aa r X i v : . [ m a t h . DG ] M a r A SCHWARZ-PICK LEMMA FOR MINIMAL MAPS
ANDREAS SAVAS-HALILAJAbstract.
In this note, we prove a Schwarz-Pick type lemmafor minimal maps between negatively curved Riemannian surfaces.More precisely, we prove that if f : M → N is a minimal mapwith bounded Jacobian between two complete negatively curvedRiemann surfaces M and N whose sectional curvatures σ M and σ N satisfy inf σ M ≥ sup σ N , then f is area decreasing. Introduction
According to the Schwarz-Pick Lemma, any non-linear holomorphicmap f : B → B from the unit disc B of the complex plane C to itselfmust be strictly distance decreasing if we endow B with the Poincar´emetric. Ahlfors [1] exposed in his generalization of the Schwarz-PickLemma the essential role played by the curvature. In the matter offact, Ahlfors generalized this lemma to holomorphic mappings betweentwo Riemann surfaces where the curvature of the domain manifold isbigger than the curvature of the target. Ahlfors’ result was extendedby Yau [24]. Yau showed that if M is a complete K¨ahler manifold withRicci curvature bounded from below by a constant and N is anotherHermitian manifold with holomorphic bisectional curvature boundedfrom above by a negative constant, then any holomorphic mappingfrom M into N decreases distances up to a constant depending only onthe curvatures of M and N . In his proof, Yau exploited a maximumprinciple at infinity for bounded functions on complete non-compactRiemannian manifolds with Ricci curvature bounded from below; seefor more details [23]. Mathematics Subject Classification.
Primary 53C40; 58J05; 53A07.
Key words and phrases.
Minimal surfaces, Omori-Yau maximum principle, areadecreasing maps.The author would like to acknowledge support by the General Secretariat forResearch and Technology (GSRT) and the Hellenic Foundation for Research andInnovation (HFRI) Grant No:133.
In this paper, we investigate minimal maps between complete Riemannsurfaces. According to the terminology introduced by Schoen [15], asmooth map f : ( M, g M ) → ( N, g N ) is called minimal , if its graphΓ( f ) := (cid:8) ( x, f ( x )) ∈ M × N : x ∈ M (cid:9) is a minimal surface of the Riemannian product ( M × N, g M × g N ).There are two important categories of minimal maps between Riemannsurfaces. The first class contains the holomorphic and anti-holomorphicmaps and the second one the minimal symplectic maps. Eells [7] provedthat a holomorphic or anti-holomorphic map is automatically minimal.Notice that when both M and N are compact, then there are non-constant holomorphic maps between them only if the genus of M isgreater or equal than the genus of N . On the other hand, the graphof a minimal symplectic map is a minimal Lagrangian surface. Schoen[15] proved an existence and uniqueness result for minimal symplecticdiffeomorphisms between hyperbolic surfaces. i.e., if g and g arehyperbolic metrics on a surface M , then there is a unique minimalmap f : ( M, g ) → ( M, g ) homotopic to the identity map. If M is a compact hyperbolic surface, due to results of Smoczyk [17] andWang [22] the mean curvature flow deforms a symplectomorphism intoa minimal Lagrangian map.Let us mention here that Aiyama, Akutagawa and Wan [3] obtaineda representation formula for a minimal diffeomorphism between twohyperbolic discs by means of the generalized Gauss map of a completemaximal surface in the anti-de Sitter 3-space.In this paper, we prove a Schwarz-Pick type lemma for minimal mapsbetween two negatively curved Riemann surfaces ( M, g M ) and ( N, g N ).Under natural assumptions, we show that a minimal map from M to N decreases two dimensional areas. This means that the absolute value | J f | of the Jacobian determinant J f := det(d f ) of f , with respect to theRiemannian metrics g M and g N , is less or equal than 1. Such maps arecalled area decreasing. In the case where the Jacobian J f is identically1, the map f is called area preserving. Theorem.
Let f : ( M, g M ) → ( N, g N ) be a minimal map, with boundedJacobian determinant, between complete negatively curved Riemanniansurfaces whose sectional curvatures σ M and σ N satisfy σ M ≥ − σ ≥ σ N ≥ − β, where σ is a positive constant. Then f is area decreasing.If, additionally, there exists a point where f is area preserving, then σ M = σ N = − σ and the graph of f is a minimal Lagrangian surface. INIMAL MAPS 3
The author and Smoczyk proved in [13] that the mean curvature flowof maps between two compact hyperbolic Riemann surfaces preservesthe graphical and the area decreasing property. Due to a result of Wan[20] minimal maps between complete Riemann surfaces satisfying theassumptions of our theorem are stable. Hence, in view of our result,the graphical mean curvature flow can be used to generate all minimalmaps between compact hyperbolic Riemann surfaces.Finally, let us mention that recently there were proved several Bernsteintype results for minimal maps. In [8, 9] it is shown that a minimalmap f : R → R with bounded Jacobian determinant must be affinelinear. This result was recently generalized by Jost, Xin and Yang in[10]. However, such a result is not true without any assumption on theJacobian determinant. For example, the map f : R → R given by f ( x, y ) = 12 (cid:0) e x − e − x (cid:1)(cid:0) cos y/ , − sin y/ (cid:1) is a minimal map whose Jacobian determinant takes every value in R . Moreover, the graph of f is not holomorphic with respect to anycomplex structure of R . Furthermore, due to a result of Torralbo andUrbano [18], the graph of any minimal map f : S → S must beholomorphic or anti-holomorphic.2. Maximum principles at infinity
The root of the maximum principle relies on the following observation:
Suppose that u : M → R is a smooth function defined on a Riemannianmanifold M and assume that it attains at a point x a local maximum.Then, |∇ u | ( x ) = 0 and ∆ u ( x ) ≤ . Such a point always exists in the case where the manifold is compact.However, the situation is different in the case of complete non-compactmanifolds. To handle the non-compact case, Omori [11] proved thefollowing criterion:
Suppose that M is a complete and non-compactRiemannian manifold, with sectional curvatures bounded from below bya constant, and u : M → R is a bounded from above smooth function.Then, there exists a sequence of points { x k } k ∈ N such that u ( x k ) ≥ sup u − /k, |∇ u | ( x k ) ≤ /k and ∆ u ( x k ) ≤ /k. (O-Y)The conclusion (O-Y) is known as the Omori-Yau maximum principle .Yau [23] showed that the conclusion (O-Y) holds under the assumptionRicci curvature is bounded from below.
ANDREAS SAVAS-HALILAJ
The result of Omori and Yau was generalized by Chen and Xin [5] toinclude cases where the Ricci curvature may decay at a certain rate.Later Pigola, Rigoli and Setti [12] realized that the validity of (O-Y)does not depend on curvature bounds as much as one would expect; formore details see [12, Theorem 1.9] and [2]. An important case wherethe Omori-Yau maximum principle can be successfully applied is forproperly immersed submanifolds F : Σ → L with bounded length ofthe mean curvature vector on a complete Riemannian manifold L withbounded sectional curvatures; see [12, Example 1.14].3. Geometry of graphs
Let us briefly review some basic facts about the geometry of maps,following closely the presentation in [13, 14].3.1.
Notation.
The graphical submanifoldΓ( f ) = { ( x, f ( x )) ∈ M × N : x ∈ M } generated by the smooth map f : M → N between the Riemannsurfaces ( M, g M ) and ( N, g N ) can be parametrized via the embedding F : M → M × N given by F = ( I, f ) , where I : M → M is theidentity map. Let us denote by π M and π N the natural projectionmaps of M × N . Then, the Riemannian metric g M × N on M × N isgiven by the formula g M × N = π ∗ M g M + π ∗ N g N . We will denote by ˜ R the curvature operator of g M × N . The inducedRiemannian metric g on the graph Γ( f ) of f is given byg = g M + f ∗ g N and its Levi-Civita connection is denoted by ∇ .Around each point x ∈ Γ( f ) we choose an adapted local orthonormalframe { e , e ; e , e } such that { e , e } is tangent and { e , e } is normalto the graph. The components of the second fundamental A of thegraph with respect to the adapted frame { e , e ; e , e } are denoted as A αij := h A ( e i , e j ) , e α i . Latin indices take values 1 and 2 while Greek indices take the values 3and 4. For instance we write the mean curvature vector in the form H = H e + H e . INIMAL MAPS 5
From the
Ricci equation we see that the curvature σ n of the normalbundle of Γ( f ) is given by the formula σ n := R ⊥ = ˜ R + A A − A A + A A − A A . The sum of the last four terms in the above formula is equal to minusthe commutator σ ⊥ of the matrices A = ( A ij ) and A = ( A ij ), i.e., σ ⊥ := h [ A , A ] e , e i = − A A + A A − A A + A A . Singular decomposition.
There is a natural way to diagonalizethe differential d f of f . Indeed, let λ ≤ µ be the eigenvalues f ∗ g N with respect to g M at a fixed point x ∈ M . The corresponding values0 ≤ λ ≤ µ are called singular values of f at x . Then there exists anorthonormal basis { α , α } of T x M , with respect to g M , and { β , β } of T f ( x ) N , with respect to the metric g N , such thatd f ( α ) = λβ and d f ( α ) = µβ . Observe that the vectors v := α √ λ and v := α p µ are orthonormal with respect to the metric g on the graph of f at x .Hence, e := 1 √ λ (cid:0) α ⊕ λβ (cid:1) and e := 1 p µ (cid:0) α ⊕ µβ (cid:1) form an orthonormal basis with respect to the metric g M × N of thetangent space d F ( T x M ) of the graph Γ( f ) at x . Moreover, e := 1 √ λ (cid:0) − λα ⊕ β (cid:1) and e := 1 p µ (cid:0) − µα ⊕ β (cid:1) form an orthonormal basis with respect to g M × N of the normal space N x M of the graph Γ( f ) at the point f ( x ).3.3. Jacobians of the projection maps.
Let ω M denote the K¨ahlerform of the Riemann surface ( M, g M ) and ω N the K¨ahler form of( N, g N ). Let us define the parallel forms ω := π ∗ M ω M and ω := π ∗ N ω N . Consider now two smooth functions u and u given by u := ∗ ( F ∗ ω ) = ∗ (cid:8) ( π M ◦ F ) ∗ ω M (cid:9) = ∗ ( I ∗ ω M )and u := ∗ ( F ∗ ω ) = ∗ (cid:8) ( π N ◦ F ) ∗ ω N (cid:9) = ∗ ( f ∗ ω N ) ANDREAS SAVAS-HALILAJ where here ∗ stands for the Hodge star operator with respect to themetric g. Note that u is the Jacobian of the projection map from Γ( f )to the first factor of M × N and u is the Jacobian of the projectionmap of Γ( f ) to the second factor of M × N . With respect to the basis { e , e , e , e } of the singular decomposition, we have u = 1 p (1 + λ )(1 + µ ) and | u | = λµ p (1 + λ )(1 + µ ) . The
Jacobian determinant J f of f , with respect to the metrics g M andg N , is the function defined by the formula f ∗ ω N = J f ω M . Therefore, J f = u u . The difference u − | u | measures how far the map f is from being areapreserving. In particular, we say that f is area decreasing if u −| u | ≥ strictly area decreasing if u − | u | >
0. If u − | u | ≡
0, then f iscalled area preserving . Clearly, in the latter situation f is symplectic.3.4. The K¨ahler angles.
There are two natural complex structuresassociated to the product space ( M × N, g M × N ), i.e., J := π ∗ M J M − π ∗ N J N and J := π ∗ M J M + π ∗ N J N , where J M and J N are the complex structures on M and N defined by ω M ( · , · ) = g M ( J M · , · ) and ω N ( · , · ) = g N ( J N · , · ) . Chern and Wolfson in [6] introduced a function which measures thedeviation of d F ( T x M ) from a complex line of the space T F ( x ) ( M × N ).More precisely, if we consider ( M × N, g M × N ) as a complex manifoldwith respect to J then its corresponding K¨ahler angle a is given bycos a = ϕ := g M × N (cid:0) J d F ( v ) , d F ( v ) (cid:1) = u − u . For our convenience we may require that a ∈ [0 , π ]. Observe that,although ϕ is smooth, in general a is not smooth at points where ϕ = ± . If there exists a point x ∈ M where a ( x ) = 0 then d F ( T x M )is a complex line of T F ( x ) ( M × N ) and x is called a complex point of F .If a ( x ) = π then d F ( T x M ) is an anti-complex line of T F ( x ) ( M × N ) and x is said anti-complex point of F . In the case where a ( x ) = π/
2, thepoint x is called Lagrangian point of the map F . In this case u = u . INIMAL MAPS 7
Similarly, if we regard ( M × N, g M × N ) as a K¨ahler manifold with respectto the complex structure J , then its corresponding K¨ahler angle a isdefined by the formulacos a = ϑ := g M × N (cid:0) J d F ( v ) , d F ( v ) (cid:1) = u + u . Notice that f is area decreasing if and only if both functions ϕ and ϑ are non-negative. Moreover, observe that there are no points on M where ϕ = − ϑ = −
1. If M is complete and non-compact, theninf ϕ = − ϑ = − λ and µ of f tends to infinity.4. Bochner formulas for the Jacobians
We will derive here the derivative and the Laplacian of a parallel 2-formon the product manifold M × N . The proofs are straightforward, makeuse of the Gauss-Codazzi equations and can be found in [22] (see also[13]). For this reason we omit them. From now we will always assumethat f is a minimal map. Lemma 4.1.
Let ω be a parallel -form on the product manifold M × N .Then the covariant derivative of the form F ∗ ω is given by ( ∇ e k F ∗ ω ) ij = X α (cid:0) A αki ω αj + A αkj ω iα (cid:1) , for any adapted orthonormal frame field { e , e ; e , e } . Again by a direct computation we can show the following formula onthe Laplacian of a parallel 2-form on the product manifold M × N . Lemma 4.2.
Let Ω be a parallel -form on the product manifold M × N .The Laplacian of the form F ∗ ω is given by the following formula − (∆ F ∗ ω ) ij = X α,k,l (cid:0) A αki A αkl ω lj + A αkj A αkl ω il (cid:1) − X α,β,k A αki A βkj ω αβ + X α,k (cid:0) ˜ R kikα ω αj + ˜ R kjkα ω iα (cid:1) where { e , e ; e , e } is an arbitrary adapted local orthonormal frame. From Lemma 4.2 we can compute the Laplacian of the Jacobians u and u . Lemma 4.3.
The Jacobian functions u and u satisfy the followingcoupled system of partial differential equations − ∆ u = k A k u + 2 σ ⊥ u + σ M (cid:0) − u − u (cid:1) u − σ N u u , − ∆ u = k A k u + 2 σ ⊥ u + σ N (cid:0) − u − u (cid:1) u − σ M u u . ANDREAS SAVAS-HALILAJ
Using the special frames introduced in subsection 3.2, from Lemma 4.1and Lemma 4.3, by a direct computation we deduce the following:
Lemma 4.4.
The gradients of the functions ϕ and ϑ are given by theequations k∇ ϕ k = (cid:0) | A | − σ ⊥ (cid:1)(cid:0) − ϕ (cid:1) & 2 k∇ ϑ k = (cid:0) | A | + 2 σ ⊥ (cid:1)(cid:0) − ϑ (cid:1) . Moreover, the functions ϕ and ϑ satisfy the following coupled systemof partial differential equations − ∆ ϕ = (cid:0) | A | − σ ⊥ (cid:1) ϕ + (cid:0) σ M ( ϕ + ϑ ) + σ N ( ϕ − ϑ ) (cid:1) (1 − ϕ ) , − ∆ ϑ = (cid:0) | A | + 2 σ ⊥ (cid:1) ϑ + (cid:0) σ M ( ϕ + ϑ ) − σ N ( ϕ − ϑ ) (cid:1) (1 − ϑ ) . Observe that away from complex or anti-complex points the secondfundamental form quantities | A | + 2 σ ⊥ and | A | − σ ⊥ are expressedin terms of the cosines of the K¨ahler angles of the graph and of theirgradients. 5. Proof of the theorem
From our assumptions, the Omori-Yau maximum principle is valid inour setting. It suffices now to prove that both inf ϕ and inf ϑ are non-negative numbers. Suppose to the contrary that inf ϕ <
0. Note thatsince by assumption J f is bounded, it follows that inf ϕ > −
1. Hencefrom the Omori-Yau maximum principle we have that there exists asequence { x k } k ∈ N , such thatlim ϕ ( x k ) = inf ϕ, lim |∇ ϕ | ( x k ) = 0 and lim ∆ ϕ ( x k ) ≥ . From Lemma 4.4 we have that − ∆ ϕ ( x k ) = 2 ϕ ( x k )1 − ϕ ( x k ) |∇ ϕ | ( x k ) + σ N ( x k ) ϕ ( x k ) (cid:0) − ϕ ( x k ) (cid:1) + 12 (cid:0) σ M ( x k ) − σ N ( x k ) (cid:1)(cid:0) ϕ ( x k ) + ϑ ( x k ) (cid:1)(cid:0) − ϕ ( x k ) (cid:1) . Note that the functions 1 − ϕ and ϕ + ϑ are positive. Hence, becauseof our curvature assumptions the last line of the above equality is non-negative. Passing to the limit we deduce that0 ≥ − σ inf ϕ (cid:0) − (inf ϕ ) (cid:1) > , which leads to a contradiction. Consequently, inf ϕ ≥
0. Similarly, weprove that inf ϑ ≥
0. Hence, the map f must be area decreasing. Thiscompletes the first part of the proof. INIMAL MAPS 9
Let us suppose now that f is an area decreasing map. Then both ϕ and ϑ are non-negative functions. Assume that there is a point x ∈ M where f is area preserving. Without loss of generality, let assume that f is orientation preserving at x . Consequently, ϕ ( x ) = 0 = min ϕ. From Lemma 4.4, we deduce that − ∆ ϕ = (cid:8) | A | − σ ⊥ + σ N (1 − ϕ ) (cid:9) ϕ + 12 (cid:0) σ M − σ N (cid:1)(cid:0) ϕ + ϑ (cid:1)(cid:0) − ϕ (cid:1) ≥ (cid:8) | A | − σ ⊥ + σ N (1 − ϕ ) (cid:9) ϕ. Then from Hopf’s strong minimum principle we deduce that ϕ mustvanish identically. Going back to the above identity we obtain that σ M = − σ = − σ N , everywhere. This completes the proof of the theorem. Remark 5.1.
Let us conclude now our paper with some final commentsand remarks.(a) It was very crucial in our proof that the second fundamental terms | A | ± σ ⊥ , were expressed as gradient terms of the cosines ϕ and ϑ of the K¨ahler angles of the graph. However, such a good structureis not available in higher dimensions and codimensions.(b) There are various Schwarz-Pick type results for harmonic maps inthe literature; see for instance [4, 16, 19]. On the other hand, aminimal map f between two Riemannian manifolds ( M, g M ) and( N, g N ) becomes harmonic if we equip M with the graphical metricg = g M + f ∗ g N . As one can see from the singular value decomposition, the map f : ( M, g) → ( N, g N ) is already length decreasing, since its singularvalues are λ √ λ and µ p µ . Hence, one cannot deduce a Schwarz-Pick type result for minimalmaps by applying directly the already known results for harmonicmaps.(c) If the map f is holomorphic or anti-holomorphic then, accordingto the result of Yau [24], it is length decreasing without imposingapriori anything on the size of the differential of f . References [1] L.V. Ahlfors,
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Andreas Savas-HalilajUniversity of IoanninaSection of Algebra & GeometryUniversity Campus45110 IoanninaGreece