IMMERSED SURFACES OF PRESCRIBED GAUSS CURVATURE

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 7, Pages 2093–2101 S 0002-9939(00)05770-1 Article electronically published on December 7, 2000

IMMERSED SURFACES OF PRESCRIBED GAUSS CURVATURE INTO MINKOWSKI SPACE YUXIN GE (Communicated by Bennett Chow)

Abstract. Given a positive real valued function k(x) on the disc, we will immerse the disc into three dimensional Minkowski space in such a way that Gauss curvature at the image point of x is −k(x). Our approach lies on the construction of Gauss map of surfaces.

1. Introduction The classical Minkowski problem is an embedding problem of differential geometry. This problem is the following: Given a positive function K(u) defined on the unit sphere, does there exist a closed convex surface in R3 having K(u) as its Gauss curvature at the point on the surface where the inner normal is u? In [11], Lewy has shown the existence of such a surface, under the condition that the function K(u) is analytic. Later, by using a similar procedure, Nirenberg [13] published a paper in which he solved the Minkowski problem under the assumption that the function K(u) possesses partial derivatives on the sphere up to second order. In [3], the author considers an analogous problem by using an approach, suggested in [8]: Given a positive real valued function k(x) on the disc, we immerse the disc in R3 in such a way that Gauss curvature at the image point of x is k(x). In this paper, we continue exploiting this method to immerse the disc into three dimensional Minkowski space. Namely we propose and use a method for constructing immersions of surfaces in the Minkowski space R2,1 by prescribing the Gauss curvature to be a negative function of the variable x in the surface. Notice that in [3] we had an analogous construction for surfaces in the Euclidean space but with a positive Gauss curvature. Let B = {x ∈ R2 , | x | < 1} be a disc in R2 . Let R2,1 be three dimensional Minkowski space with the standard metric g = (dx1 )2 + (dx2 )2 − (dx3 )2 . Let H2 = {x ∈ R3 , g(x, x) = −1} be the unit hyperboloid of two sheets and let H2+ = {x ∈ H2 , x3 > 0} be the upper sheet contained in the half-space {x3 > 0}. Let l : ∂B −→ H2+ be a prescribed C 2,γ mapping with γ > 0. We consider the space Hl1 (B, H2+ ) of functions u in H 1 (B, R3 ) satisfying that u ∈ H2+ a.e. and u = l

Received by the editors April 29, 1999 and, in revised form, October 20, 1999. 2000 Mathematics Subject Classification. Primary 53C42, 53B25. Key words and phrases. Gauss curvature, surfaces, Minkowski space, harmonic maps. c

2000 American Mathematical Society

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on ∂B. We define on Hl1 (B, H2+ ) the following energy functional E: Z X 2 ∂u ∂u 1 aij (x)g , (1) dx, E(u) = 2 B i,j=1 ∂xi ∂xj where aij (x) satisfy the following conditions: (3)

∃ α > 0, such that aij (x)ξ i ξ j ≥ α| ξ |2 , ¯ R), ∀ 1 ≤ i, j ≤ 2; aij (x) ∈ C 1,γ (B,

(4)

aij = aji ,

(2)

∀ x ∈ B, ∀ ξ ∈ R2 ;

∀ 1 ≤ i, j ≤ 2.

Here, it is easy to check that the critical points of E satisfy in the sense of distributions the following Euler equation:  2   X ∂ a (x) ∂u + λu = 0, in B, ij (5) ∂xi ∂xj   i,j=1 u = l, on ∂B, 2 X ∂u ∂u aij (x)g (x), (x) . where λ = − ∂x ∂x i j i,j=1 Notice that λ < 0. We deduce from (5) the following equality:   2 2 X X ∂u  ∂  aij (x) (6) u× = 0, ∂x ∂x i j i=1 j=1 where ξ × η = (ξ2 η3 − ξ3 η2 , ξ3 η1 − ξ1 η3 , ξ2 η1 − ξ1 η2 ) for all ξ, η ∈ R2,1 is vectorial product in R2,1 . Assume that u is an immersion. Thus, we obtain a new immersion G from B to R2,1 satisfying X ∂u ∂G = a1j (x)u × , ∂x2 ∂x j j=1 2

(7)

2 X ∂G ∂u =− a2j (x)u × . ∂x1 ∂x j j=1

Our aim here is to prove that G has the prescribed Gauss curvature. More precisely, we will show the following theorem. Theorem. Under the above assumptions, the metric induced by g on G(B) is Riemannian. Moreover, G(B) has the Gauss curvature equal to −det(aij )−1 at each point G(x). This paper is organized as follows. We first prove that there exists a solution of (5) in the C 2,γ norm. Then, we show that the solution u is unique. By the same strategy as in [9], we deduce that u is a diffeomorphism. Hence, using the above approach, we will establish our result. 2. Existence and regularity Let us first give the existence and regularity results. 2 ) Proposition 1. Under the above hypothesis, there exists a minimum u ∈ Hl1 (B, H+ of E which satisfies (5). Furthermore, one has the estimate:

(8)

kukC 2,γ ≤ C1 (kukH 1 + klkC 2,γ ),

where C1 is a constant depending only on α, γ and kaij kC 1,γ .


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Remark. Notice that the metric induced by g on H2+ is Riemannian. So it is natural for us to look for the minimum of E. Proof. We will make use of the stereographic projection: P : H2+ (9)

−→ B,

(x, y, z) 7−→

x y , 1+z 1+z

.

With these stereographic coordinates, we can write the the functional E as follows: Z X 2 aij (x) ∂v ∂v , (10) dx, E(v) = 2 2 B i,j=1 (1 − | v | )2 ∂xi ∂xj where v ∈ Hh1 (B, R2 ) with h = P ◦ l and h, i denotes the standard Euclidian inner product. Assume that |h| ≤ r with some r < 1. Let f : R+ −→ R be a decreasing continuous map satisfying  1  if 0 ≤ z ≤ r;   (1 − z 2 )2 , (11) f (z) =  1   , if z ≥ r. (1 − r2 )2 Consider the second energy functional E1 Z X 2 ∂v ∂v E1 (v) = 2 aij (x)f (|v|) , dx, ∂xi ∂xj B i,j=1 where v ∈ Hh1 (B, R2 ). Obviously, E1 (v) ≤ E(v). By coerciveness and lower semi-continuity of E1 (see [5] and [15]), it is clear that ˜ by there exists w ∈ Hh1 (B, R2 ) minimizing E1 . We define w  if |wi (x)| ≤ r;   wi (x), (12) w ˜i (x) = r, if wi (x) ≥ r;   −r, if wi (x) ≤ −r, ˜ ≤ E1 (w). Thus, | wi (x) |≤ r a.e. for i = 1, 2. Obviously, w ˜ ∈ Hh1 (B, R2 ) and E1 (w) for i = 1, 2. Replacing w by (w1 cosθ − w2 sinθ, w1 sinθ + w2 cosθ) for any θ ∈ R, we deduce that | w(x) |≤ r a.e. So w is also a minimizer of E. Thanks to a result due to Jost and Meier [10], Lemma 1 (see also [3], Lemma 1), we conclude that there exists q > 2 such that kwkW 1,q (B,R2 ) ≤ C4 (kwkH 1 (B,R2 ) + khkC 1 ),

(13)

where the constants C4 and q depend only on α and kaij kC 1 . Now we consider u = P −1 ◦ w and return to equation (5). From Lp -estimates and using Sobolev embedding theorem, we have kuk

W

1,

2q 4−q

≤ CkukW 2, q2 ≤ C(kukH 1 + klkC 2 ),

if q < 4.

Iterating the above procedure and using Schauder estimates, we complete the proof (cf. [5]).


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3. Uniqueness In this part, our main result is the following: ¯ H2 ) is unique. Proposition 2. The solution for equation (5) in C 2 (B, + Remark. This result and the proof we propose generalize an analogous result for harmonic maps due independently to [6] and [1]. ¯ H2+ ) Denote ∇ the Levi-Civita connection on H2+ for the metric g. Let u1 ∈ C 2 (B, ¯ be a map with the same boundary condition as u. For any x ∈ B, let γx (s) denote the unique geodesic arc in H2+ parametrized with constant speed (depending on x) for s ∈ [0, 1], and connecting u(x) with u1 (x). The uniqueness of γx (s) follows from H2+ having nonpositive curvature and simply connected. Define a C 2 map ¯ H2 ) be given by ¯ × [0, 1] −→ H2 by F (x, s) = γx (s) and let us ∈ C 2 (B, F : B + + us (x) = F (x, s). Then, F is a deformation of u. We will write the first and second variations of the energy E (see [2]). Lemma 1. Under the above hypothesis, we have the following formulas:   Z 2 X ∂u ∂u dE(us ) s s =− ,∇ ∂ ( g aij ) (14) ∂xi ds ∂s ∂x j B i,j=1 and d2 E(us ) =− ds2

Z

2 X

aij R(

B i,j=1

(15)

Z

2 X

+

∂F ∂F ∂F ∂F , ) , , ∂s ∂xi ∂xj ∂s

aij g ∇

B i,j=1

∂ ∂xi

∂us ∂us ,∇ ∂ ∂xj ∂s ∂s

,

where R is the curvature of H2+ . Proof. First, we suppose that F is C ∞ . By definition, 1 E(us ) = 2

Z

2 X

B i,j=1

aij (x)g(

∂us ∂us , )dx. ∂xi ∂xj

Differentiating under the integral sign and using the symmetry of the Riemannian connection, we obtain dE(us ) ds

∂us ∂us aij (x)g ∇ ∂ , dx ∂s ∂x i ∂xj B i,j=1 Z X 2 ∂us ∂us , aij (x)g ∇ ∂ dx = ∂xi ∂s ∂xj B i,j=1   Z 2 X ∂u ∂u s s  ,∇ ∂ ( g aij ) dx, =− ∂xi ∂s ∂x j B i,j=1 Z

=

2 X



since have

∂F ∂s

2097

= 0 on ∂B. Therefore, we obtain (14). Taking the derivative of (14), we d2 E(us ) ds2

 2 X ∂u ∂u s s ,∇ ∂ ∇ ∂ ( = − g aij ) dx ∂s ∂xi ∂s ∂x j B i,j=1 Z X 2 ∂F ∂F ∂F ∂F , ) aij R( , , =− ∂s ∂xi ∂xj ∂s B i,j=1   Z 2 X ∂u ∂u s s ,∇ ∂ ∇ ∂ ( aij ) dx − g ∂xi ∂s ∂s ∂x j B i,j=1 Z X 2 ∂F ∂F ∂F ∂F , ) aij R( , , =− ∂s ∂xi ∂xj ∂s B i,j=1 Z X 2 ∂us ∂us ,∇ ∂ aij g ∇ ∂ dx. + ∂xi ∂s ∂xj ∂s B i,j=1 Z



Thus, we establish (15). By density, we finish the proof. ¯ H2+ ) is another solution for equaProof of Proposition 2. Suppose that u1 ∈ Cl2 (B, tion (5). Putting s = 0 and s = 1 in (14), we obtain dE(us ) |s=0,1 = 0. ds On the other hand, we have d2 E(us ) ≥0 ds2 ∂F ∂F , ·, ·, ) is a positive quadratic form, that is, E(us ) is convex. Thus, ∂s ∂s E(us ) ≡ E(u0 ). Thanks to formula (15), we infer that ∂F ∂s ≡ 0. This contradiction completes the proof. since −R(

4. The diffeomorphism property Let l : ∂B −→ H2 ∩ {x3 = α1 , α1 > 1} be a C 2 diffeomorphism with deg(l, ∂B) = 1. We will prove the following result. Proposition 3. Under the above assumptions, the unique minimizer u of E is a ¯ diffeomorphism and rank(∇u(x)) = 2 for all x ∈ B. The proof here is the same as in [3]. To prove this fact, we will consider the following energy functional: Z X 2 h i ∂u ∂u 1 , (1 − t)δij + taij (x) g (16) dx. Et (u) = 2 B i,j=1 ∂xi ∂xj Let It = inf v∈Hl1 (B,H2+ ) Et (v). Denote ut ∈ Hl1 (B, H2+ ) the unique minimum of Et in Hl1 (B, H2+ ) given by Propositions 1 and 2, then ut satisfies:  2 i ∂ut X ∂ h    (1 − t)δij + taij (x) + λt ut = 0, in B, ∂x ∂x i j (17) i,j=1    t on ∂B, u = l,


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where λt = −

YUXIN GE

2 h i ∂ut X ∂ut ¯ H2+ ) by (x), (x) . ut is in C 2 (B, (1 − t)δij + taij (x) g ∂x ∂x i j i,j=1

Proposition 1. Define a mapping F∗ : ¯ H2+ ), F∗ : [0, 1] −→ C 2 (B, t t 7−→ u . We need also several technical lemmas. Lemma 2. With the above notations, we have rank(∇ut (x)) = 2, for any t ∈ [0, 1] and x ∈ ∂B. The proof is the same as that of Lemma 5 in [3]. Lemma 3. F∗ is continuous. Proof. First we notice that It is continuous. Indeed, for some fixed v ∈ Hl1 (B, H2+ )   2 X 1 kaij kC 0  k∇vk2L2 ≤ C, ∀0 ≤ t ≤ 1. 4+ 0 ≤ It ≤ 2 i,j=1 On the other hand, we have that for any 0 ≤ t, t0 ≤ 1   2 X 1 |t − t0 | 4 + kaij kC 0  max{It , It0 }. |It − It0 | ≤ min(1, α) i,j=1 Then the claim yields. Now let t be fixed. Assume that {tn }n∈N is a sequence con¯ H2+ ). verging to t. It follows from Proposition 1 that {utn }n∈N is compact in C 2 (B, tn 2 ¯ 2 −→ u in C (B, H+ ) for u ∈ Modulo a subsequence, we can assume that u ¯ H2 ) ∩ H 1 (B, H2 ). Clearly, C 2 (B, + + l Et (u) = It . Now by Proposition 2, we terminate the proof. Proof of Proposition 3. We define a set T1 = {t ∈ [0, 1], ut is a diffeomorphism}. Step 0 : T1 is not empty. In view of Theorem 5.1.1 in [9] (see also [3], Lemma 7), we have 0 ∈ T1 . Step 1 : T1 is open. Let t1 ∈ T1 . Applying Lemmas 2 and 3, we get ∃ τ1 > 0, s.t. ∀ t ∈ ]t1 − τ1 , t1 + τ1 [ ∩ [0, 1] =⇒ rank(∇ut (x)) = 2,

¯ ∀x ∈ B.

Now the claim follows from a result in [14] (see also [3], Lemma 6). Step 2 : T1 is also closed. Let {tn }n∈N be a sequence converging to t. Assume that utn are diffeomorphisms, ∀ n ∈ N. We suppose that ¯ ∃ x0 ∈ B,

s.t.

det(∇(P ◦ ut )(x0 )) = 0.



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Denote v = P ◦ ut and choose θ ∈ R such that ((∇v1 ) cos θ + (∇v2 ) sin θ)(x0 ) = 0. Define ω∗ = ω1 cos θ + ω2 sin θ for all continous functions ω : B −→ R2 . Thanks to the results of Hartman and Wintner [7], Theorems 1 and 2 (see also [3], Lemma 9), there exists some n1 ≥ 1 and a ∈ C∗ such that ∂z v∗ (z) = a(z − z0 )

n1

n

+ o(| z − z0 | 1 )

where z0 = (x0 )1 + i(x0 )2 . Therefore, there exists r0 > 0 and some n sufficiently large such that deg(∂z (P ◦ utn )∗ |∂B(z0 ,r0 ) , 0) = deg(∂z v∗ |∂B(z0 ,r0 ) , 0) = n1 ≥ 1. However, by property of degree, this contradicts that utn is a diffeomorphism. Hence, Proposition 3 is proved.

5. Proof of the Theorem Now, with the above results and preceding method, we can prove our main result. ∂G , u(x)) = 0 for i = 1, 2. That is, u(x) is the normal vector on Note first that g( ∂x i ¯ On the other hand, we have G(B) at point G(x) for all x ∈ B. g(u × ux1 , u × ux1 )

= (u3 )2 (u2x1 )2 + (u2 )2 (u3x1 )2 − 2u2 u2x1 u3 u3x1 + (u3 )2 (u1x1 )2 +(u1 )2 (u3x1 )2 − 2u1 u1x1 u3 u3x1 − (u1 )2 (u2x1 )2 − (u2 )2 (u1x1 )2 +2u2 u2x1 u1 u1x1 = (u3 )2 ((u2x1 )2 + (u1x1 )2 ) + (u2 )2 ((u3x1 )2 − (u1x1 )2 ) +(u1 )2 ((u3x1 )2 − (u2x1 )2 ) − 2(u3 )2 (u3x1 )2 + 2u2 u2x1 u1 u1x1 ,

to since g(u, ux1 ) = 0 (here subscripts denote partial p differentiation with respect 3 2 3 1 2 2 2 coordinates). With help of the equalities u = 1 + (u ) + (u ) and (ux1 ) = (u1 u1x1 + u2 u2x1 )2 , we deduce 1 + (u1 )2 + (u2 )2 g(u × ux1 , u × ux1 ) = g(ux1 , ux1 ). Replacing x1 by x2 and x1 + x2 , implies g(u × ux2 , u × ux2 ) = g(ux2 , ux2 )

and

g(u × ux1 , u × ux2 ) = g(ux1 , ux2 ).

Therefore, we conclude that G is an immersion and that the metric induced on G(B) is Riemannian. Now we will calculate the curvature of G(B). Denote D (resp. ∇) the Levi-Civita connection on R2,1 (resp. G(B)) and R the curvature. Obviously, we have ∇X Y = DX Y + g(DX Y, u)u,


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YUXIN GE

where X and Y are vector fields on G(B). So, this implies ∂G ∂G ∂G ∂G , , , R ∂x1 ∂x2 ∂x2 ∂x1 ∂G ∂G ∂G −∇ ∂ ∇ ∂ , = g ∇ ∂ ∇ ∂ ∂x2 ∂x1 ∂x 2 ∂x1 ∂x1 ∂x2 ∂x2 ∂G ∂G ∂G −D ∂ ∇ ∂ , = g D ∂ ∇ ∂ 2 ∂x2 ∂x1 ∂x 2 ∂x 1 ∂x1 ∂x2 ∂x ∂G ∂G ∂G ∂G , u g D ∂ u, , u g D ∂ u, −g D ∂ = g D ∂ ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂G ∂G ∂G ∂G g D ∂ u, +g D ∂ u, g D ∂ u, = −g D ∂ u, ∂x2 ∂x ∂x ∂x 1 1 2 ∂x2 ∂x1 ∂x2 ∂x1 = (−a11 a22 + a212 )g(u × ux1 , ux2 )2 , since g(u × uxi , uxi ) = 0 for i = 1, 2. On the other hand,

=

g(Gx1 , Gx1 )g(Gx2 , Gx2 ) − g(Gx1 , Gx2 )2 2 2 2 2 X X X X g( a2j u × uxj , a2k u × uxk )g( a1j u × uxj , a1k u × uxk ) j=1 2 X

−g(

k=1 2 X

a1j u × uxj ,

j=1

j=1

k=1

a2k u × uxk )2

k=1

=

2 2 2 2 2 2 X X X X X X g( a2j uxj , a2k uxk )g( a1j uxj , a1k uxk )−g( a1j uxj , a2k uxk )2

=

det(aij )2 (g(ux1 , ux1 )g(ux2 , ux2 ) − g(ux1 , ux2 )2 )

=

−det(aij )2 g(ux1 × ux2 , ux1 × ux2 )

=

det(aij )2 g(u × ux1 , ux2 )2 .

j=1

k=1

j=1

k=1

j=1

k=1

Hence, K(G(x)) = −det(aij (x))−1 . References [1] S. I. Al’ber, Spaces of mappings into a manifold with negative curvature, Dokl. Akad. Nauk. SSSR. Tom 178 (1968), No. 1. MR 37:5817 [2] J. Eells and L. Lemaire, A report on the harmonic maps, Bull. London. Math. Soc 10 (1978), 1-68. [3] Y. Ge, An elliptic variational approach to immersed surfaces of prescribed Gauss curvature, Calc. Var. 7 (1998) 173-190. [4] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Studies. 105, Princeton. Univ. Press, Princeton (1983). MR 86b:49003 [5] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren. 224, Spinger, Berlin-Heidelberg-New York-Tokyo (1983). MR 86c:35035 [6] P. Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967) 673-687. MR 35:4856 [7] P. Hartman and A. Wintner, On the local behavior of solutions of nonparabolic partial differential equations, Amer. J. Math. 75 (1953) 449-476. MR 15:318b [8] F. H´ elein, Applications harmoniques, lois de conservation et rep` ere mobile, Diderot ´editeur, Paris-New York-Amsterdam (1996). [9] J. Jost, Two-dimensional geometric variational problems, Wiley (1991). MR 92h:58045 [10] J. Jost and M. Meier, Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262 (1983) 549-561. MR 84i:35051 [11] H. Lewy, On differential geometry in the large, I (Minkowski’s problem), Trans. Amer. Math. Soc. 43 (1938) 258-270. CMP 95:18



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[12] C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Grundlehren. 130, New York (1966). MR 34:2380 [13] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1966), 337-394. MR 15:347b [14] Sto¨ılow, Le¸cons sur les principes topologiques de la th´ eorie des fonctions analytiques, Paris (1938), Gauthier-Villars, p. 130. [15] M. Struwe, Variational Methods, Springer, Berlin-Heidelberg-New York-Tokyo (1990). D´ epartement de Math´ ematiques, Facult´ e de Sciences et Technologie, Universit´ e Paris XII-Val de Marne, 61 avenue du G´ en´ eral de Gaulle, 94010 Cr´ eteil Cedex, France C.M.L.A., E.N.S de Cachan, 61, avenue du Pr´ esident Wilson, 94235 Cachan Cedex, France E-mail address: [email protected]
