Bernstein theorems for length and area decreasing minimal maps

Transcription

1 Noname manuscript No. will be inserted by the editor Bernstein theorems for length and area decreasing minimal maps Andreas Savas-Halilaj Knut Smoczyk Received: date / Accepted: date Abstract In this article we prove Liouville and Bernstein theorems in higher codimension for length and area decreasing maps between two Riemannian manifolds. The proofs are based on a strong elliptic maximum principle for sections in vector bundles, which we also present in this article. Keywords Strong maximum principle sections tensors minimal maps parallel mean curvature Bernstein theorems Mathematics Subject Classification C40 53A07 35J47 58J05 1 Introduction The famous Bernstein theorem [4] states that all complete minimal graphs in the three dimensional euclidean space are generated by affine maps. This result cannot be extended to complete minimal graphs in any euclidean space without imposing further assumptions. There is a very rich and long literature concerning complete minimal graphs which are generated by maps between euclidean spaces, marked by works of W. Fleming [16], S.-S. Chern and R. Osserman [8], J. Simons [30], E. Bombieri, E. de Giorgi and E. Giusti [6], R. Schoen, L. Simon and S.-T. Yau [29], S. Hildebrandt, J. Jost and K.-O. The first author is supported financially by the grant EΣΠA: PE1-417 A. Savas-Halilaj K. Smoczyk Leibniz Universität Hannover, Institut für Differentialgeometrie and Riemann Center for Geometry and Physics, Welfengarten 1, Hannover, Germany savasha@math.uni-hannover.de K. Smoczyk smoczyk@math.uni-hannover.de

2 2 Andreas Savas-Halilaj, Knut Smoczyk Widmann [21] and many others. In the last decade there have been obtained several Bernstein theorems in higher codimension, see for instance [31], [25], [37] [20,19] and [23]. The generalized Bernstein problem that we are investigating is to determine under which additional geometric conditions minimal graphs that are generated by maps f : M N, where M, N are Riemannian manifolds, are totally geodesic. There are several recent results involving mean curvature flow in the case where both M and N are compact. For instance, we mention [38, 39,36], [32], [34] and [24]. In these papers the authors prove that the mean curvature flow of graphs, generated by smooth maps f : M N satisfying suitable conditions, evolves f to a constant map or an isometric immersion as time tends to infinity. This implies in particular Bernstein results for minimal graphs satisfying the same conditions as the initial map. Let M, g M and N, g N be Riemannian manifolds of dimensions m and n respectively. For any smooth map f : M N its differential df induces a map Λ k df : Λ k T M Λ k T N given by Λ k df v 1,..., v k := dfv 1 dfv k, for any smooth vector fields v 1,..., v k T M. The map Λ k df is called the k-jacobian of f. Definition 1.1 The map f : M N is called weakly k-volume decreasing resp. strictly k-volume decreasing; k-volume preserving if Λ k df v 1,..., v k gn 1 resp. < 1; = 1, for all {v 1,..., v k } T M orthonormal with respect to g M. As usual for k = 1 we use the term length instead of 1-volume and if k = 2 we use the term area instead of 2-volume. The map f : M N is called an isometric immersion, if f g N = g M. A smooth map f : M N is called minimal, if its graph Γ f := {x, fx M N : x M} is a minimal submanifold of M N, g M N = g M g N. One of the main objectives in the present article is to prove the following results: Theorem A. Let M, g M and N, g N two Riemannian manifolds. Suppose that M is compact, m = dim M 2 and that there exists a constant σ > 0 such that the sectional curvatures σ M of M and σ N of N and the Ricci curvature Ric M of M satisfy σ M > σ and Ric M m 1σ m 1σ N. If f : M N is a minimal map that is weakly length decreasing, then one of the following holds:

3 Bernstein theorems for minimal maps 3 a f is constant. In particular, any strictly length decreasing minimal map is constant. b f is an isometric totally geodesic immersion, M has constant sectional curvature σ and the restriction of σ N to dft M is equal to σ. Moreover, Γ f is a totally geodesic submanifold of M N. A similar statement holds in the case of weakly area decreasing maps. Theorem B. Let M, g M and N, g N two Riemannian manifolds. Suppose that M is compact, m = dim M 2 and that there exists a constant σ > 0 such that the sectional curvatures σ M of M and σ N of N and the Ricci curvature Ric M of M satisfy σ M > σ and Ric M m 1σ m 1σ N. If f : M N is a smooth minimal map that is weakly area decreasing, then one of the following holds: a f is constant. In particular, any strictly area decreasing minimal map is constant. b There exists a non-empty and closed set D such that f is an isometric immersion on D and f is strictly area decreasing on the complement of D. Moreover, M is Einstein on D with Ric M = m 1σ and the restriction of σ N to dft D is equal to σ. Furthermore, if D has an interior point then Γ f is totally geodesic. In the special case where the manifold N is one-dimensional we have the following stronger theorem: Theorem C. Let M, g M and N, g N two Riemannian manifolds. Suppose that M is compact, dim M 2, Ric M > 0 and that dim N = 1. Then any smooth minimal map f : M N is constant. As pointed out in the final remarks of Section 3, Theorems A, B and C are optimal in various ways. We include some examples and remarks concerning the imposed assumptions at the end of the paper. The paper is organized as follows. In Section 2 we recall the strong maximum principle for uniformly elliptic systems of second order and give a strong elliptic maximum principle for sections in vector bundles. The geometry of graphs will be treated in Section 3, where we also derive the crucial formula needed in the proof of Theorems A, B and C. 2 Strong elliptic maximum principles for sections in vector bundles In this section we shall derive a strong elliptic maximum principle for sections in Riemannian vector bundles. The original idea goes back to the fundamental

4 4 Andreas Savas-Halilaj, Knut Smoczyk work of Hamilton [17,18] on the Ricci flow, where a strong parabolic maximum principle for symmetric tensors and weak parabolic maximum principles for sections in vector bundles were proven. 2.1 Convex sets In this subsection we review the basic facts about the geometry of convex sets in euclidean space such as supporting half-spaces, tangent cones and normal vectors. A brief exposition can be found in the book by Andrews and Hopper [3, Appendix B]. Recall that a subset K of R n is called convex if for any pair of points z, w K, the segment E z,w := {tz + 1 tw R n : t 0, 1} is contained in K. The set K is said to be strictly convex, if for any pair z, w K the segment E z,w belongs to the interior of K. A convex set K R n may have non-smooth boundary. Hence, there is no well-defined tangent or normal space of K in the classical sense. However, there is a way to generalize these important notions for closed convex subsets of R n. This difficulty can be overcome by using the property that points lying outside of the given set can be separated from the set itself by half-spaces. This property, leads to the notion of generalized tangency. Let K be a closed convex subset of the euclidean space R n. A supporting half-space of the set K is a closed half-space of R n which contains K and has points of K on its boundary. A supporting hyperplane of K is a hyperplane which is the boundary of a supporting half-space of K. The tangent cone C y0 K of K at y 0 K is defined as the intersection of all supporting half-spaces of K that contain y 0. We may also introduce the notion of normal vectors to the boundary of a closed convex set. Let K R n be a closed convex subset and y 0 K. Then, a non-zero vector ξ is called normal vector of K at y 0, if ξ is normal to a supporting hyperplane of K passing through y 0. This normal vector is called inward pointing, if it points into the half-space containing the set K. A vector η is called inward pointing at y 0 K, if ξ, η 0 for any inward pointing normal vector ξ at y 0. Here,, denotes the usual inner product in R n. 2.2 Maximum principles for systems In the case of smooth functions, E. Hopf [22] established a strong maximum principles for a wide class of general second order differential equations. For example, he proved that if a solution u of the uniformly elliptic differential

5 Bernstein theorems for minimal maps 5 equation L u := a ij iju 2 + b j j u = 0, i,j=1 j=1 attains its supremum or infimum at an interior point of its domain of definition D, then it must be constant. The strong elliptic maximum principle of Hopf can be interpreted as follows: If a solution u of L u = 0 maps an interior point of D to the boundary of the set K = inf D u, sup D u, then u maps any point of D to the boundary of K and hence it must be constant. For the proof of this strong maximum principle Hopf used the Hopf Lemma, which implies that the subset B D consisting of points where u attains a value in K is open. Since by continuity B is also closed, one has B = D, if D is connected and B is non-empty. The generalization of Hopf s maximum principle to elliptic and semi-linear parabolic systems has been first considered by H. Weinberger [41], where he established a strong maximum principle for vector valued maps with values in a convex set K R n whose boundary K satisfies regularity conditions that he called slab conditions. Inspired by the ideas of Weinberger, X. Wang [40] gave a geometric proof of the strong maximum principle, in the case where the boundary of K is of class C 2. The idea of Wang was to apply the classical maximum principle of Hopf to the function du : D R, whose value at x is equal to the distance of ux from the boundary K of K. Very recently, L.C. Evans [14] was able to remove all additional regularity requirements on the boundary of the convex set K by showing that even if du is not twice differentiable, it is still a viscosity super-solution of an appropriate partial differential equation. The argument of Evans is completed by applying a strong maximum principle due to F. Da Lio [11] for viscosity super-solutions of partial differential equations. Theorem 2.1 Weinberger-Evans Let K be a closed, convex set of R n and u : D R m K R n a solution of the uniformly elliptic system of partial differential equations L ux + Ψx, ux = 0, x D, where D is a domain of R m, Ψ : D R n R n is a continuous map that is locally Lipschitz continuous in the second variable and L is a uniformly elliptic operator given in. Suppose that a there is a point x 0 in the interior of D such that ux 0 K, b for any x, y D K, the vector Ψx, y points into K at the point y K. Then ux K for any x D. If K is strictly convex, then u is constant.

6 6 Andreas Savas-Halilaj, Knut Smoczyk Remark 2.1 The above maximum principle is not valid without the convexity assumption. We illustrate this by an example. Let D = {x, y R 2 : x 2 + y 2 < 1} be the unit open disc in R 2 and let h : D Γ R 2 be a continuous function that maps D onto the upper semicircle Γ = {x, y R 2 : x 2 + y 2 = 1 and y 0}. Denote by u : D R 2 the solution of the Dirichlet problem with boundary data given by the function h. Let us examine the image of the harmonic map u. We claim at first that the image of u is contained in the convex hull CΓ of the upper semicircle. That is, K := u D CΓ = {x, y R 2 : x 2 + y 2 1 and y 0}. Arguing indirectly, let us assume that this is not true. The convex hull CK of K contains CΓ. Since K is compact, the set CK is also compact. Consequently, there exist a point x 0, y 0 in D such that ux 0, y 0 CK and ux 0, y 0 CΓ. Then, from the maximum principle of Weinberger-Evans we deduce that ux, y CK for any x, y D. This contradicts with the boundary data imposed by the Dirichlet condition. Therefore, K is contained in CΓ. From Theorem 2.1, we conclude that there is no common point of K with the x-axes. Hence, K is not convex. The same argument yields that there is no point of D which is mapped to Γ via u. On the other hand, because K is compact, there are infinitely many points of D which are mapped to the boundary of K. Furthermore, we claim that the set K has non-empty interior. In order to show this, suppose to the contrary that K \ K =. Then, rankdu 1 which implies that the closure of the set ud is a continuous curve L joining the points 1, 0 and 1, 0. But then, the continuity of u leads to a contradiction. Indeed, for any sequence {p k } k N of points of D converging to a point p u 1 0, 1, we have lim up k 0, Maximum principles for sections in vector bundles Here we give the strong elliptic maximum principle for smooth sections in Riemannian vector bundles. This maximum principle is in the most general form and contains all the previous results by Hopf, Weinberger, Evans and it also contains the elliptic version of Hamilton s parabolic maximum principle. It turns out that this principle is extremely powerful and we apply it to derive optimal Bernstein results for minimal graphs generated from smooth maps between Riemannian manifolds.

7 Bernstein theorems for minimal maps 7 In order to state the elliptic version of the strong maximum principle for smooth sections in vector bundles, we must introduce an appropriate notion of convexity for subsets of Riemannian vector bundles. In [17] Hamilton gave the following definition: Definition 2.1 Hamilton Suppose that E, π, M is a vector bundle over the manifold M and let K be a closed subset of E. a The set K is said to be fiber-convex or convex in the fiber, if for each point x of M, the set K x := K E x is a convex subset of the fiber E x = π 1 x. b The set K is said to be invariant under parallel transport, if for every smooth curve γ : [0, b] M and any vector v K γ0, the unique parallel section vt E γt, t [0, b], along γt with v0 = v, is contained in K. c A fiberwise map Ψ : E E is a map such that π Ψ = π, where π denotes the bundle projection. Let K be a fiber-convex closed subset of E. We say a fiberwise map Ψ points into K or is inward pointing, if for any x M and any ϑ K x the vector Ψϑ belongs to the tangent cone C ϑ K x of K x at ϑ. Let E, π, M be a vector bundle of rank k over a smooth manifold M. Suppose g E is a bundle metric on E and that is a metric connection on E. In this paper we consider uniformly elliptic operators L on Γ E of second order that are given locally by L = a ij 2 e i,e j + b j ej, i,j=1 j=1 where a Γ T M T M is a symmetric, uniformly positive definite tensor and b Γ T M is a smooth vector field such that and a = a ij e i e j i,j=1 b = b j e j j=1 in a local frame field {e 1,..., e k } of T M. Next we prove a strong elliptic maximum principle for smooth sections in Riemannian vector bundles. We would like to point out that our approach is inspired by ideas developed by Weinberger [41] and Hamilton [17, 18]. For the proof of the strong maximum principle we will use the following beautiful result due to C. Böhm and B. Wilking [5, Lemma 1.2, p. 670]. Throughout the paper all the manifolds will be considered smooth and connected.

8 8 Andreas Savas-Halilaj, Knut Smoczyk Lemma 2.1 Böhm-Wilking Suppose that M is a Riemannian manifold and that E, π, M is a Riemannian vector bundle over M equipped with a metric connection. Let K be a closed and fiber-convex subset of the bundle E that is invariant under parallel transport. If φ is a smooth section with values in K then, for any x M and v T x M, the Hessian 2 v,vφ = v v φ vvφ belongs to the tangent cone of K x at the point φx. The following result is an immediate consequence of the above lemma. Lemma 2.2 Suppose that M is a Riemannian manifold and that E, π, M is a Riemannian vector bundle over M equipped with a metric connection. Let K be a closed and fiber-convex subset of E that is invariant under parallel transport. If φ is a smooth section with values in K then, for any x M, the vector L φx belongs to the tangent cone C φx K x, for any operator L of the form given in. Now we prove the strong maximum principle for sections in vector bundles. Theorem 2.2 Strong Elliptic Maximum Principle Let M, g M be a Riemannian manifold and E, π, M a vector bundle over M equipped with a Riemannian metric g E and a metric connection. Let K be a closed fiberconvex subset of the bundle E that is invariant under parallel transport and let φ Γ E, φ : M K, be a smooth section such that L φ + Ψφ = 0, where here L is a uniformly elliptic operator of second order of the form given in and Ψ is a smooth fiberwise map that points into K. If there exists a point x 0 in the interior of M such that φx 0 K x0, then φx K x for any point x M. If, additionally, K x0 is strictly convex in a neighborhood of φx 0, then φ is a parallel section. Proof Let {φ 1,..., φ k } be a local geodesic orthonormal frame field of smooth sections in E, which is defined in a sufficiently small neighborhood U of a local trivialization around x 0 M. Hence, φ = k u i φ i i=1 where u i : U R, i {1,..., k}, are smooth functions. With respect to this frame we have k L φ = {L u i + k } gradient terms of u i + u j g E L φ j, φ i φ i i=1 = k g E Ψφ, φ i φ i. i=1 j=1

9 Bernstein theorems for minimal maps 9 Therefore, the map u : U R k, u = u 1,..., u k, satisfies a uniformly elliptic system of second order of the form L u + Φu = 0, 2.1 where here Φ : R k R k, Φ := Φ 1,..., Φ k, is given by k Φ i u = g E Ψ u j φ j + k u j L φ j, φ i, 2.2 for any i {1,..., k}. Consider now the convex set j=1 K := {y 1,..., y k R k : j=1 k y i φ i x 0 K x0 }. i=1 Claim 1: For any point x U we have ux K. Indeed, fix a point x U and let γ : [0, 1] U be the geodesic curve joining the points x and x 0. Denote by θ the parallel section which is obtained by the parallel transport of φx along the geodesic γ. Then, θ γ = k y i φ i γ, i=1 where y i : [0, 1] R, i {1,..., k}, are smooth functions. Because, θ and φ i, i {1,..., k} are parallel along γ, it follows that 0 = t θ γ = k y itφ i γt. Hence, y i t = y i 0 = u i x, for any t [0, 1] and i {1,..., k}. Therefore, θγ1 = θx 0 = i=1 k u i xφ i x 0. Since by our assumptions K is invariant under parallel transport, it follows that θx 0 K x0. Hence, uu K and this proves Claim 1. Claim 2: For any point y K the vector Φy as defined in 2.2 points into K at y. First note that the boundary of each slice K x is invariant under parallel transport. From 2.2 we deduce that it suffices to prove that both terms appearing on the right hand side of 2.2 point into K. The first term points into K by assumption on Ψ. The second term is inward pointing due to Lemma 2.2 by Böhm and Wilking. This completes the proof of Claim 2. The solution of the uniformly second order elliptic partial differential system 2.1 satisfies all the assumptions of Theorem 2.1. Therefore, because ux 0 i=1

10 10 Andreas Savas-Halilaj, Knut Smoczyk K it follows that uu is contained in the boundary K of K. Consequently, φx K for any x U. Since M is connected, we deduce that φm K. Note, that if K is additionally strictly convex at ux 0, then the map u is constant. This implies that φx = k u i x 0 φ i x i=1 for any x U. Consequently, φ is a parallel section taking all its values in K. This completes the proof of Theorem 2.2. Remark 2.2 Suppose the fibers of the set K are strictly convex, have C 2 - smooth boundaries and contain 0. Then, from Theorem 2.2 we deduce that if φx = 0 in a point x M, then φ vanishes everywhere. For the classification of minimal maps between Riemannian manifolds we will later use a special case of Theorem 2.2 for smooth, symmetric 2-tensors φ SymE E. Before stating the result let us recall the following definition due to Hamilton [18, Section 9]. Definition 2.2 Hamilton A fiberwise map Ψ : SymE E SymE E is said to satisfy the null-eigenvector condition, if whenever ϑ is a nonnegative symmetric 2-tensor at a point x M and if v T x M is a nulleigenvector of ϑ, then Ψϑv, v 0. The next theorem is the elliptic analogue of the maximum principle of Hamilton [17, Lemma 8.2, p. 174]. More precisely: Theorem 2.3 Let M, g M be a Riemannian manifold and let E, π, M a Riemannian vector bundle over M equipped with a metric connection. Suppose that φ SymE E is non-negative definite and satisfies L φ + Ψφ = 0, where Ψ is a smooth fiberwise map satisfying the null-eigenvector condition. If there is an interior point of M where φ has a null-eigenvalue, then φ must have a null-eigenvalue everywhere. Proof Let K be the set of non-negative definite symmetric 2-tensors on M, that is K := {ϑ SymE E : ϑ 0}. Each fiber K x is a closed convex cone that is strictly convex at 0. Moreover, K is invariant under parallel transport. The set of all boundary points of K x is given by K x = {ϑ K x : a non-zero v T x M such that ϑv, v = 0}.

11 Bernstein theorems for minimal maps 11 It is a classical fact in Convex Analysis see for example the book [3, Appendix B], that the tangent cone of K x at a point ϑ of its boundary is given by C ϑ K x = {ψ SymE x E x : ψv, v 0, v E x with ϑv, v = 0}. Thus ψ is in the tangent cone of K x at ϑ, if and only if it satisfies the nulleigenvector condition of Hamilton given in Definition 2.2. By Theorem 2.2 we immediately get the proof of Theorem A second derivative criterion for symmetric 2-tensors For φ SymE E a real number λ is called eigenvalue of φ with respect to g E at the point x M, if there exists a non-zero vector v E x, such that φv, w = λg E v, w, for any w E x. The linear subspace Eig λ,φ x of E x given by Eig λ,φ x := {v E x : φv, w = λg E v, w, for any w E x }, is called the eigenspace of λ at x. Since the tensor φ is symmetric it admits k real eigenvalues λ 1 x,..., λ k x at each point x M. We will always arrange the eigenvalues such that λ 1 x λ k x. The following result is due to Hamilton [18]. To make the paper self-contained we include a proof of it. Theorem 2.4 Second Derivative Criterion Suppose that M, g M is a Riemannian manifold and E, π, M a Riemannian vector bundle of rank k over M equipped with a metric connection. Let φ SymE E is a smooth symmetric 2-tensor. If the biggest eigenvalue λ k of φ admits a local maximum λ at an interior point x 0 M, then φv, v = 0 and L φv, v 0, for all vectors v in the eigenspace of λ at x 0 and for all uniformly elliptic second order operators L. Remark. Replacing φ by φ in Theorem 2.4 gives a similar result for the smallest eigenvalue λ 1 of φ. Proof Let v Eig λ,φ x 0 be a unit vector and V Γ E a smooth section such that V x 0 = v and V x 0 = 0. Define the symmetric 2-tensor S given by S := φ λg E. From our assumptions, the symmetric 2-tensor S is non-positive definite in a small neighborhood of

12 12 Andreas Savas-Halilaj, Knut Smoczyk x 0. Moreover, the biggest eigenvalue of S at x 0 equals 0. Consider the smooth function f : M R, given by fx := S V x, V x. The function f is non-positive in the same neighborhood around x 0 and attains a local maximum in an interior point x 0. In particular, fx 0 = 0, dfx 0 = 0 and L fx 0 0. Consider now a local orthonormal frame field {e 1,..., e m } with respect to g M defined in a neighborhood of the point x 0 and assume that the expression of L with respect to this frame is L = A simple calculation yields a ij 2 e i,e j + b j ej. i,j=1 j=1 ei f = dfe i = ei S V, V + 2 S ei V, V. Taking into account that g E is parallel, we deduce that Furthermore, 0 = fx 0 = Sv, v = φv, v. 2 e i,e j f = 2 e i,e j SV, V + 2 SV, 2 e i,e j V +2 ei S ej V, V + 2 ej S ei V, V +2 S ei V, ej V. Bearing in mind the definition of S and using the fact that g E is parallel with respect to, we obtain L f = L φv, V + 2 SV, L V + 2a ij { S ei V, ej V + 2 ei S ej V, V } i,j=1 = L φv, V + 2 SV, L V + 2a ij { S ei V, ej V + 2 ei S ej V, V }. i,j=1 Estimating at x 0 and taking into account that V x 0 = v is a null eigenvector of S at x 0, we get 0 L fx 0 = L φv, v. This completes the proof.

13 Bernstein theorems for minimal maps 13 In order to demonstrate how to apply the strong elliptic maximum principle and the second derivative criterion, we shall give here an example in the case of hypersurfaces in euclidean space. Let M be an oriented m-dimensional hypersurface of R m+1. Denote by ξ a unit normal vector field along the hypersurface. The most natural symmetric 2-tensor on M is the scalar second fundamental form h of the hypersurface with respect to the unit normal direction ξ, that is for any v, w T M. The eigenvalues hv, w := dξv, w, λ 1 λ m of h with respect to the induced metric g are called the principal curvatures of the hypersurface. The quantity H given by H := λ λ m is called the scalar mean curvature of the hypersurface and the function h given by h 2 := λ λ 2 m is called the norm of the second fundamental form with respect to the metric g. It is well known that if h is non-negative definite, then M is locally the boundary of a convex subset of R m+1. For this reason, the hypersurface M is called convex whenever its scalar second fundamental form is non-negative definite. Example 2.1 We will give an alternative short proof of a well-known theorem, first proven by W. Süss [33]. More precisely, we shall show that: Any closed and convex hypersurface M in R m+1 with constant mean curvature is a round sphere. The Laplacian of the second fundamental form h with respect to the induced Riemannian metric g, is given by Simons identity [30] h + h 2 h Hh 2 = 0, 2.3 where h 2 v, w := trace hv, hw,. Since the manifold M is closed, there exists an interior point x 0 M, where the smallest principal curvature λ 1 attains its global minimum λ min. Recall that by convexity we have that λ min 0. The fiberwise map Ψ given by Ψϑ = ϑ 2 ϑ Hϑ 2, obviously satisfies the null-eigenvector condition.

14 14 Andreas Savas-Halilaj, Knut Smoczyk If λ min = 0, then due to Theorem 2.3, the smallest principal curvature of M vanishes everywhere. Hence, rank h < m. It is a well known fact in Differential Geometry that the set M 0 := {x M : rank h x = max z M rank h z }, is open and dense in M a standard reference is [15]. From the Codazzi equation, it follows that the nullity distribution D := {v T M 0 : hv, w = 0, for all w T M 0 }, is integrable and its integrals are totally geodesic submanifolds of M. On the other hand, the Gauß formula says that these submanifolds are totally geodesic in R m+1. Moreover, because M is complete it follows that these submanifolds must be also complete. This contradicts with the assumption of compactness. Consequently, the minimum λ min of the smallest principal curvature must be strictly positive. Let v be a unit eigenvector of h corresponding to λ min at x 0. Applying Theorem 2.4, we obtain 0 h 2 x 0 λ min Hλ 2 min = λ min h 2 x 0 Hλ min, Because h 2 H 2 /m, we deduce that h 2 x 0 Hλ min H H/m λ min 0. Consequently, and so 0 λ min H H/m λ min 0, H/m = λ min. On the other hand λ min is the global minimum of all principal curvatures on M and H is constant. This shows that the smallest principal curvature λ 1 x at an arbitrary point x M satisfies λ min λ 1 x H/m = λ min. Therefore M is everywhere umbilic. It is well-known that the only closed and totally umbilic hypersurfaces are the round spheres. 3 Bernstein theorems for minimal maps In this section we shall develop the relevant geometric identities for graphs induced by smooth maps f : M N. Moreover, we will derive estimates that will be crucial in the proofs of Theorems A, B and C.

15 Bernstein theorems for minimal maps Geometry of graphs Let M, g M and N, g N be Riemannian manifolds of dimension m and n, respectively. The induced metric on the product manifold will be denoted by g M N = g M g N. A smooth map f : M N defines an embedding F : M M N, by F x = x, fx, x M. The graph of f is defined to be the submanifold Γ f := F M. Since F is an embedding, it induces another Riemannian metric on M. The two natural projections g := F g M N π M : M N M, π N : M N N are submersions, that is they are smooth and have maximal rank. Note that the tangent bundle of the product manifold M N, splits as a direct sum T M N = T M T N. The four metrics g M, g N, g M N and g are related by g M N = π M g M + π Ng N, 3.1 Additionally, let us define the symmetric 2-tensors g = F g M N = g M + f g N. 3.2 s M N := π M g M π Ng N, 3.3 s := F s M N = g M f g N. 3.4 Note that s M N is a semi-riemannian metric of signature m, n on M N. The Levi-Civita connection g M N associated to the Riemannian metric g M N on M N is related to the Levi-Civita connections g M on M, gm and g N on N, gn by g M N = π M g M π N g N. The corresponding curvature operator R M N on M N with respect to the metric g M N is related to the curvature operators R M on M, g M and R N on N, g N by R M N = π M R M π NR N. Denote the Levi-Civita connection on M with respect to the induced metric g = F g M N simply by and the curvature tensor by R. On the manifold M there are many interesting bundles. The most important one is the pull-back bundle F T M N. The differential df of the map F is

16 16 Andreas Savas-Halilaj, Knut Smoczyk a section in F T M N T M. The covariant derivative of it is called the second fundamental tensor A of the graph. That is, Av, w := df v, w = g M N df v df w df vw where v, w T M, is the induced connection on F T M N T M and is the Levi-Civita connection associated to the Riemannian metric g := F g M N. The trace of A with respect to the metric g is called the mean curvature vector field of Γ f and it will be denoted by H := trace A. Note that H is a section in the normal bundle of Γ f. If H vanishes identically the graph is said to be minimal. Following Schoen s [28] terminology, a map f : M N between Riemannian manifolds is called minimal if its graph Γ f is a minimal submanifold of the product space M N, g M N. By Gauß equation the curvature tensors R and R M N are related by the formula Rv 1, w 1, v 2, w 2 = F R M N v 1, w 1, v 2, w 2 +g M N Av1, v 2, Aw 1, w 2 g M N Av1, w 2, Aw 1, v 2, 3.5 for any v 1, v 2, w 1, w 2 T M. Moreover, the second fundamental form satisfies the Codazzi equation u Av, w v Au, w = R M N df u, df v df w for any u, v, w on T M. df Ru, vw, Singular decomposition In this subsection we closely follow the notations used in [34]. For a fixed point x M, let λ 2 1x λ 2 mx be the eigenvalues of f g N with respect to g M. The corresponding values λ i 0, i {1,..., m}, are usually called singular values of the differential df of f and give rise to continuous functions on M. Let r = rx = rank dfx. Obviously, r min{m, n} and λ 1 x = = λ m r x = 0. At the point x consider an orthonormal basis {α 1,..., α m r ; α m r+1,..., α m } with respect

17 Bernstein theorems for minimal maps 17 to g M which diagonalizes f g N. Moreover, at fx consider an orthonormal basis {β 1,..., β n r ; β n r+1,..., β n } with respect to g N such that dfα i = λ i xβ n m+i, for any i {m r + 1,..., m}. The above procedure is called the singular decomposition of the differential df. Now we are going to define a special basis for the tangent and the normal space of the graph in terms of the singular values. The vectors α i, 1 i m r, e i := 1 α 1+λ 2 i λ i x β n m+i, m r + 1 i m, i x 3.7 form an orthonormal basis with respect to the metric g M N of the tangent space df T x M of the graph Γ f at x. Moreover, the vectors β i, 1 i n r, ξ i := 1 λ 1+λ 2 i+m nxα i+m n β i, n r + 1 i n, i+m n x 3.8 give an orthonormal basis with respect to g M N of the normal space N x M of the graph Γ f at the point fx. From the formulas above, we deduce that s M N e i, e j = 1 λ2 i x 1 + λ 2 i xδ ij, 1 i, j m. 3.9 Consequently, the eigenvalues of the 2-tensor s with respect to g, are 1 λ 2 1x 1 + λ 2 1 x 1 λ2 m 1x 1 + λ 2 m 1 x 1 λ2 mx 1 + λ 2 mx. Moreover, δ ij, 1 i n r s M N ξ i, ξ j = 1 λ2 i+m n x 1 + λ 2 i+m n xδ ij, n r + 1 i n and s M N e m r+i, ξ n r+j = 2λ m r+ix 1 + λ 2 m r+i xδ ij, 1 i, j r. 3.11

18 18 Andreas Savas-Halilaj, Knut Smoczyk 3.3 Area decreasing maps Recall that a map f : M N is weakly area decreasing if Λ 2 df 1 and strictly area decreasing if Λ 2 df < 1. The above notions are expressed in terms of the singular values by the inequalities λ 2 i xλ 2 jx 1 and λ 2 i xλ 2 jx < 1, for any 1 i < j m and x M, respectively. On the other hand, the sum of two eigenvalues of the tensor s with respect to g equals 1 λ 2 i 1 + λ 2 i + 1 λ2 j 1 + λ 2 j = 21 λ2 i λ2 j 1 + λ 2 i 1 + λ2 j. Hence, the strictly area-decreasing property of the map f is equivalent to the 2-positivity of the symmetric tensor s. The weakly 2-positivity of a symmetric tensor T SymT M T M can be expressed as the weakly positivity of another symmetric tensor T [2] SymΛ 2 T M Λ 2 T M. Indeed, let P and Q be two symmetric 2-tensors. Then, the map P Q given by P Qv 1 w 1, v 2 w 2 = Pv 1, v 2 Qw 1, w 2 + Pw 1, w 2 Qv 1, v 2 Pw 1, v 2 Qv 1, w 2 P v 1, w 2 Qw 1, v 2 gives rise to an element of SymΛ 2 T M Λ 2 T M. The operator is called the Kulkarni-Nomizu product. Now we assign to each symmetric 2-tensor T SymT M T M an element T [2] of the bundle SymΛ 2 T M Λ 2 T M, by setting T [2] := T g. The Riemannian metric G of the bundle Λ 2 T M is related to the Riemannian metric g on the manifold M by the formula G = 1 2 g g = 1 2 g[2]. The relation between the eigenvalues of T and the eigenvalues of T [2] is explained in the following lemma: Lemma 3.1 Suppose that T is a symmetric 2-tensor with eigenvalues µ 1 µ m and corresponding eigenvectors {v 1,..., v m } with respect to g. Then the eigenvalues of the symmetric 2-tensor T [2] with respect to G are with corresponding eigenvectors µ i + µ j, 1 i < j m, v i v j, 1 i < j m.

19 Bernstein theorems for minimal maps A Bochner-Weitzenböck formula Our next goal is to compute the Laplacians of the tensors s and s [2]. The next computations closely follow those for a similarly defined tensor in [32]. In order to control the smallest eigenvalue of s, let us define the symmetric 2-tensor where c is a non-negative constant. Φ c := s 1 c 1 + c g, At first let us compute the covariant derivative of the tensor Φ c. Since g = 0 and g M N sm N = 0, we have v Φ c u, w = v su, w = g M N df v s M N df u, df w for any tangent vectors u, v, w T M. +s M N Av, u, df w + sm N df u, Av, w = s M N Av, u, df w + sm N df u, Av, w, Now let us compute the Hessian of Φ c. Differentiating once more gives 2 v1,v 2 Φ c u, w = s M N v1 Av 2, u, df w + s M N Av2, u, Av 1, w +s M N Av1, u, Av 2, w + s M N df u, v1 Av 2, w, for any tangent vectors v 1, v 2, u, w T M. Applying Codazzi s equation 3.6 and exploiting the symmetry of A and s M N, we derive 2 v1,v 2 Φ c u, w = s M N u Av 1, v 2 + R M N df v1, df u df v 2, df w +s M N w Av 1, v 2 + R M N df v1, df w df v 2, df u +s M N Av1, u, Av 2, w + s M N Av1, w, Av 2, u s Rv 1, uv 2, w s Rv 1, wv 2, u. The decomposition of the tensors s M N and R M N, implies s M N R M N df v1, df u df v 2, df w = πm g M R M N df v1, df u df v 2, df w π N g N R M N df v1, df u df v 2, df w = g M RM v1, u v 2, w g N RN dfv1, dfu dfv 2, dfw = R N dfv1, dfu, dfv 2, dfw R M v1, u, v 2, w.

20 20 Andreas Savas-Halilaj, Knut Smoczyk Gauß equation 3.5 gives s Rv 1, uv 2, w = Φ c Rv1, uv 2, w 1 c 1 + c g Rv 1, uv 2, w = Φ c Rv1, uv 2, w + 1 c 1 + c R v 1, u, v 2, w = Φ c Rv1, uv 2, w + 1 c 1 + c { gm N Av 1, v 2, Au, w g M N Av 1, w, Av 2, u } + 1 c 1 + c R M v 1, u, v 2, w + 1 c 1 + c R N dfv 1, dfu, dfv 2, dfw. In the sequel consider any local orthonormal frame field {e 1,..., e m } with respect to the induced metric g on M. Then, taking a trace, we derive the Laplacian of the tensor Φ c. Let us now summarize the previous computations in the next lemma: Lemma 3.2 For any smooth map f : M N, the symmetric tensor Φ c satisfies the identity Φc v, w = sm N v H, df w + s M N w H, df v where +2 1 c 1 + c g M N H, Av, w +Φ c Ric v, w + Φc Ric w, v +2 sm N 1 c 1 + c g M N Aek, v, Ae k, w c f R N e k, v, e k, w cr M e k, v, e k, w, Ric v := Re k, ve k is the Ricci operator on M, g and {e 1,..., e m } is any orthonormal frame with respect to the induced metric g. The expressions of the covariant derivative and the Laplacian of a symmetric 2-tensor T [2] are given in the following Lemma. The proof follows by a straightforward computation and for that reason will be omitted. Lemma 3.3 Any symmetric 2-tensor T satisfies the identities, a v T [2] = v T [2],

21 Bernstein theorems for minimal maps 21 b 2 v,v T [2] = 2 v,v T [2], c T [2] = T [2], for any vector v on the manifold M. 3.5 Proofs of the Theorems A, B and C We will first show the following lemma. Lemma 3.4 Let f : M N be weakly length decreasing. Suppose {e 1,..., e m } is orthonormal with respect to g such that it diagonalizes the tensor s. Then for any e l we have 2 R M e k, e l, e k, e l f R N e k, e l, e k, e l = λ 2 k l k +σ g M e l, e l f g N e l, e l } + Ric M e l, e l m 1σg M e l, e l + 1 λ 2 k 1 + λ 2 σ M e k e l + σ g M e l, e l, k l k λ 2 k { σ σn dfe k dfe l f g N e l, e l where Ric M stands for the Ricci curvature with respect to g M and σ M e k e l the sectional curvatures of the 2-planes e k e l on M, g M. Here, the terms σ N dfe k dfe l are zero if dfe k, dfe l are collinear and otherwise denote the sectional curvatures on N, g N of the planes dfe k dfe l. Proof In terms of the singular values we get Since we derive and se k, e k = g M e k, e k f g N e k, e k = 1 λ2 k 1 + λ 2. k 1 = ge k, e k = g M e k, e k + f g N e k, e k g M e k, e k = λ 2, f g N e k, e k = λ2 k k 1 + λ 2 k 2g M e k, e k = 1 λ2 k 1 + λ 2 + 1, 2f g N e k, e k = 1 λ2 k k 1 + λ 2 1. k Note also that for any k l we have g M e k, e l = f g N e k, e l = ge k, e l = 0.

22 22 Andreas Savas-Halilaj, Knut Smoczyk We compute 2 R M e k, e l, e k, e l f R N e k, e l, e k, e l = 2 k l σ M e k e l g M e k, e k g M e l, e l 2 k l σ N dfe k dfe l f g N e k, e k f g N e l, e l. Hence the formula for g M e k, e k implies 2 = k l R M e k, e l, e k, e l f R N e k, e l, e k, e l λ2 k 1 + λ 2 σ M e k e l g M e l, e l k +2 k l f g N e k, e k { σ σn dfe k dfe l f g N e l, e l 2σ k l f g N e k, e k g M e l, e l = k l λ2 k 1 + λ 2 σ M e k e l g M e l, e l k +σ g M e l, e l f g N e l, e l } +2 k l f g N e k, e k { σ σn dfe k dfe l f g N e l, e l +σ g M e l, e l f g N e l, e l } +σ 1 λ 2 k 1 + λ 2 1 g M e l, e l. k l k We may then continue to get 2 R M e k, e l, e k, e l f R N e k, e l, e k, e l = 2 k l f g N e k, e k { σ σn dfe k dfe l f g N e l, e l +σ g M e l, e l f g N e l, e l } + Ric M e l, e l m 1σ g M e l, e l + 1 λ 2 k 1 + λ 2 σ M e k e l + σ g M e l, e l. k l k

23 Bernstein theorems for minimal maps 23 This completes the proof. Proof of Theorem A. Suppose that f : M N is weakly length decreasing. Then the tensor s satisfies s = g M f g N 0. In case s > 0 the map f is also strictly area decreasing. Thus in such a case the statement follows from Theorem B which we will prove further below. It remains to show that s vanishes identically, if s admits a null-eigenvalue somewhere. Claim 1. If s has a null-eigenvalue at one point of M, then s has a nulleigenvalue at any point of M. Since s = Φ 1, from Lemma 3.2 we get where s +Ψs = 0, Ψϑ v, w = ϑ Ric v, w ϑ Ric w, v 2 +2 s M N Aek, v, Ae k, w R M e k, v, e k, w f R N e k, v, e k, w. Let v be a null-eigenvector of the symmetric, positive semi-definite tensor ϑ. Since f is weakly length decreasing, equation 3.10 shows that s M N is nonpositive definite on the normal bundle of the graph. Hence m Ψϑ v, v 2 R M e k, v, e k, v f R N e k, v, e k, v 0, where we have used Lemma 3.4 and the curvature assumptions on M, g M, N, g N respectively. This shows that Ψ satisfies the null-eigenvector condition and Claim 1 follows from the strong maximum principle in Theorem 2.3. Claim 2. If s admits a null-eigenvalue at some point x M, then s vanishes at x. We already know that the tensor s admits a null-eigenvalue everywhere on M. Since s 0 we may then apply the test criterion Theorem 2.4 to the tensor s at an arbitrary point x M. At x consider a basis {e 1,..., e m }, orthonormal

24 24 Andreas Savas-Halilaj, Knut Smoczyk with respect to g consisting of eigenvectors of s, such that v := e m is a nulleigenvector of s and λ 2 m = 1. From Lemma 3.4, we conclude 0 Ψs e m, e m 2 λ 2 { k σ 1 + λ 2 σn dfe k dfe m f g k m k }{{} N e l, e l }{{} 0 0 +σ g M e m, e m f g N e m, e m }{{} =0 + Ric M e m, e m m 1σ g M e m, e m }{{} 0 + k m 1 λ 2 k 1 + λ 2 k }{{} 0 σ M e k e m + σ }{{} >0 g M e m, e m }{{} = 1 2 } = 0, 3.12 because the curvature assumptions on M, g M and N, g N imply that the right hand side is a sum of non-negative terms and thus we conclude that all of them must vanish. In particular, σ M e k e m + σ > 0 implies λ 2 k = 1 for all k. Now we can finish the proof of Theorem A. Claim 1 and 2 imply that a weakly length decreasing map f which is not strictly length decreasing must be an isometric immersion. A simple computation shows that the second fundamental form A of F is given by A = A I A f, where A I is the second fundamental form of the map I : M, g M, g M and A f for the second fundamental form of f : M, g N, g N. Because f is an isometric immersion we get that g = 2g M. Hence, A I is identically zero. Moreover, since F is minimal immersion, we deduce that f is minimal immersion too. Once we know that all tangent vectors at x are null-eigenvectors of s, we may choose e m in 3.12 arbitrarily. Then, and Ric M v, v = m 1σg M v, v σ = σ N dfv dfw for all linearly independent vector fields v, w T M. According to the Gauß equation for the isometric immersion f, we deduce that it must be totally geodesic. This implies that Γ f is also totally geodesic. Again by the Gauß equation, it follows that M has constant sectional curvature σ. This completes the proof of Theorem A.

25 Bernstein theorems for minimal maps 25 Proof of Theorem B. Since the manifold M is compact, there exists a point x 0 where the smallest eigenvalue of s [2] with respect to the metric G attains its minimum. Let us denote this value by ρ 0. Note that in terms of the singular values λ 2 1 λ 2 m we must have ρ 0 = 1 λ2 mx λ 2 mx λ2 m 1x λ 2 m 1 x 0 0. For simplicity we set κ := λ 2 m 1x 0 and µ := λ 2 mx 0. Hence, 1 κµ ρ 0 = κ1 + µ. Claim 3. If µ = 0, then the map f is constant. In this case we have ρ 0 = 2. Because ρ 0 is the minimum of the smallest eigenvalue of the symmetric tensor s [2], we obtain 1 λ 2 i 1 xλ2 j x 1 + λ 2 i x1 + λ2 j x, for any x M and 1 i < j m. From the above inequality one can readily see that all the singular values of f vanish everywhere. Thus, in this case f is constant. This completes the proof of Claim 3. Since we are assuming that f is weakly area decreasing, we deduce that κµ 1. Consider now the symmetric 2-tensor Φ := Φ 2 ρ 0 2+ρ 0 = s ρ 0 2 g. According to Lemma 3.3, Φ [2] = Φ [2]. At x 0 consider an orthonormal bases {e 1,..., e m } with respect to g such that s becomes diagonal and According to Theorem 2.4, we obtain se k, e k = 1 λ2 k 1 + λ 2. k 0 Φ [2] e m 1 e m, e m 1 e m = Φ e m 1, e m 1 + Φ e m, e m.

26 26 Andreas Savas-Halilaj, Knut Smoczyk In view of Lemma 3.2, we deduce that 0 2ΦRic e m 1, e m 1 + 2ΦRic e m, e m +2 s M N ρ 0 2 g M N Ae k, e m 1, Ae k, e m ρ 0 2 ρ ρ 0 2 ρ 0 s M N ρ 0 2 g M N Ae k, e m, Ae k, e m f R N e k, e m 1, e k, e m 1 R M e k, e m 1, e k, e m 1 f R N e k, e m, e k, e m R M e k, e m, e k, e m Because e m is an eigenvector of s with respect to g, we have ΦRic e m, e m = κ µ 1 + κ1 + µ gric e m, e m. From the Gauß equation 3.5 and the minimality of the graph, we obtain that Hence, gric e m, e m = ΦRic e m, e m = + R M e k, e m, e k, e m + f R N e k, e m, e k, e m g M N Ae k, e m, Ae k, e m. κ µ 1 + κ1 + µ κ µ 1 + κ1 + µ κ µ 1 + κ1 + µ R M e k, e m, e k, e m 3.14 f R N e k, e m, e k, e m g M N Ae k, e m, Ae k, e m.

27 Bernstein theorems for minimal maps 27 Similarly, ΦRic e m 1, e m 1 = + µ κ 1 + κ1 + µ µ κ 1 + κ1 + µ µ κ 1 + κ1 + µ R M e k, e m 1, e k, e m f R N e k, e m 1, e k, e m 1 g M N Ae k, e m 1, Ae k, e m 1. In view of 3.14 and 3.15, the inequality 3.13 can be written equivalently in the form 0 s M N 1 µ 1 + µ g M N Ae k, e m, Ae k, e m + s M N 1 κ 1 + κ g M NAe k, e m 1, Ae k, e m µ κ f R N µr M e k, e m, e k, e m f R N κr M e k, e m 1, e k, e m Claim 4. The sum A of the first two terms on the right hand side of inequality 3.16 is non-positive. Indeed, if µ = 0, then f is constant by Claim 3 and thus A = 0. So, let us consider the case where µ > 0. From Theorem 2.4 again, we have 0 = ek s [2] ρ 0 Ge m e m 1, e m e m 1 = 2 ek se m, e m + 2 ek se m 1, e m 1 = 4s M N Ae k, e m, e m + 4s M N Ae k, e m 1, e m for any k. Since dimn = 1 implies that rankdf 1, from 3.11 we obtain 0 = A ξn e k, e m s M N ξ n, e m + A ξn e k, e m 1 s M N ξ n, e m 1 }{{} µ = 2A ξn e k, e m, 1 + µ }{{} >0 =0 where here A ξ v, w := g N Av, w, ξ, v, w T x M,

28 28 Andreas Savas-Halilaj, Knut Smoczyk stands for the second fundamental form of the graph Γ f in the normal direction ξ and the normal basis {ξ 1,..., ξ n } is chosen as in Hence, by the weakly area decreasing property of f, we get A = 1 µ 1 + µ + 1 κ A 2 ξ 1 + κ n e k, e m 1 0. In the case dimn 2, from 3.10, 3.17 and the weakly area decreasing condition we obtain A = s M N 1 µ 1 + µ g M N Ae k, e m, Ae k, e m + s M N 1 κ 1 + κ g M N Ae k, e m 1, Ae k, e m µ 1 + µ A 2 ξ n e k, e m 2 1 κ 1 + κ In view of equations 3.17 and 3.11, we have Hence, A 2 ξ n 1 e k, e m 1. 0 = s M N Ae k, e m, e m + s M N Ae k, e m 1, e m 1 µ κ = µ A ξ n e k, e m κ A ξ n 1 e k, e m 1. A 2 ξ n e k, e m = Because, κ µ and κµ 1, we deduce that κ1 + µ2 µ1 + κ 2 A2 ξ n 1 e k, e m 1. κ1 + µ 2 µ1 + κ 2 1. This proves our assertion. Now it is clear that the quantity A is always nonpositive which proves Claim 4. Claim 5. The sum B of the last two terms on the right hand side of inequality 3.16 is non-positive. We have, B = µ κ 2 f R N µr M e k, e m, e k, e m } {{ } =:B 1 2 f R N κr M e k, e m 1, e k, e m 1 } {{ } =:B 2

29 Bernstein theorems for minimal maps 29 From the identities 3.2 and 3.4, we deduce that g M = 1 g + s 2 and f g N = 1 g s. 2 Since {e 1,..., e m } diagonalizes g and s, it follows that it diagonalizes g M and f g N as well. In fact, for any i {1,..., m}, we have f g N e i, e i = λ 2 i g M e i, e i. Proceeding exactly as in the proof of Lemma 3.4, but using µσ M instead of σ M and µσ instead of σ, we obtain that B 1 = 2 f R N e k, e m, e k, e m µr M e k, e m, e k, e m k m = 2µ f g 1 + µ N e k, e k σ σ N dfe k dfe m k m µ Ric M e m, e m m 1σ g M e m, e m µ 1 λ 2 k 1 + µ 1 + λ 2 σ M e k e m + σ. k m k Similarly, B 2 = 2 k m 1 f R N e k, e m 1, e k, e m 1 κr M e k, e m 1, e k, e m 1 = 2κ f g 1 + κ N e k, e k σ σ N dfe k dfe m 1 k m 1 κ Ric M e m 1, e m 1 m 1σ g M e m 1, e m 1 κ 1 λ 2 k 1 + κ 1 + λ 2 σ M e k e m 1 + σ. k k m 1

30 30 Andreas Savas-Halilaj, Knut Smoczyk Taking into account that λ 2 1 λ 2 m 2 1, we deduce that B = 2µ 1 + µ 2 k m f g N e k, e k σ σ N dfe k dfe m 2κ 1 + κ 2 f g N e k, e k σ σ N dfe k dfe m 1 k m 1 µ Ric M e m, e m m 1σ g 1 + µ M e m, e m κ Ric M e m 1, e m 1 m 1σ g 1 + κ M e m 1, e m 1 m 2 µ 1 λ 2 k 1 + µ λ 2 σ M e k e m + σ k κ 1 + κ 2 m 2 1 λ 2 k 1 + λ 2 σ M e k e m + σ k κ + µ1 κµ 1 + κ µ 2 σ M e m 1 e m + σ This completes the proof of Claim 5. Now we shall distinguish two cases. Case 1. Assume at first that f is strictly area decreasing. Hence, κµ < 1. In view of our curvature assumptions, Claim 4, Claim 5, 3.18 and from inequality 3.16 we deduce that κ = µ = 0. Hence, from Claim 3 the map f must be constant. Case 2. Suppose that κµ = 1. From Claim 1, 3.18 and inequality 3.16 we deduce that at x 0 we must have 1 = λ 2 1x 0 = = λ 2 m 2x 0 κ 1. Hence, κ = 1 and so µ = 1. Therefore, at each point x where Φ [2] has a zero eigenvalue, all the singular values of f are equal to 1. Thus, the set D := {x M : Λ 2 df = 1}, is closed, non-empty and moreover D = {x M : f g N = g M }. Obviously, the map f is strictly area decreasing on the complement of D. Moreover, by 3.18, Ric M = m 1σ at any point of D and the restriction of σ N to dft D is equal to σ. If D has an interior point then, proceeding in the same way as in the proof of Theorem A, we deduce that in an open neighborhood where F is totally geodesic. According to the Analytic Continuation Theorem cf. [26, Theorem 1, p.213] the map F is everywhere totally geodesic. This completes the proof of Theorem B.

31 Bernstein theorems for minimal maps 31 Proof of Theorem C. Note that in this case the singular values of the map f are 0 = λ 2 1 = = λ 2 m 1 = κ µ. Hence, automatically, f is strictly area decreasing. From Claim 4, Claim 5, inequality 3.13 and 3.18, we deduce that 0 2µ 1 + µ Ric M e m, e m 0. Thus µ = 0 and the map f must be constant. This completes the proof of Theorem C. 3.6 Final remarks We end this paper with some examples and remarks concerning the imposed assumptions in Theorems A, B and C. Remark 3.1 In several cases, graphical submanifolds over the manifold M, g M with parallel mean curvature, i.e., H = 0, where stands for the connection of the normal bundle, must be minimal. This problem was first considered by Chern in [9]. So, whenever graphs with parallel mean curvature vector are minimal we can immediately apply Theorems A, B and C. For example this can be done for graphs considered in the paper by G. Li and I.M.C. Salavessa [25]. Remark 3.2 The reason that the result of Theorem B is weaker than that of Theorem A is due to the fact that in Theorem B we cannot apply the strong elliptic maximum principle stated in Theorem 2.3. In fact, the null-eigenvector condition of the corresponding tensor Ψϑ [2] in the equation of s [2] seems to hold only for some weakly 2-positive definite tensors ϑ, including s. Remark 3.3 In some situations, a smooth minimal map f : M N satisfying the assumptions in Theorem B can only be constant. For instance, if dim M > dim N the map f cannot be an isometric immersion since rankdf < dim M. Moreover, if M is not Einstein or the sectional curvature of N is strictly less than σ, then any such map must be constant. Remark 3.4 In this remark we show that the assumptions on the curvatures of M and N in Theorems A and B are sharp. a Scaling. Suppose that f : M N is a smooth map between two Riemannian manifolds M, g M and N, g N, and assume that there exists a