The correspondance is established between the sigma models, the minimal
surfaces and the Monge-Ampere equation. The Lax -Pairs of the minimality
condition of the minimal surfaces and the Monge-Ampere equations are given.
Existance of infinitely many nonlocal conservation laws is shown and some
Backlund transformations are also given.Comment: Latex, 13pp, to be published in Letters in Mathematical Physic

Gurses, Metin

English

arXiv

Correspondence is established between sigma models, minimal surfaces and the Monge-Ampére equation. The Lax pairs of the minimality condition of the minimal surfaces and the Monge-Ampére equations are given. Existence of infinitely many nonlocal conservation laws is shown and some Bäcklund transformations are also given

Gürses, Metin

Bilkent University Institutional Repository

See	discussions,	stats,	and	author	profiles	for	this	publication	at:	https://www.researchgate.net/publication/290266970Sigma	Models	and	Minimal	SurfacesArticle	·	April	1998CITATION1READS231	author:Some	of	the	authors	of	this	publication	are	also	working	on	these	related	projects:Nonlocal	Reduction	of	Integrable	Equations	View	projectİntegrable	Surfaces	View	projectMetin	GürsesBilkent	University173	PUBLICATIONS			1,688	CITATIONS			SEE	PROFILEAll	content	following	this	page	was	uploaded	by	Metin	Gürses	on	18	October	2016.The	user	has	requested	enhancement	of	the	downloaded	file.Letters in Mathematical Physics44: 1–8, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.1Sigma Models and Minimal SurfacesMETIN GÜRSESDepartment of Mathematics, Faculty of Science, Bilkent University, 06533 Ankara, Turkeye-mail: gurses@fen.bilkent.edu.tr(Received: 30 May 1997)Abstract. Correspondence is established between sigma models, minimal surfaces and the Monge–Ampére equation. The Lax pairs of the minimality condition of the minimal surfaces and the Monge–Ampére equations are given. Existence of infinitely many nonlocal conservation laws is shown andsome Bäcklund transformations are also given.Mathematics Subject Classifications (1991):81T20, 53A07, 53A10, 35Q58.Key words: sigma models, minimal surfaces, integrability, conservation laws, Bäcklund transforma-tions.1. In a recent paper [1], we investigated the classical integrability of the sigmamodels in a non-Riemannian background and gave their one-soliton Bäcklundtransformations. In particular, two-dimensional sigma models with a Wess–Zuminoterm have been studied in detail.Let M be a two-dimensional manifold with local coordinatesxµ = (t, x) and3µν be the components of a tensor field inM. Let P be a 2×2 matrix with det(P ) =1. We assume thatP is a Hermitian (P † = P ) matrix. Then the sigma model weconsider is given as∂∂xα(3αβP−1∂ P∂xβ)= 0. (1)The integrability of the above equation has been studied in [1]. The uniquenessof the solutions of these equations under certain boundary conditions is given in[2]. In these works, the matrix functionP and the tensor3α β were consideredto be independent. We have classified possible forms of the tensor3α β under thecondition of integrability.In some cases, these two quantities may be related. Such a relation may providesome interesting equations. In this Letter, we are interested in the integrabilityproperty of such cases. As an example, letP = g, whereg is matrix repre-mat97069.tex; 20/04/1998; 8:42; p.1(Kb. 6) INTERPRINT: S.A. PIPS Nr.: 160275 (mathkap:mathfam) v.1.152 METIN GÜRSESsenting the metricgα β symmetric with respect to the lower indices. Also letting3αβ = gα β , the inverse components of the metricgα β , then (1) becomes∂∂xα(gαβg−1∂ g∂xβ)= 0. (2)In the theory of surfaces inR3 there is a class, the minimal surfaces of whichhave special importance both in physics and mathematics [3, 4]. LetS = {(t, x, z) ∈R3; z = h(t, x)} define a surfaceS ∈ R3 which is the graph of a differentiablefunctionh(t, x). This surface is called minimal ifh satisfies the condition(1+ h,2x ) h,tt −2h,x h,t h,xt +(1+ h,2t ) h,xx = 0, (3)The Gaussian curvatureK of the surfaceS is given byK = h,xx h,tt −h,2xt(1+ h,2x +h,2t )2. (4)2. The sigma model equation (1) is integrable for certain choices of the tensor field3αβ . In two dimensions, the integrability conditions on this tensor are given by∂α(1σ3α β ∂β σ)= 0, ∂α(1σ3β α ∂β φ)= 0, (5)whereσ is the determinant andφ is its antisymmetric part of the tensor field3α β .Hence, by letting3αβ = gα β , the above conditions are trivially satisfied becauseσ = 1 andφ = 0. Then using the approach developed in [1], it is straightforwardto show that (2) is also integrable. This leads to the following proposition:PROPOSITION 1.The Lax pair of (2) isεα β∂∂xβ9 = 1k2+ 1 (k gα β − εα β) g−1 ∂ g∂xβ9, (6)provideddet(g) = 1 and gα β is symmetric. Herek is an arbitrary constant (thespectral parameter) andεα β is the Levi-Civita tensor withε12 = 1.A standard parametrization ofgα β may be given asds2 = gα β dxα dxβ = 1w[(1+ a2)dt2 + 2a b dx dt + (1+ b2)dx2], (7)wherexα = (t, x), a andb are differentiable functions oft andx andw2 = 1+ a2+ b2. (8)mat97069.tex; 20/04/1998; 8:42; p.2SIGMA MODELS AND MINIMAL SURFACES 3PROPOSITION 2.Leth be a differentiable function oft andx and leta = h,t andb = h,x, then the minimality condition (3) solves the sigma model equation (2).This result is interesting and also very important. We shall give the Lax pair (6) in amore detailed way, but before that we write the minimality condition in a covariantway. The metric on this minimal two-dimensional surfaceS is(ds)2m = g(m)µ ν dxµ dxν (9)= (1+ h,2t )dt2 + 2h,t h,x dx dt + (1+ h,2x )dx2. (10)Then the minimality condition (3) may be written covariantly asgα β(m) ∂α ∂β h = 0. (11)Sinceg(m) µν = δµν + h,µ h,ν , whereδµν is the Kronecker delta symbol, (11) isalso equivalent to∂α (√g(m) gα β(m)) = 0, (12)whereg(m) is the determinant of the metricg(m) α β on S. S is embedded in aflat three-dimensional Euclidean spaceR3 with metric ds2 = dt2 + dx2 + dz2.The minimality conditions (11) and(12) are equivalent to the harmonicity of thefunctionh(t, x) with respect to the metric ofS∂α (√g(m) gα β(m) ∂β h) = 0. (13)In the language of harmonic mappings of Riemannian manifolds [5], Equations(11), (12), and (13) imply that the mappingxα : S → S is harmonic. Here wewould like remark that the nonlinear partial differential equation (3) describingthe minimality condition of a two-dimensional surfaceS is a special case of thesigma model equation (2). Hence, it straightforward to conclude that Equation(3) is integrable and its Lax pair is given in (6). We shall now give this Lax pairmore explicitly. LetA = g−1 ∂t g andB = g−1 ∂x g be two 2× 2 matrices withcomponentsA11 =1w2[p(1+ q2)r − q(1+ p2)s], (14)A12 =1w2[q(1+ q2)r + p(1− q2)s], (15)A21 =1w2[q(1− p2)r + p(1+ p2)s], (16)A22 = −1w2[p(1+ q2)r − q(1+ p2)s], (17)mat97069.tex; 20/04/1998; 8:42; p.34 METIN GÜRSESB11 =1w2[p(1+ q2)s − q(1+ p2)t], (18)B12 =1w2[q(1+ q2)s + p(1− q2)t], (19)B21 =1w2[q(1− p2)s + p(1+ p2)t], (20)B22 = −1w2[p(1+ q2)s − q(1+ p2)t], (21)where we have used the same notation as used in [4]p = ht, q = hx, r = htt, s = htx, t = hxx, (22)w2 = 1+ p2+ q2. (23)Then the Lax pair becomes9,x = − 1k2+ 1 [k(−r′A+ q ′ B)+ B]9, (24)9,t = − 1k2+ 1 [k(−q′A+ p′ B)+A]9, (25)wherek is the spectral parameterp′, q ′ andr ′ are given byp′ = 1+ p2w, q ′ = p qw, r ′ = 1+ q2w. (26)Integrability of Equations (24) and (25) gives(r ′A− q ′ B),t + (p′ B − q ′A),x = 0, (27)A,x − B,t = [A,B]. (28)The first of the above equations is identical with the minimality condition (3) andthe second one is a trivial identity.3. From the Lie symmetries of the minimality condition, it may be possible to findsome conservation laws. Some of these are given by [4]( qw),x+(pw),t= 0, (29)mat97069.tex; 20/04/1998; 8:42; p.4SIGMA MODELS AND MINIMAL SURFACES 5(p qw),x+(−(1+ q2)w),t= 0, (30)((1+ p2)w),x+(−p qw),t= 0. (31)These conservation laws are local in the following sense. In general, any conserva-tion law can be written asX,x = T,t , whereX andT are functions ofh, p, q, r, s, t ,and higher derivatives of these functions with respectx and t . Such conservationlaws are the local ones. In the case of nonlocal conservation laws, the functionsX and T depend, in addition toh, p, q, r, s, t , and higher derivatives of thesefunctions with respectx andt , upon the integrals of these variables with respect tox and t . One can find such conservation laws in this case as well. Let us assumethat the function9 in (24)–(25) is analytic in the parameterk and can be expandedas9 = 90+ k 91+ k292+ · · · . (32)Then Equations (24)–(25) imply90 = g−1, (33)(g 91),x = −gM g−1, (34)(g 91),t = −g N g−1, (35)(g 92),x = gx g−1− gM g−1D−1x g M g−1, (36)(g 92),t = gt g−1− g N g−1D−1x g N g−1, (37). . . . . . . . . . . . . . . . . . . ,whereD−1x andD−1t are, respectively, the inverse operators of the total derivativeswith respect tox andt andM = −r ′ g−1g,t + q ′ g−1g,x, N = −q ′ g−1gt + p′ g−1gx. (38)Hence, we have now infintely many conservation laws with functionsXn andTn forall n = 0,1,2, . . .. The first two members may be given from the above equations:X0 = M, T0 = N, (39)X1 = g−1 g,x + (D−1x M)M, T1 = g−1 g,t + (D−1t N)N, (40). . . . . . . . . . . . . . . . . . . .mat97069.tex; 20/04/1998; 8:42; p.56 METIN GÜRSESIn this way, one can find infinitely many nonlocal conservation laws.4. The Bäcklund transformation obtainable from the Lax pair (24)–(25) is notsuitable because the correspondence between the new and old solutions will beof the same degree as of that of the minimality condition. Hence, one has tosolve a second-order differential equation which is as hard as the original equation.Instead, we shall mention two interesting nonauto-Bäcklund transformations.The solution of (3) can be expressed in terms of two harmonic functions.PROPOSITION 3.Let x and t be harmonic functions ofu andv and let a differ-entiable functionh(t, x) be defined by[1+ p2] t,u= −w x,v −q p x,u , [1+ p2] t,v = −w x,u−q p x,vThen the functionh(t, x) is a harmonic function ofu andv if and only if it satisfiesthe minimality condition (3).This proposition implies that the functionh(t, x) can be constructed in terms oftwo harmonic functionst (u, v) andx(u, v). The functionh(t, x) obtained in thisway automatically satisfies the minimality condition (3). In this case, the metric(10) on the two-dimensional surfaceS takes the conformally flat formds2(m) = w2(x,2u+x,2v1+ p2)(du2+ dv2). (41)Here we understand that the minimality condition (3) arises from a sigma modelso that the target and base space metrics are the same. Such a sigma model has aLax pair defined in the linear equation (6) in Proposition 1 (or in (24) – (25)). ThisLax pair may be used to construct Bäcklund transformation for Equation (3) (theminimality condition). Instead of following such a direction, we find the Bäcklundtransformation by defining a new 2× 2 matrix function Q,gα β g−1 ∂β g = εα β ∂β Q (42)PROPOSITION 4. (a)Equation corresponding to the matrixQ is∂α (gα β ∂β Q)− εα β ∂α Q ∂β Q = 0. (43)(b) The corresponding linear equation isεα β ∂β 9 = 1k2+ 1 (k εα β + gα β) ∂βQ 9 (44)There is a second Bäcklund transformation for Equation (3) obtainable simply byusing either (6) or (44).mat97069.tex; 20/04/1998; 8:42; p.6SIGMA MODELS AND MINIMAL SURFACES 7PROPOSITION 5.Letz = h(t, x) define a minimal surface embedded in the threedimensional Euclidean spaceR3. The following transformation:h,xw= ψ,t , h,tw= −ψ,x , (45)maps the minimality condition(3) to the equation(1− ψ,2x ) ψ,tt +2ψ,x ψ,t ψ,xt +(1− ψ,2t ) ψ,xx = 0. (46)This equation defines a minimal surfaceS′ = ((t, x,w′) : w′ = ψ(t, x)). S′is embedded in a three-dimensional Minkowski spaceM3 with the metric ds2 =dt2 + dx2 − dw′ 2. The metric onS′ is given byds′ 2(m) = g′(m) α β dxα dxβ= (1− ψ,2t )d t2 − 2ψ,t ψ,x dx d t + (1− ψ,2x )dx2. (47)The minimality condition (46) for the surfaceS′ may be written asg′ α β(m) ψ,α β = 0. (48)As an illustration to the above transformation (45), we can give the followingnontrivial examples. The minimal surfacesψ = 1λ[ln cosh(λ t)− ln cosh(λ x)]andh = 1λcos−1 [sinh(λ t) sinh(λ x)]are transformable to each other. Hereλ is a nonvanishing constant.Finally we would like to mention another Bäcklund transformation which mapssolutions of the minimality condition to the solutions of the Monge–Ampere equa-tion. This is given by the following proposition:PROPOSITION 6.Let the functionh(t, x), with enough differentiability, satisfythe minimality condition (3), then the metricgµν = (1/w) g(m)µ ν satisfies thecondition∂α gµν = ∂ν gµα, (49)which also implies thatgµν = ∂µ ∂ν u, (50)mat97069.tex; 20/04/1998; 8:42; p.78 METIN GÜRSESwhereu(t, x) is enough differentiable function oft, x satisfying the equationDet(∂µ ∂ν u) = u,tt u,xx − u2tx = 1. (51)This is the equation known as the Monge–Ampére equation which is also integrableand its Lax Pair can be easily obtained by using (50) in (6) or in (24)–(25). Hyper-bolic minimal surfaces also have similar correspondence with the Monge–Ampéreequation. Using (46) and (47) we haveg′µν = ∂µ ∂ν u, (52)withDet(∂µ ∂ν u) = u,tt u,xx − u2tx = 1. (53)which does not give the hyperbolic Monge–Ampére equation as expected. Thecorrespondence between the minimal surfaces inR3 and the Monge–Ampére equa-tion is mentioned in [6, 7]. The correspondence between the Born–Infeld and thehyperbolic Monge–Ampére equation is mentioned in [8].AcknowledgementThis work is partially supported by the Scientific and Technical Research Councilof Turkey (TÜḂITAK) and Turkish Academy of Sciences (TÜBA).References1. Gürses, M. and Karasu, A.:Internat J. Modern Phys. A.6 (1991), 487.2. Gürses, M.:Lett. Math. Phys26 (1992), 265.3. do Carmo, M.:Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976.4. Dierken, U., Hildebrandt, S., Künster, A. and Wohlrab, O.:Minimal Surfaces I, GrundlehrenMath. Wiss. 295, Springer-Verlag, Berlin, Heidelberg, 1992.5. Eells, J. and Sampson, J. H.:Amer. J. Math.86 (1964), 109.6. Jörgens., K.:Math. Annal.127(1954), 130.7. Heinz, E.:Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl.IIa (1952), 51.8. Mokhov, O. I. and Nutku, Y.:Lett. Math. Phys.32 (1994), 121.mat97069.tex; 20/04/1998; 8:42; p.8View publication stats

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