (1 , R )) | = | ∂F ( A (1 , R )) | .We now prove that the solution of E[ R , ǫ , C ] also gives rise to a free boundaryminimal annulus in a ball: Theorem 4.1.
Let v be a solution of E[ R , ǫ , C ] . There exist a minimal immersion X : A (1 − ǫ, R + ǫ ) → R and a unit sphere where they intersect orthogonally alonglevel curves | z | = 1 and | z | = R . Moreover, the solution v satisfies v = log | z | λ ,where λ is a metric factor induced by X . J. LEE AND E. YEON
Proof.
By applying the same method in [5], we observe that the solution v is givenby v = log 4 | h z | (1 + C | h | ) (4.1)for some locally univalent meromorphic function h in A (1 − ǫ, R + ǫ ).Let us consider a minimal immersion X : A (1 − ǫ, R + ǫ ) → R defined by theWeierstrass data ( g, ω ), where g = C h, ω = 1 z h z d z. Later we will prove that it has no real period so that X is well-defined.The induced metric is given by λ | d z | = 14 | ω | (1 + | g | ) = (1 + C | h | ) | h z | | z | | d z | , which implies that λ = (1 + C | h | ) | h z | | z | . Hence we obtain log 1 | z | λ = log 4 | h z | (1 + C | h | ) = v, and it follows easily from the second and third equations of E[ R , ǫ , C ] that thegeodesic curvature of level curves | z | = 1 and | z | = R are equal to 1. Moreover, thesecond fundamental form is Re { d g · ω } = Re (cid:26) C z d z (cid:27) so that each level curve of | z | is a line of curvature on the minimal surface.Now it is possible to apply Proposition 3.3. We first show that the Weierstrassdata ( g, ω ) does not have a real period. Indeed, if there was a real period, the datagives rise to a periodic minimal surface. Consider a minimal surface correspondingto the triple of the period. Then Proposition 3.3 implies that the level curve | z | = 1lies on a sphere. It is impossible since three points corresponding to the image of z = 1 are collinear, but every lines can only intersect a sphere in at most two points.Therefore there is no real period and the immersion X : A (1 − ǫ, R + ǫ ) → R iswell-defined.Again by Proposition 3.3, there exist unit spheres S O , centered at O , and S O ,centered at O , such that S O and S O intersect the minimal surface orthogonallyalong level curves | z | = 1 and | z | = R , respectively. As h is locally univalent, thereis no umbilic point and the last condition of E[ R , ǫ , C ] gives 2 | X ( A (1 , R )) | = | ∂X ( A (1 , R )) | . Then, by Lemma 4.2 below, we can conclude that O = O and wefinish the proof. (cid:3) HE FRASER-LI CONJECTURE AND THE LIOUVILLE EQUATION 9
Lemma 4.2.
Let Σ be a minimal annulus in R with ∂ Σ = Γ ∪ Γ . Suppose thatthere exist unit spheres S O and S O , with centers O and O , respectively, such thateach S O k intersects Σ orthogonally along Γ k . If | Σ | = | ∂ Σ | and the boundary doesnot contain umbilic points, then O = O .Proof. Let ρ := O − O and let Y be the position vector with the origin at O . Thedivergence theorem and the minimality of Σ imply that2 | Σ | = Z Σ div Y d A = Z Γ Y · ν d s + Z Γ Y · ν d s, where ν k ’s are outward unit conormal vectors. Since the surface and spheres intersectorthogonally, we have Y · ν = 1 and Y · ν = 1 + ρ · ν . Hence2 | Σ | = | Γ | + | Γ | + ρ · Z Γ ν d s = | ∂ Σ | + ρ · Flux(Σ)and we obtain ρ · Flux(Σ) = 0.On the other hand, as the torque with respect to O is zero at Γ , it should alsobe zero at Γ : 0 = Z Γ Y ∧ ν d s = ρ ∧ Z Γ ν d s = ρ ∧ Flux(Σ) . Since the boundary does not contain umbilic points, its curvature on each spheredoes not change the sign. Therefore Flux(Σ) = 0 and we have ρ = 0. (cid:3) Since the rotationally symmetric free boundary minimal annulus in a ball is knownto be the critical catenoid, it follows from Theorem 4.1 that solving the Fraser-Liconjecture in real analytic boundaries is equivalent to showing that the solution ofE[ R , ǫ , C ] has a radial symmetry, whenever it exists, for arbitrary R >
1, 0 < ǫ < C . References [1] S. Brendle,
Embedded minimal tori in S and the Lawson conjecture . Acta Math., (2013),no. 2, 177–190.[2] T. H. Colding and W. P. Minicozzi II, A course in minimal surfaces, Grad. Stud. Math . ,Amer. Math. Soc., Providence, RI, 2011.[3] A. Fraser and M. Li, Compactness of the space of embedded minimal surfaces with free boundaryin three-manifolds with nonnegative Ricci curvature and convex boundary . J. Differential Geom., (2014), no. 2, 183–200.[4] D. Hoffman and H. Karcher, Complete embedded minimal surfaces of finite total curvature . In:Geometry V (edited by R. Osserman), Encyclopaedia Math. Sci., Springer, Berlin, Heidelberg, (1997), 5–93.[5] A. Jim´enez, The Liouville equation in an annulus . Nonlinear Anal., (2012), no. 4, 2090–2097.[6] M. Spivak, A comprehensive introduction to differential geometry Vol. III . Publish of Perish,Berkeley, (1979).
Jaehoon Lee, Department of Mathematical Sciences, Seoul National University,Seoul 08826, Korea
E-mail address : [email protected] Eungbeom Yeon, Department of Mathematical Sciences, Seoul National University,Seoul 08826, Korea
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