This note gives sufficient conditions (isothermic or totally nonisothermic)
for an immersion of a compact surface to have no Bonnet mate.Comment: 7 pages, LaTeX2

Jensen, Gary R.

Musso, Emilio

Nicolodi, Lorenzo

English

arXiv

This note gives sufficient conditions (isothermic or totally nonisothermic) for an immersion of a compact surface to have no Bonnet mat

NICOLODI, Lorenzo

Archivio istituzionale della Ricerca -  Università degli Studi di Parma

10 December 2022University of Parma Research RepositoryCompact surfaces with no Bonnet mate / Jensen, Gary R.; Musso, Emilio; Nicolodi, Lorenzo. - In: THEJOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 28:3(2018), pp. 2644-2652.[https://doi.org/10.1007/s12220-017-9924-y]OriginalCompact surfaces with no Bonnet matePublisher:PublishedDOI:https://doi.org/10.1007/s12220-017-9924-yTerms of use:openAccessPublisher copyright(Article begins on next page)Anyone can freely access the full text of works made available as "Open Access". Works made availableAvailability:This version is available at: 11381/2830057 since: 2018-09-11T08:34:14ZSpringer New York LLCThis is the peer reviewd version of the followng article:COMPACT SURFACES WITH NO BONNET MATEGARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODIAbstract. This note gives sufficient conditions (isothermic or totally non-isothermic) for an immersion of a compact surface to have no Bonnet mate.1. IntroductionConsider a smooth immersion x : M → R3 of a connected, orientable surface M ,with unit normal vector field e3. Its induced metric I = dx ·dx and the orientationof M induced by e3 from the standard orientation of R3 induce a complex structureon M , which provides a decomposition into bidegrees of the second fundamentalform II of x relative to e3,−de3 · dx = II = II2,0 +HI + II0,2.Here H is the mean curvature of x relative to e3 and II2,0 = II0,2 is the Hopfquadratic differential of x. Relative to a complex chart (U, z) in M ,(1) I = e2udzdz̄, II2,0 =12he2udzdz,where the conformal factor eu, the Hopf invariant h, and the mean curvature Hsatisfy the structure equations on U relative to z,−4e−2uuzz̄ = H2 − |h|2 Gauss equation(e2uh)z̄ = e2uHz Codazzi equationand the Gauss curvature is K = H2 − |h|2. See [JMN16, page 212].In 1867 Bonnet [Bon67] began an investigation into the problem of whether thereexist noncongruent immersions x, x̃ : M → R3 with the same induced metric, I = Ĩ,and the same mean curvature, H = H̃. 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 → R3 is Bonnet if there is a noncongruentimmersion x̃ : M → R3 such that Ĩ = I and H̃ = H. Then x̃ is called a Bonnetmate of x and (x, x̃) form a Bonnet pair.A constant mean curvature (CMC) immersion x : M → R3, 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, page2000 Mathematics Subject Classification. 53C42, 53A10, 53A05.Authors partially supported by MIUR (Italy) under the PRIN project Varietà reali e complesse:geometria, topologia e analisi armonica; and by the GNSAGA of INdAM.12 GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI302]. The local problem is thus to determine if an immersion 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 → R3 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 II2,0. For the following definitions we assume thatx has no umbilics in the domain U . If (U, z) is a complex coordinate chart in M ,then the local coefficient e2uh of 2II2,0 in U has the polar representatione2uh = eG+ig,for a smooth function G : U → R and a smooth map eig : U → S1. The functiong : U → R is defined only locally, up to an additive integral multiple of 2π. Ifw = w(z) is another complex coordinate in U , and if the invariants relative to itare denoted by û and ĥ, thene2uh = e2ûĥ(w′)2,where w′ = dwdz is a nowhere zero holomorphic function of z. Setting e2ûĥ = eĜ+iĝon U , we find by an elementary calculation(2) gz̄z = ĝz̄zon U . The Laplace-Beltrami operator of (M, I) is given in the local chart (U, z) by∆ = 4e−2u ∂2∂z∂z̄ . We conclude from (2) that ∆g = ∆ĝ on U , and therefore that ∆gis a globally defined smooth function on M away from the umbilic points of x.Definition 3. A surface immersion x : M → R3 is called isothermic if it hasan atlas of charts (U, (x, y)) each of which satisfies I = e2u(dx2 + dy2) and II =eu(adx2 + cdy2) [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 → R3 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 → R3 for whichM 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 immersionx : M → R3 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 → R3 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 conditionCOMPACT SURFACES WITH NO BONNET MATE 3preventing 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 → R3 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 → R3 is isothermic, then x : M → R3 has no Bonnet mate.(2) If x : M \ D → R3 is totally nonisothermic, then x : M → R3 has noBonnet mate.2. The deformation quadratic differentialFrom the Gauss equation above, the Hopf invariants h and h̃ relative to a complexcoordinate z of two immersions with the same induced metric and the same meancurvatures must satisfy|h̃| = |h|,since ũ = u. Hence, the only possible difference in the invariants of two suchimmersions must be in the arguments of the complex valued functions h and h̃.Moreover, taking the difference of their Codazzi equations, we get(e2uh̃− e2uh)z̄ = e2u(Hz −Hz) = 0,at every point of the domain U of the complex coordinate z. This means that thefunctionF = e2u(h̃− h) : U → Cis holomorphic.Definition 5. If x, x̃ : M → R3 are immersions that induce the same complexstructure on M , then their deformation quadratic differential isQ = ĨI2,0− II2,0.If x and x̃ have the same induced metric and mean curvature, then the expressionfor Q relative to a complex coordinate z is(3) Q = 12e2u(h̃− h)dzdz = 12Fdzdz,which shows that Q is a holomorphic quadratic differential on M , and(4) |F + e2uh| = |e2uh̃| = |e2uh|on U , since |h̃| = |h|. Q is identically zero on M if and only if h̃ = h in anycomplex coordinate system. Therefore, by Bonnet’s Congruence Theorem, Q = 0if and only if the immersions x and x̃ are congruent in the sense that there exists arigid motion (y, A) ∈ E(3) such that x̃ = y + Ax : M → R3. Thus, an immersionx̃ : M → R3 is a Bonnet mate of x : M → R3 if it induces the same metric andmean curvature and the deformation quadratic differential is not identically zero.Proposition 6. If an immersion x : M → R3 possesses a Bonnet mate x̃ : M →R3, then the umbilics of x must be isolated and coincide with those of x̃.4 GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODIProof. Under the given assumptions, the holomorphic quadratic differential Q isnot identically zero. Therefore, in any complex coordinate chart (U, z), we haveQ = 12Fdzdz, 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 ifh̃(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 . Let x : M → R3 be an immersion with a Bonnet mate x̃ : M → R3. Let (U, z)be a complex coordinate chart in M and let h and h̃ be the Hopf invariants of xand x̃, 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 andh̃ = hA,for a smooth function A : U \ D → S1, where S1 ⊂ C is the unit circle. OnU \ D then, the difference of the Hopf differentials is the holomorphic quadraticdifferentialQ = ĨI2,0 − II2,0 = II2,0(A− 1).This shows that A : M \ D → S1 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 II2,0 would then be holomorphic and thus H would be constantby the Codazzi equation.Proposition 8 (Sabitov[Sab12]). If an immersion x : M → R3 possesses a Bonnetmate x̃ : M → R3, then the deformation quadratic differential Q of x is zero onlyat the umbilics of x. Therefore, A : M \ D → S1 never takes the value 1 ∈ S1.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(m0) = 0 for some point m0 ∈ M \ D. SinceQ is holomorphic, and not identically zero, its zeros are isolated. Let (U, z) be acomplex coordinate chart of M \D centered at m0, containing no other zeros of Q,and such that z(U) is an open disk of C. Now A(m0) = 1 and A is continuous,so we may assume U chosen small enough that A never takes the value −1 on U .Then there exists a smooth map v : U → R such that −π < v < π and A = eiv onU . Since A = 1 on U only at m0, it follows that(5) v(U \ {m0}) ⊂ (−π, 0) or v(U \ {m0}) ⊂ (0, π).Let e2u and h be the conformal factor and Hopf invariant of x relative to z. Thenh never zero on U implies it has a polar representation h = ef+ig, for some smoothfunctions f, g : U → R. Now Q = 12Fdzdz, whereF = e2uef+ig(eiv − 1) = e2u+f (ei(g+v) − eig) : U → Cis holomorphic. Using the identityei(g+v) − eig = ei(2g+v)/2(eiv/2 − e−iv/2) = 2iei(g+v/2) sin(v/2),we getF = 2ie2u+f+i(g+v/2) sin(v/2)COMPACT SURFACES WITH NO BONNET MATE 5on U . The contour integral of d logF about any circle in U centered at m0 is 2πitimes the number of zeros of F inside the circle. By assumption, this integral isnot zero. But,d logF = d(2u+ f + i(g + v/2)) + d log(| sin(v/2)|),and the contour integral of the right hand side is zero, since these are exact differ-entials on U \ {m0}. In fact, the values of v/2 on U \ {m0} lie entirely in (0, π/2)or entirely in (−π/2, 0), so sin(v/2) is never zero. This is the desired contradictionto our assumption that Q has a zero in M \ D. As a consequence of this Proposition, the smooth map A : M \ D → S1 nevertakes the value 1 ∈ S1, so there exists a smooth mapr : M \ D → (0, 2π) ⊂ R,such that A = eir on M \ D.3. Proof of the TheoremProof. Seeking a contradiction, we suppose that x possesses a Bonnet mate x̃ :M → R3. Let II2,0 and ĨI2,0 be the Hopf quadratic differentials of x and x̃,respectively. By the preceding propositions, the quadratic differential ĨI2,0 − II2,0is holomorphic on M , and on M \ DĨI2,0 − II2,0 = II2,0(eir − 1),where the function r : M\D → (0, 2π) is smooth. Let (U, z) be a complex coordinatechart inM \D. Let h and eu be the Hopf invariant and conformal factor of x relativeto z. Then h = ef+ig on U , for some smooth functions f : U → R and eig : U → S1.1). If x is isothermic, then gz̄z = 0 identically on U . Let G = f + 2u : U → R.Then (eG+ig(eir − 1))z̄ = 0 implies(6) rz̄ = i(G+ ig)z̄(1− e−ir)on U . Applying ∂z to this, and using that rz is the complex conjugate of rz̄, wefind(7) rz̄z = 0on U . Hence, r : M \D → (0, 2π) 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 ≤ 0 or ∆g ≥ 0 on M \ D.To be specific, let us suppose that ∆g ≤ 0 on M \ D. Now (6) holds and by theproof of Theorem 10.13 on pages 303-304 of [JMN16], we have(8) eir = 1 +−2gz̄zD(gz̄z + iL),on U , where L = |Gz̄ + igz̄|2−Gz̄z and D = g2z̄z +L2. Applying ∂z to (6) and using(8), we find(9) rz̄z = −2gz̄z,on U . Therefore, ∆r = −2∆g ≥ 0 on M \ D.6 GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODIRecall [HK76, Def. §2.1, pages 40-41] that a function 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 z0 is any point of V then there exist arbitrarily small positive values ofR such thatv(z0) ≤12πR∫ 2π0v(z0 +Reit)dt.If v is of class C2 in V , then v is subharmonic in V if and only if vz̄z ≥ 0 in 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−1 : 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 ≥ 0on 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 m0 ∈ D, andsmall enough that no other point of D lies in it. Then r ◦ z−1 is subharmonic onthe open set z(U) \ {0}, 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 z0 ∈ V such that v(z) ≤ v(z0) 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 m0 ∈ M . Let (U, z) be a complex coordinate chartcentered at m0. Choose R > 0 such that the disk 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−1 must be constantly equal to r(m0) on D(0, R). Itfollows thatE = {m ∈ M : r(m) = r(m0)}is an open subset of M . ButE = M \ {m ∈ M : r(m) < r(m0)}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.References[BE98] Alexander Bobenko and Ulrich Eitner. Bonnet surfaces and Painlevé equations. J. ReineAngew. Math., 499:47–79, 1998.[BE00] Alexander I. Bobenko and Ulrich Eitner. Painlevé equations in the differential geometryof surfaces, volume 1753 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2000.[Bia09] Luigi Bianchi. Lezioni di geometria differenziale, volume 1-3. E. Spoerri, Pisa, 1903-1909. Seconda edizione riveduta e considerevolmente aumentata.COMPACT SURFACES WITH NO BONNET MATE 7[Bob08] Alexander I. Bobenko. Exploring surfaces through methods from the theory of integrablesystems: the Bonnet problem. In Surveys on geometry and integrable systems, volume 51of Adv. Stud. Pure Math., pages 1–53. Math. Soc. Japan, Tokyo, 2008.[Bon67] Pierre Ossian Bonnet. Mémoire sur la théorie des surfaces applicables sur une surfacedonnée, deuxième partie. Journal de l’Ecole Polytechnique, 42:1 – 151, 1867.[Car42] Elie Cartan. Sur les couples de surfaces applicables avec conservation des courburesprincipales. Bull. Sc. Math., 66:55–85, 1942. Oeuvres Complète, Partie III, Volume 2,pp. 1591–1620.[Che85] Shiing-Shen Chern. Deformation of surfaces preserving principal curvatures. In I. Chaveland H.M. Farkas, editors, Differential Geometry and Complex Analysis: A Volume Ded-icated to the Memory of Harry Ernest Rauch, pages 155 – 163, Berlin, 1985. Springer-Verlag.[Gra24] W. C. Graustein. Applicability with preservation of both curvatures. Bull. Amer. Math.Soc., 30(1-2):19–23, 1924.[HK76] W. K. Hayman and P. B. Kennedy. Subharmonic functions. Vol. I. Academic Press[Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathemati-cal Society Monographs, No. 9.[JMN16] Gary R. Jensen, Emilio Musso, and Lorenzo Nicolodi. Surfaces in classical geometries.Universitext. Springer, Cham, 2016. A treatment by moving frames.[KPP98] George Kamberov, Franz Pedit, and Ulrich Pinkall. Bonnet pairs and isothermic surfaces.Duke Math. J., 92(3):637–644, 1998.[LT81] H. Blaine Lawson, Jr. and Renato de Azevedo Tribuzy. On the mean curvature functionfor compact surfaces. J. Differential Geom., 16(2):179–183, 1981.[RH90] Ioannis M. Roussos and Gastón E. Hernández. On the number of distinct isometricimmersions of a Riemannian surface into R3 with given mean curvature. Amer. J. Math.,112(1):71–85, 1990.[Sab12] I. Kh. Sabitov. Isometric surfaces with common mean curvature and the problem ofBonnet pairs. Sbornik: Mathematics, 203(1):111–152, 2012.(G. R. Jensen) Department of Mathematics, Washington University, One BrookingsDrive, St. Louis, MO 63130, USAE-mail address: gary@wustl.edu(E. Musso) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Ducadegli Abruzzi 24, I-10129 Torino, ItalyE-mail address: emilio.musso@polito.it(L. Nicolodi) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Univer-sità di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, ItalyE-mail address: lorenzo.nicolodi@unipr.it

Compact surfaces with no Bonnet mate

This note gives sufficient conditions (isothermic or totally nonisothermic) for an immersion of a compact surface to have no Bonnet mate

PORTO Publications Open Repository TOrino

Compact Surfaces with No Bonnet Mate

MUSSO, EMILIO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

09 November 2022POLITECNICO DI TORINORepository ISTITUZIONALECompact Surfaces with No Bonnet Mate / Jensen, Gary R.; Musso, Emilio; Nicolodi, Lorenzo. - In: THE JOURNAL OFGEOMETRIC ANALYSIS. - ISSN 1050-6926. - ELETTRONICO. - (2018), pp. 1-9. [10.1007/s12220-017-9924-y]OriginalCompact Surfaces with No Bonnet MateSpringer postprint/Author's Accepted ManuscriptPublisher:PublishedDOI:10.1007/s12220-017-9924-yTerms of use:openAccessPublisher copyrightThis version of the article has been accepted for publication, after peer review (when applicable) and is subject toSpringer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements,or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s12220-017-9924-y(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2685796 since: 2017-10-16T19:48:19ZSpringer New York LLCCOMPACT SURFACES WITH NO BONNET MATEGARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODIAbstract. This note gives sufficient conditions (isothermic or totally non-isothermic) for an immersion of a compact surface to have no Bonnet mate.1. IntroductionConsider a smooth immersion x : M → R3 of a connected, orientable surface M ,with unit normal vector field e3. Its induced metric I = dx ·dx and the orientationof M induced by e3 from the standard orientation of R3 induce a complex structureon M , which provides a decomposition into bidegrees of the second fundamentalform II of x relative to e3,−de3 · dx = II = II2,0 +HI + II0,2.Here H is the mean curvature of x relative to e3 and II2,0 = II0,2 is the Hopfquadratic differential of x. Relative to a complex chart (U, z) in M ,(1) I = e2udzdz̄, II2,0 =12he2udzdz,where the conformal factor eu, the Hopf invariant h, and the mean curvature Hsatisfy the structure equations on U relative to z,−4e−2uuzz̄ = H2 − |h|2 Gauss equation(e2uh)z̄ = e2uHz Codazzi equationand the Gauss curvature is K = H2 − |h|2. See [JMN16, page 212].In 1867 Bonnet [Bon67] began an investigation into the problem of whether thereexist noncongruent immersions x, x̃ : M → R3 with the same induced metric, I = Ĩ,and the same mean curvature, H = H̃. 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 → R3 is Bonnet if there is a noncongruentimmersion x̃ : M → R3 such that Ĩ = I and H̃ = H. Then x̃ is called a Bonnetmate of x and (x, x̃) form a Bonnet pair.A constant mean curvature (CMC) immersion x : M → R3, 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, page2000 Mathematics Subject Classification. 53C42, 53A10, 53A05.Authors partially supported by MIUR (Italy) under the PRIN project Varietà reali e complesse:geometria, topologia e analisi armonica; and by the GNSAGA of INdAM.12 GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI302]. The local problem is thus to determine if an immersion x with nonconstantmean curvature has a Bonnet mate. By nonconstant mean curvature H we meanthat dH 6= 0 on a dense, open subset of M .Definition 2. A Bonnet immersion x : M → R3 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 II2,0. For the following definitions we assume thatx has no umbilics in the domain U . If (U, z) is a complex coordinate chart in M ,then the local coefficient e2uh of 2II2,0 in U has the polar representatione2uh = eG+ig,for a smooth function G : U → R and a smooth map eig : U → S1. The functiong : U → R is defined only locally, up to an additive integral multiple of 2π. Ifw = w(z) is another complex coordinate in U , and if the invariants relative to itare denoted by û and ĥ, thene2uh = e2ûĥ(w′)2,where w′ = dwdz is a nowhere zero holomorphic function of z. Setting e2ûĥ = eĜ+iĝon U , we find by an elementary calculation(2) gz̄z = ĝz̄zon U . The Laplace-Beltrami operator of (M, I) is given in the local chart (U, z) by∆ = 4e−2u ∂2∂z∂z̄ . We conclude from (2) that ∆g = ∆ĝ on U , and therefore that ∆gis a globally defined smooth function on M away from the umbilic points of x.Definition 3. A surface immersion x : M → R3 is called isothermic if it hasan atlas of charts (U, (x, y)) each of which satisfies I = e2u(dx2 + dy2) and II =eu(adx2 + cdy2) [JMN16, Definition 9.5, page 277].Definition 3 is equivalent to the first statement of the following definition if thereare no umbilics [JMN16, Corollary 9.14, page 280].Definition 4. An umbilic free immersion x : M → R3 of an oriented connectedsurface is isothermic if ∆g = 0 identically on M . It is totally nonisothermic if∆g 6= 0 on a connected, open, dense subset of M .The following is known about the local situation. Suppose that x : M → R3is an umbilic free immersion for which M is simply connected and possesses acomplex coordinate z : M → C. Cartan [Car42] proved that if x is proper Bonnet,then it has a 1-parameter family of distinct mates [JMN16, Theorem 10.42, pages340-342]. Graustein [Gra24] proved that if x is isothermic and Bonnet, then itis proper Bonnet. The present authors [JMN16, Theorem 10.13, pages 303-304]proved that if x is totally nonisothermic, then it has a unique Bonnet mate. Thiscontrasts emphatically with the case when M is compact, as stated in item (2) ofthe following Theorem.What is the global situation? Lawson–Tribuzy [LT81] proved that x : M → R3cannot be proper Bonnet if M is compact. Since then the question whether thereexist Bonnet pairs for a compact surface M of genus g > 0 has been open.” Roussos–Hernandez [RH90] proved that x : M → R3 has no Bonnet mate if M is compactand x is a surface of revolution with nonconstant mean curvature. Sabitov [Sab12,COMPACT SURFACES WITH NO BONNET MATE 3Theorem 13, page 144] gives a sufficient condition preventing the existence of aBonnet mate when the mean curvature is nonconstant and M is compact. He givesno geometric interpretation of his condition. It is known, and proved in the nextsection, that a necessary condition that x be Bonnet is that its set of umbilics is adiscrete subset of M .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 → R3 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 → R3 is isothermic, then x : M → R3 has no Bonnet mate.(2) If x : M \ D → R3 is totally nonisothermic, then x : M → R3 has noBonnet mate.2. The deformation quadratic differentialFrom the Gauss equation above, the Hopf invariants h and h̃ relative to a complexcoordinate z of two immersions with the same induced metric and the same meancurvatures must satisfy|h̃| = |h|,since ũ = u. Hence, the only possible difference in the invariants of two suchimmersions must be in the arguments of the complex valued functions h and h̃.Moreover, taking the difference of their Codazzi equations, we get(e2uh̃− e2uh)z̄ = e2u(Hz −Hz) = 0,at every point of the domain U of the complex coordinate z. This means that thefunctionF = e2u(h̃− h) : U → Cis holomorphic.Definition 5. If x, x̃ : M → R3 are immersions that induce the same complexstructure on M , then their deformation quadratic differential isQ = ĨI2,0− II2,0.If x and x̃ have the same induced metric and mean curvature, then the expressionfor Q relative to a complex coordinate z is(3) Q = 12e2u(h̃− h)dzdz = 12Fdzdz,which shows that Q is a holomorphic quadratic differential on M , and(4) |F + e2uh| = |e2uh̃| = |e2uh|on U , since |h̃| = |h|. Q is identically zero on M if and only if h̃ = h in anycomplex coordinate system. Therefore, by Bonnet’s Congruence Theorem, Q = 0if and only if the immersions x and x̃ are congruent in the sense that there exists arigid motion (y, A) ∈ E(3) such that x̃ = y + Ax : M → R3. Thus, an immersionx̃ : M → R3 is a Bonnet mate of x : M → R3 if it induces the same metric andmean curvature and the deformation quadratic differential is not identically zero.4 GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODIProposition 6. If an immersion x : M → R3 possesses a Bonnet mate x̃ : M →R3, then the umbilics of x must be isolated and coincide with those of x̃.Proof. Under the given assumptions, the holomorphic quadratic differential Q isnot identically zero. Therefore, in any complex coordinate chart (U, z), we haveQ = 12Fdzdz, 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 ifh̃(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 . Let x : M → R3 be an immersion with a Bonnet mate x̃ : M → R3. Let (U, z)be a complex coordinate chart in M and let h and h̃ be the Hopf invariants of xand x̃, 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 andh̃ = hA,for a smooth function A : U \ D → S1, where S1 ⊂ C is the unit circle. OnU \ D then, the difference of the Hopf differentials is the holomorphic quadraticdifferentialQ = ĨI2,0− II2,0 = II2,0(A− 1).This shows that A : M \ D → S1 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 II2,0 would then be holomorphic and thus H would be constantby the Codazzi equation.Proposition 8 (Sabitov[Sab12]). If an immersion x : M → R3 possesses a Bonnetmate x̃ : M → R3, then the deformation quadratic differential Q of x is zero onlyat the umbilics of x. Therefore, A : M \ D → S1 never takes the value 1 ∈ S1.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(m0) = 0 for some point m0 ∈M \ D. SinceQ is holomorphic, and not identically zero, its zeros are isolated. Let (U, z) be acomplex coordinate chart of M \D centered at m0, containing no other zeros of Q,and such that z(U) is an open disk of C. Now A(m0) = 1 and A is continuous,so we may assume U chosen small enough that A never takes the value −1 on U .Then there exists a smooth map v : U → R such that −π < v < π and A = eiv onU . Since A = 1 on U only at m0, it follows that(5) v(U \ {m0}) ⊂ (−π, 0) or v(U \ {m0}) ⊂ (0, π).Let e2u and h be the conformal factor and Hopf invariant of x relative to z. Thenh never zero on U implies it has a polar representation h = ef+ig, for some smoothfunctions f, g : U → R. Now Q = 12Fdzdz, whereF = e2uef+ig(eiv − 1) = e2u+f (ei(g+v) − eig) : U → Cis holomorphic. Using the identityei(g+v) − eig = ei(2g+v)/2(eiv/2 − e−iv/2) = 2iei(g+v/2) sin(v/2),COMPACT SURFACES WITH NO BONNET MATE 5we getF = 2ie2u+f+i(g+v/2) sin(v/2)on U . The contour integral of d logF about any circle in U centered at m0 is 2πitimes the number of zeros of F inside the circle. By assumption, this integral isnot zero. But,d logF = d(2u+ f + i(g + v/2)) + d log(| sin(v/2)|),and the contour integral of the right hand side is zero, since these are exact differ-entials on U \ {m0}. In fact, the values of v/2 on U \ {m0} lie entirely in (0, π/2)or entirely in (−π/2, 0), so sin(v/2) is never zero. This is the desired contradictionto our assumption that Q has a zero in M \ D. As a consequence of this Proposition, the smooth map A : M \ D → S1 nevertakes the value 1 ∈ S1, so there exists a smooth mapr : M \ D → (0, 2π) ⊂ R,such that A = eir on M \ D.3. Proof of the TheoremProof. Seeking a contradiction, we suppose that x possesses a Bonnet mate x̃ :M → R3. Let II2,0 and ĨI2,0be the Hopf quadratic differentials of x and x̃,respectively. By the preceding propositions, the quadratic differential ĨI2,0− II2,0is holomorphic on M , and on M \ DĨI2,0− II2,0 = II2,0(eir − 1),where the function r : M\D → (0, 2π) is smooth. Let (U, z) be a complex coordinatechart in M \D. Let h and eu be the Hopf invariant and conformal factor of x relativeto z. Then h = ef+ig on U , for some smooth functions f : U → R and eig : U → S1.1). If x is isothermic, then gz̄z = 0 identically on U . Let G = f + 2u : U → R.Then (eG+ig(eir − 1))z̄ = 0 implies(6) rz̄ = i(G+ ig)z̄(1− e−ir)on U . Applying ∂z to this, and using that rz is the complex conjugate of rz̄, wefind(7) rz̄z = 0on U . Hence, r : M \D → (0, 2π) 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 ≤ 0 or ∆g ≥ 0 on M \ D.To be specific, let us suppose that ∆g ≤ 0 on M \ D. Now (6) holds and by theproof of Theorem 10.13 on pages 303-304 of [JMN16], we have(8) eir = 1 +−2gz̄zD(gz̄z + iL),on U , where L = |Gz̄ + igz̄|2−Gz̄z and D = g2z̄z +L2. Applying ∂z to (6) and using(8), we find(9) rz̄z = −2gz̄z,6 GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODIon U . Therefore, ∆r = −2∆g ≥ 0 on M \ D.Recall [HK76, Def. §2.1, pages 40-41] that a function 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 z0 is any point of V then there exist arbitrarily small positive values ofR such thatv(z0) ≤12πR∫ 2π0v(z0 +Reit)dt.If v is of class C2 in V , then v is subharmonic in V if and only if vz̄z ≥ 0 in 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−1 : 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 ≥ 0on 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 m0 ∈ D, andsmall enough that no other point of D lies in it. Then r ◦ z−1 is subharmonic onthe open set z(U) \ {0}, 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 z0 ∈ V such that v(z) ≤ v(z0) 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 m0 ∈ M . Let (U, z) be a complex coordinate chartcentered at m0. Choose R > 0 such that the disk 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−1 must be constantly equal to r(m0) on D(0, R). Itfollows thatE = {m ∈M : r(m) = r(m0)}is an open subset of M . ButE = M \ {m ∈M : r(m) < r(m0)}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.References[BE98] Alexander Bobenko and Ulrich Eitner. Bonnet surfaces and Painlevé equations. J. ReineAngew. Math., 499:47–79, 1998.[BE00] Alexander I. Bobenko and Ulrich Eitner. Painlevé equations in the differential geometryof surfaces, volume 1753 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2000.[Bia09] Luigi Bianchi. Lezioni di geometria differenziale, volume 1-3. E. Spoerri, Pisa, 1903-1909. Seconda edizione riveduta e considerevolmente aumentata.COMPACT SURFACES WITH NO BONNET MATE 7[Bob08] Alexander I. Bobenko. Exploring surfaces through methods from the theory of integrablesystems: the Bonnet problem. In Surveys on geometry and integrable systems, volume 51of Adv. Stud. Pure Math., pages 1–53. Math. Soc. Japan, Tokyo, 2008.[Bon67] Pierre Ossian Bonnet. Mémoire sur la théorie des surfaces applicables sur une surfacedonnée, deuxième partie. Journal de l’Ecole Polytechnique, 42:1 – 151, 1867.[Car42] Elie Cartan. Sur les couples de surfaces applicables avec conservation des courburesprincipales. Bull. Sc. Math., 66:55–85, 1942. Oeuvres Complète, Partie III, Volume 2,pp. 1591–1620.[Che85] Shiing-Shen Chern. Deformation of surfaces preserving principal curvatures. In I. Chaveland H.M. Farkas, editors, Differential Geometry and Complex Analysis: A Volume Ded-icated to the Memory of Harry Ernest Rauch, pages 155 – 163, Berlin, 1985. Springer-Verlag.[Gra24] W. C. Graustein. Applicability with preservation of both curvatures. Bull. Amer. Math.Soc., 30(1-2):19–23, 1924.[HK76] W. K. Hayman and P. B. Kennedy. Subharmonic functions. Vol. I. Academic Press[Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathemati-cal Society Monographs, No. 9.[JMN16] Gary R. Jensen, Emilio Musso, and Lorenzo Nicolodi. Surfaces in classical geometries.Universitext. Springer, Cham, 2016. A treatment by moving frames.[KPP98] George Kamberov, Franz Pedit, and Ulrich Pinkall. Bonnet pairs and isothermic surfaces.Duke Math. J., 92(3):637–644, 1998.[LT81] H. Blaine Lawson, Jr. and Renato de Azevedo Tribuzy. On the mean curvature functionfor compact surfaces. J. Differential Geom., 16(2):179–183, 1981.[RH90] Ioannis M. Roussos and Gastón E. Hernández. On the number of distinct isometricimmersions of a Riemannian surface into R3 with given mean curvature. Amer. J. Math.,112(1):71–85, 1990.[Rou99] Ioannis M. Roussos. Global results on Bonnet surfaces. J. Geom., 65(1-2):151–168, 1999.[Sab12] I. Kh. Sabitov. Isometric surfaces with common mean curvature and the problem ofBonnet pairs. Sbornik: Mathematics, 203(1):111–152, 2012.(G. R. Jensen) Department of Mathematics, Washington University, One BrookingsDrive, St. Louis, MO 63130, USAE-mail address: gary@wustl.edu(E. Musso) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Ducadegli Abruzzi 24, I-10129 Torino, ItalyE-mail address: emilio.musso@polito.it(L. Nicolodi) Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Univer-sità di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, ItalyE-mail address: lorenzo.nicolodi@unipr.it

Compact surfaces with no Bonnet mate

Abstract

Similar works

Full text

Available Versions

Archivio istituzionale della Ricerca - Università degli Studi di Parma

PORTO Publications Open Repository TOrino

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

Crossref