Compact surfaces with no Bonnet mate
aa r X i v : . [ m a t h . DG ] M a r COMPACT SURFACES WITH NO BONNET MATE
GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI
Abstract.
This note gives sufficient conditions (isothermic or totally non-isothermic) for an immersion of a compact surface to have no Bonnet mate. Introduction
Consider a smooth immersion x : M → R of a connected, orientable surface M ,with unit normal vector field e . Its induced metric I = d x · d x and the orientationof M induced by e from the standard orientation of R induce a complex structureon M , which provides a decomposition into bidegrees of the second fundamentalform II of x relative to e , − d e · d x = II = II , + HI + II , . Here H is the mean curvature of x relative to e and II , = II , is the Hopfquadratic differential of x . Relative to a complex chart ( U, z ) in M ,(1) I = e u dzd ¯ z, II , = 12 he u dzdz, where the conformal factor e u , the Hopf invariant h , and the mean curvature H satisfy the structure equations on U relative to z , − e − u u z ¯ z = H − | h | Gauss equation( e u h ) ¯ z = e u H z Codazzi equationand the Gauss curvature is K = H − | h | . See [JMN16, page 212].In 1867 Bonnet [Bon67] began an investigation into the problem of whether thereexist noncongruent immersions x , ˜ x : M → R with the same induced metric, I = ˜ I ,and the same mean curvature, H = ˜ H . This Bonnet Problem has been studiedby Bianchi [Bia09], Graustein [Gra24], Cartan [Car42], Lawson–Tribuzy [LT81],Chern [Che85], Kamberov–Pedit–Pinkall [KPP98], Bobenko–Eitner [BE98, BE00],Roussos–Hernandez [RH90], Sabitov [Sab12], the present authors [JMN16], andmany others cited in these references.
Definition 1.
An immersion x : M → R is Bonnet if there is a noncongruentimmersion ˜ x : M → R such that ˜ I = I and ˜ H = H . Then ˜ x is called a Bonnetmate of x and ( x , ˜ x ) form a Bonnet pair. A constant mean curvature (CMC) immersion x : M → R , for which M issimply connected and x is not totally umbilic, admits a 1-parameter family ofBonnet mates, which are known as the associates of x [JMN16, Example 10.11, page Mathematics Subject Classification.
Variet`a reali e complesse:geometria, topologia e analisi armonica ; and by the GNSAGA of INdAM. x with nonconstantmean curvature has a Bonnet mate. By nonconstant mean curvature H we meanthat dH = 0 on a dense, open subset of M . Definition 2.
A Bonnet immersion x : M → R is proper if its mean curvatureis nonconstant and there exist at least two noncongruent Bonnet mates. It is known [JMN16, page 211] that the umbilics of x are precisely the zeros ofits Hopf quadratic differential II , . For the following definitions we assume that x has no umbilics in the domain U . If ( U, z ) is a complex coordinate chart in M ,then the local coefficient e u h of 2 II , in U has the polar representation e u h = e G + ig , for a smooth function G : U → R and a smooth map e ig : U → S . The function g : U → R is defined only locally, up to an additive integral multiple of 2 π . If w = w ( z ) is another complex coordinate in U , and if the invariants relative to itare denoted by ˆ u and ˆ h , then e u h = e u ˆ h ( w ′ ) , where w ′ = dwdz is a nowhere zero holomorphic function of z . Setting e u ˆ h = e ˆ G + i ˆ g on U , we find by an elementary calculation(2) g ¯ zz = ˆ g ¯ zz on U . The Laplace-Beltrami operator of ( M, I ) is given in the local chart (
U, z ) by∆ = 4 e − u ∂ ∂z∂ ¯ z . We conclude from (2) that ∆ g = ∆ˆ g on U , and therefore that ∆ g is a globally defined smooth function on M away from the umbilic points of x . Definition 3.
A surface immersion x : M → R is called isothermic if it hasan atlas of charts ( U, ( x, y )) each of which satisfies I = e u ( dx + dy ) and II = e u ( adx + cdy ) [JMN16, Definition 9.5, page 277] . Definition 3 is equivalent to the following definition if there are no umbilics[JMN16, Corollary 9.14, page 280].
Definition 4.
An umbilic free immersion x : M → R of an oriented connectedsurface is isothermic if ∆ g = 0 identically on M . It is totally nonisothermic if ∆ g = 0 on a connected, open, dense subset of M . The following is known about umbilic free immersions x : M → R for which M is simply connected. Cartan [Car42] proved that if x is proper Bonnet, thenit has a 1-parameter family of distinct mates [JMN16, Theorem 10.42, pages 340-342]. Graustein [Gra24] proved that if x is isothermic and Bonnet, then it is properBonnet. The present authors [JMN16, Theorem 10.13, pages 303-304] proved thatif x is totally nonisothermic, then it has a unique Bonnet mate.What is the global situation? In particular, if M is compact, can an immersion x : M → R have a Bonnet mate? It is known, and proved in the next section,that a necessary condition that x be Bonnet is that its set of umbilics is a discretesubset of M . Lawson–Tribuzy [LT81] proved that x cannot be proper Bonnetif M is compact. Roussos–Hernandez [RH90] proved that x : M → R has noBonnet mate if M is compact and x is a surface of revolution with nonconstantmean curvature. Sabitov [Sab12, Theorem 13, page 144] gives a sufficient condition OMPACT SURFACES WITH NO BONNET MATE 3 preventing the existence of a Bonnet mate when the mean curvature is nonconstantand M is compact. He gives no geometric interpretation of his condition.The goal of this paper is to prove the following result. It generalizes the Roussos–Hernandez result, since a surface of revolution is isothermic [JMN16, Example 9.7,page 277]. It also gives a geometrical clarification of the Sabitov result. Theorem.
Let x : M → R be a smooth immersion with nonconstant mean cur-vature H of a compact, connected surface, and suppose that D , the set of umbilicsof x , is a discrete subset of M . (1) If x : M \ D is isothermic, then x has no Bonnet mate. (2) If x is totally nonisothermic, then it has no Bonnet mate. The deformation quadratic differential
From the Gauss equation above, the Hopf invariants h and ˜ h relative to a complexcoordinate z of two immersions with the same induced metric and the same meancurvatures must satisfy | ˜ h | = | h | , since ˜ u = u . Hence, the only possible difference in the invariants of two suchimmersions must be in the arguments of the complex valued functions h and ˜ h .Moreover, taking the difference of their Codazzi equations, we get( e u ˜ h − e u h ) ¯ z = e u ( H z − H z ) = 0 , at every point of the domain U of the complex coordinate z . This means that thefunction F = e u (˜ h − h ) : U → C is holomorphic. Definition 5. If x , ˜ x : M → R are immersions that induce the same complexstructure on M , then their deformation quadratic differential is Q = f II , − II , . If x and ˜ x have the same induced metric and mean curvature, then the expressionfor Q relative to a complex coordinate z is(3) Q = 12 e u (˜ h − h ) dzdz = 12 F dzdz, which shows that Q is a holomorphic quadratic differential on M , and(4) | F + e u h | = | e u ˜ h | = | e u h | on U , since | ˜ h | = | h | . Q is identically zero on M if and only if ˜ h = h in anycomplex coordinate system. Therefore, by Bonnet’s Congruence Theorem, Q = 0if and only if the immersions x and ˜ x are congruent in the sense that there exists arigid motion ( y , A ) ∈ E (3) such that ˜ x = y + A x : M → R . Thus, an immersion˜ x : M → R is a Bonnet mate of x : M → R if it induces the same metric andmean curvature and the deformation quadratic differential is not identically zero. Proposition 6.
If an immersion x : M → R possesses a Bonnet mate ˜ x : M → R , then the umbilics of x must be isolated and coincide with those of ˜ x . GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI
Proof.
Under the given assumptions, the holomorphic quadratic differential Q isnot identically zero. Therefore, in any complex coordinate chart ( U, z ), we have Q = F dzdz , where F is a nonzero holomorphic function of z . Its zeros must beisolated. A point m ∈ U is an umbilic of x if and only if h ( m ) = 0 if and only if˜ h ( m ) = 0, by (4). In either case F ( m ) = 0 by (4). Therefore, the set of umbilicpoints is a subset of the set of zeros of Q , which is a discrete subset of M . (cid:3) Let x : M → R be an immersion with a Bonnet mate ˜ x : M → R . Let ( U, z )be a complex coordinate chart in M and let h and ˜ h be the Hopf invariants of x and ˜ x , respectively, relative to z on U . Let D be the set of umbilics of x , necessarilya discrete subset of M . On U \ D we have h never zero and˜ h = hA, for a smooth function A : U \ D → S , where S ⊂ C is the unit circle. On U \ D then, the difference of the Hopf differentials is the holomorphic quadraticdifferential Q = ] II , − II , = II , ( A − . This shows that A : M \ D → S is a well-defined smooth map on all of M \ D . Remark 7.
Under our assumption of nonconstant H , the map A cannot be con-stant, for otherwise II , would then be holomorphic and thus H would be constantby the Codazzi equation. Proposition 8 (Sabitov[Sab12]) . If an immersion x : M → R possesses a Bonnetmate ˜ x : M → R , then the deformation quadratic differential Q of x is zero onlyat the umbilics of x . Therefore, A : M \ D → S never takes the value ∈ S .Proof. This is Theorem 1, pages 113ff of [Sab12]. He says the result is stated in[Bob08], but he believes the proof there is inadequate. Sabitov’s proof uses resultsfrom the Hilbert boundary-value problem. The following proof is essentially thesame as Sabitov’s, but avoids use of the Hilbert boundary-value problem.Seeking a contradiction, suppose Q ( m ) = 0 for some point m ∈ M \ D . Since Q is holomorphic, and not identically zero, its zeros are isolated. Let ( U, z ) be acomplex coordinate chart of M \ D centered at m , containing no other zeros of Q ,and such that z ( U ) is an open disk of C . Now A ( m ) = 1 and A is continuous,so we may assume U chosen small enough that A never takes the value − U .Then there exists a smooth map v : U → R such that − π < v < π and A = e iv on U . Since A = 1 on U only at m , it follows that(5) v ( U \ { m } ) ⊂ ( − π,
0) or v ( U \ { m } ) ⊂ (0 , π ) . Let e u and h be the conformal factor and Hopf invariant of x relative to z . Then h never zero on U implies it has a polar representation h = e f + ig , for some smoothfunctions f, g : U → R . Now Q = F dzdz , where F = e u e f + ig ( e iv −
1) = e u + f ( e i ( g + v ) − e ig ) : U → C is holomorphic. Using the identity e i ( g + v ) − e ig = e i (2 g + v ) / ( e iv/ − e − iv/ ) = 2 ie i ( g + v/ sin( v/ , we get F = 2 ie u + f + i ( g + v/ sin( v/ OMPACT SURFACES WITH NO BONNET MATE 5 on U . The contour integral of d log F about any circle in U centered at m is 2 πi times the number of zeros of F inside the circle. By assumption, this integral isnot zero. But, d log F = d (2 u + f + i ( g + v/ d log( | sin( v/ | ) , and the contour integral of the right hand side is zero, since these are exact differ-entials on U \ { m } . In fact, the values of v/ U \ { m } lie entirely in (0 , π/ − π/ , v/
2) is never zero. This is the desired contradictionto our assumption that Q has a zero in M \ D . (cid:3) As a consequence of this Proposition, the smooth map A : M \ D → S nevertakes the value 1 ∈ S , so there exists a smooth map r : M \ D → (0 , π ) ⊂ R , such that A = e ir on M \ D .3. Proof of the Theorem
Proof.
Seeking a contradiction, we suppose that x possesses a Bonnet mate ˜ x : M → R . Let II , and ] II , be the Hopf quadratic differentials of x and ˜ x ,respectively. By the preceding propositions, the quadratic differential ] II , − II , is holomorphic on M , and on M \ D ] II , − II , = II , ( e ir − , where the function r : M \D → (0 , π ) is smooth. Let ( U, z ) be a complex coordinatechart in M \D . Let h and e u be the Hopf invariant and conformal factor of x relativeto z . Then h = e f + ig on U , for some smooth functions f : U → R and e ig : U → S .1). If x is isothermic, then g ¯ zz = 0 identically on U . Let G = f + 2 u : U → R .Then ( e G + ig ( e ir − ¯ z = 0 implies(6) r ¯ z = i ( G + ig ) ¯ z (1 − e − ir )on U . Applying ∂ z to this, and using that r z is the complex conjugate of r ¯ z , wefind(7) r ¯ zz = 0on U . Hence, r : M \ D → (0 , π ) is a bounded harmonic function. Since the pointsof D are isolated and r is bounded, we know that r extends to a harmonic functionon all of M . But then r must be constant, since M is compact. This contradictsour assumption of nonconstant H , by Remark 7.2). If x is totally nonisothermic, we have either ∆ g ≤ g ≥ M \ D .To be specific, let us suppose that ∆ g ≤ M \ D . Now (6) holds and by theproof of Theorem 10.13 on pages 303-304 of [JMN16], we have(8) e ir = 1 + − g ¯ zz D ( g ¯ zz + iL ) , on U , where L = | G ¯ z + ig ¯ z | − G ¯ zz and D = g zz + L . Applying ∂ z to (6) and using(8), we find(9) r ¯ zz = − g ¯ zz , on U . Therefore, ∆ r = − g ≥ M \ D . GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI
Recall [HK76, Def. § v : V → R ∪ {−∞} on adomain V ⊂ C is subharmonic if(1) −∞ ≤ v ( z ) < + ∞ in V .(2) v is upper semi-continuous in V . (This means that for any c ∈ R , the set { z ∈ U : v ( z ) < c } is open in V.)(3) If z is any point of V then there exist arbitrarily small positive values of R such that v ( z ) ≤ πR Z π v ( z + Re it ) dt. If v is of class C in V , then v is subharmonic in V if and only if v ¯ zz ≥ V [HK76, Example 3, page 41].If M is a connected Riemann surface, we define a function v : M → R ∪ {−∞} to be subharmonic if for any complex coordinate chart ( U, z ) of M , the local repre-sentative v ◦ z − : z ( U ) → R is subharmonic. This is well-defined by the Corollaryto Theorem 2.8 on page 53 of [HK76].We conclude from (9) that r is subharmonic on M \ D . In the event that ∆ g ≥ M \ D , we conclude that − r is subharmonic and continue as below with − r .Suppose ( U, z ) is a complex coordinate chart centered at a point m ∈ D , andsmall enough that no other point of D lies in it. Then r ◦ z − is subharmonic onthe open set z ( U ) \ { } , so it extends uniquely to a subharmonic function on z ( U ),by Theorem 5.8 on page 237 of [HK76]. It follows that r extends uniquely to asubharmonic function on M .By Theorem 1.2 on page 4 of [HK76], if v : V → R ∪ {−∞} is upper semi-continuous on a nonempty compact domain V ⊂ C , then v attains its maximumon V ; i.e., there exists z ∈ V such that v ( z ) ≤ v ( z ) for all z ∈ V . The sameproof shows that this is true for an upper semi-continuous function on a compactRiemann surface. Thus, the subharmonic function r : M → R ∪ {−∞} attainsits maximum at some point m ∈ M . Let ( U, z ) be a complex coordinate chartcentered at m . Choose R > D (0 , R ) = { z ∈ C : | z | ≤ R } iscontained in z ( U ). By the maximum principle for subharmonic functions [HK76,Theorem 2.3, page 47], r ◦ z − must be constantly equal to r ( m ) on D (0 , R ). Itfollows that E = { m ∈ M : r ( m ) = r ( m ) } is an open subset of M . But E = M \ { m ∈ M : r ( m ) < r ( m ) } is closed, since r is upper semi-continuous. We conclude that r is constant on M ,which is our sought for contradiction, by Remark 7. (cid:3) References [BE98] Alexander Bobenko and Ulrich Eitner. Bonnet surfaces and Painlev´e equations.
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E-mail address : [email protected] (E. Musso) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Ducadegli Abruzzi 24, I-10129 Torino, Italy E-mail address : [email protected] (L. Nicolodi) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Univer-sit`a di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy E-mail address ::