ISOMETRIC IMMERSIONS OF SURFACES WITH TWO CLASSES OF METRICS AND NEGATIVE GAUSS CURVATURE. 1. Introduction

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1 ISOMETRIC IMMERSIONS OF SURFACES WITH TWO CLASSES OF METRICS AND NEGATIVE GAUSS CURVATURE WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG Abstract. The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negatie Gauss curature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in L are obtained through the anishing iscosity method and the compensated compactness framework. The L uniform estimate and H 1 compactness are established through a transformation of state ariables and construction of proper inariant regions for two types of gien metrics including the catenoid type and the helicoid type. The global weak solutions in L to the Gauss-Codazzi equations yield the C 1,1 isometric immersions of surfaces with the gien metrics. 1. Introduction Isometric embedding or immersion into R 3 of two-dimensional Riemannian manifolds (or surfaces) M 2 is a well known classical problem in differential geometry (cf. Cartan [6] in 1927, Codazzi [12] in 1860, Janet [30] in 1926, Mainardi [34] in 1856, Peterson [41] in 1853), with emerging applications in shell theory, computer graphics, biological leaf growth and protein folding in biology and so on (cf. [21, 49]). The classical surface theory indicates that for the gien metric, the isometric embedding or immersion can be realized if the first fundamental form and the second fundamental form satisfy the Gauss-Codazzi equations (cf. [5, 35, 36, 44]). There hae been many results for the isometric embedding of surfaces with positie Gauss curature, which can be studied by soling an elliptic problem of the Darboux equation or the Gauss-Codazzi equations; see [27] and the references therein. When the Gauss curature is negatie or changes signs, there are only a few studies in literature. The case where the Gauss curature is negatie can be formulated into a hyperbolic problem of nonlinear partial differential equations, and the case in which the Gauss curature changes signs becomes soling nonlinear partial differential equations of mixed elliptic-hyperbolic type. Han in [26] obtained al isometric embedding of surfaces with Gauss curature changing sign cleanly. Hong in [28] proed the isometric immersion in R 3 of completely negatie cured surfaces with the negatie Gauss curature decaying at a certain rate in the time-like direction, and the C 1 norm of initial data is small so that he can obtain the smooth solution. Recently, Chen-Slemrod-Wang in [8] deeloped a general method, which combines a fluid dynamic formulation of conseration laws for the Gauss-Codazzi system with a compensated compactness framework, to realize the isometric immersions in R 3 with negatie Gauss curature. Christoforou in [11] obtained 2000 Mathematics Subject Classification. 53C42, 53C21, 53C45, 58J32, 35L65, 35M10, 35B35. Key words and phrases. Isometric immersion, L estimate, H 1 compactness, compensated compactness, catenoid metric, helicoid metric. 1

2 2 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG the small BV solution to the Gauss-Codazzi system with the same catenoid type metric as in [8]. See [18, 19, 20, 28, 31, 44, 46, 50] for other related results on surface embeddings. For the higher dimensional isometric embeddings we refer the readers to [2, 3, 4, 9, 10, 22, 25, 38, 39, 40, 43] and the references therein. Recall that in Chen-Slemrod-Wang [8], the C 1,1 isometric immersion of surfaces was obtained for the catenoid type metric: and negatie Gauss curature ds 2 = E(y)dx 2 + ce(y)dy 2 K(y) = k 0 E(y) β2 with index β > 2, c = 1 and k 0 > 0. The goal of this paper is to study the isometric immersion of surfaces with negatie Gauss curature and more general metrics. To this end, we shall apply directly the artificial anishing iscosity method ([13]) and introduce a transformation of state ariables (see e.g. [27]). The anishing iscosity limit will be obtained through the compensated compactness method ([1, 7, 14, 15, 16, 23, 29, 32, 33, 37, 48]). We shall establish the L estimate of the iscous approximate solutions by studying the Riemann inariants to find the inariant region (see e.g [47]) in the new state ariables. Then we proe the H 1 compactness of the iscous approximate solutions, and finally apply the compensated compactness framework in [8] to obtain the weak solution of the Gauss-Codazzi equations, and thus realize the isometric immersion of surfaces into R 3. In the new state ariables, using the anishing artificial iscosity method, we are able to obtain the weak solution of the Gauss-Codazzi system for two classes of metrics, that is, for the catenoid type metric with β 2 and c > 0, and for the helicoid type metric ds 2 = E(y)dx 2 + dy 2 and K(y) = k 0 E(y) a with a 2 and k 0 > 0. An important step to achiee this deelopment is to find the inariant regions of the iscous approximate solutions for the wider classes of metrics, for which the introduction of new state ariables (u, ) plays a crucial role. We note that the L solution of the Gauss-Codazzi equations for the gien metric in C 1,1 yields the C 1,1 isometric immersion from the fundamental theorem of surfaces by Mardare [35, 36]. The paper is organized as follows. In Section 2, we introduce a new formulation of the Gauss-Codazzi system and proide the iscous approximate solutions. In Section 3, we establish the L estimate for the two types of metrics to get the global existence of the equations with iscous terms. In Section 4, we proe the H 1 compactness. In Section 5, combining the aboe two estimates and the compensated compactness framework we state and proe our theorems on the existence of weak solutions for the surfaces with the catenoid type and helicoid type metrics. 2. Reformulation of the Gauss-Codazzi System As in [27] and [8], the isometric embedding problem for the two-dimensional Riemannian manifolds (or surfaces) in R 3 can be formulated through the Gauss-Codazzi system using the fundamental theorem of surface theory (cf. Mardare [35, 36]).

3 ISOMETRIC IMMERSIONS OF SURFACES 3 Let Ω R 2 be an open set and (x, y) Ω. For a two-dimensional surface defined on Ω with the gien metric in the first fundamental form: I = Edx 2 + 2F dxdy + Gdy 2, where E, F, G are differentiable functions of (x, y) in Ω. If the second fundamental form of the surface is II = h 11 dx 2 + 2h 12 dxdy + h 22 dy 2, where h 11, h 12 = h 21, h 22 are also functions of (x, y) in Ω, then the Gauss-Codazzi system has the following form: M x L y = Γ 2 22L 2Γ 2 12M + Γ 2 11N, N x M y = Γ 1 22L + 2Γ 1 12M Γ 1 11N, (2.1) LN M 2 = K. Here and in the rest of the paper x, y stand for the partial deriaties of corresponding function with respect to x, y respectiely. In (2.1), L = h 11 g, M = h 12 g, N = h 22 g, g = EG F 2 > 0, K = K(x, y) is the Gauss curature which is determined by E, F, G according to the Gauss s Theorem Egregium ([17, 46]), Γ k ij (i, j, k = 1, 2) are the Christoffel symbols gien by the following formulas ([27]): Γ 1 11 = GE x 2F F x + F E y 2(EG F 2 ) Γ 1 12 = Γ 1 21 = GE y F G x 2(EG F 2 ), Γ 1 22 = 2GF y GG x F G x 2(EG F 2 ), Γ 2 11 = 2EF x EE y F E x 2(EG F 2, ) Γ2 12 = Γ 2 21 = EG x F E y 2(EG F 2 ),, Γ 2 22 = EG y 2F F y + F G x 2(EG F 2. ) (2.2) As in Mardare [35, 36] and Chen-Slemrod-Wang [8], the fundamental theorem of surface theory holds when (h ij ) or L, M, N are in L for the gien E, F, G C 1,1, and the immersion surface is C 1,1. Therefore it suffices to find the solutions L, M, N in L to the Gauss-Codazzi system to realize the surface with the gien metric. In this paper, we consider the isometric immersion into R 3 of a two-dimensional Riemanian manifold with negatie Gauss curature. We write the negatie Gauss curature as K = γ 2 (2.3) with γ > 0 and γ C 1,1 (Ω), and rescale L, M, N as L = L γ, M = M γ, Ñ = N γ. (2.4) Then the third equation (or the Gauss equation) of system (2.1) becomes LÑ M 2 = 1. (2.5)

4 4 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG The other two equations (or the Codazzi equations) of system (2.1) become where M x L y = Γ 2 L 22 2 Γ 2 M Γ 11Ñ, Ñ x M y = Γ 1 L Γ 1 M 12 Γ 1 11Ñ, (2.6) Γ 2 22 = Γ γ y γ, Γ 1 22 = Γ 1 22, Γ2 12 = Γ γ x 2γ, Γ2 11 = Γ 2 11, Γ1 12 = Γ γ y 2γ, Γ1 11 = Γ γ x γ. (2.7) Consider the following iscous approximation of system (2.5)-(2.6) with artificial iscosity: L y M x = ε L xx Γ 2 L Γ 2 M 12 Γ 2 11Ñ, M y Ñx = ε M xx + Γ 1 L 22 2 Γ 1 M 12 + Γ 1 11Ñ, (2.8) LÑ M 2 = 1, where ε > 0. Our goal is to apply the anishing iscosity method to the smooth solutions of (2.8) to obtain the L solution of (2.5)-(2.6). The eigenalues of the system (2.6) for L 0 are λ ± = M ± 1, L and the right eigenectors are r ± = (1, λ ± ) T. A direct calculation shows λ ± r ± = M ± 1 L 2 thus we can take λ ± as the Riemann inariants. then Introduce the new ariables: Thus L = 1, L M ± 1 = 0, L u := M L, := 1 L, (2.9) u M =, Ñ = u2 2. (2.10) and L y = y 2, Lxx = 22 x xx 3, Ñ x = (2uu x 2 x ) (u 2 2 ) x 2 = 2uu x 2 x u 2 x 2, M y = u y u y 2, Mx = u x u x 2, M xx = u xx u xx 2 2u2 x 2u x x 3. (2.11) (2.12)

5 ISOMETRIC IMMERSIONS OF SURFACES 5 Substituting (2.11) and (2.12) into the system (2.8), we get and y 2 u x u x 2 = 2ε2 x ε xx 3 Γ Γ 2 u 12 Γ 2 u , (2.13) u y u y 2 = εu xx εu xx 2 2uu x 2 x u 2 x 2 Multiplying (2.13) by 2, we get 2εu2 x 2εu x x 3 + Γ Γ 1 u 12 + Γ 1 u (2.14) y + u x u x = ε xx 2ε2 x and multiplying (2.14) by, one has + Γ Γ 2 12u + Γ 2 11(u 2 2 ), (2.15) u y + u y + 2uu x x u2 x = εu xx εu xx 2εu x x + 2εu2 x 2 Γ Γ 1 12u Γ 1 11(u 2 2 ). Substituting (2.15) into (2.16), we obtain (2.16) u y + uu x x = εu xx 2ε xu x Γ Γ 1 12u Γ 1 11(u 2 2 ) + Γ 2 22u + 2 Γ 2 12u 2 + Γ 2 11(u 2 2 )u = εu xx 2ε xu x Γ ( Γ Γ 1 12)u + (2 Γ 2 12 Γ 1 11)u 2 + Γ Γ 2 11(u 2 2 )u. Therefore we hae the following system in the ariables (u, ): u y + (uu x x ) =εu xx 2ε xu x Γ ( Γ Γ 1 12)u + (2 Γ 2 12 Γ 1 11)u 2 + Γ Γ 2 11(u 2 2 )u, (2.17) y + (u x u x ) =ε xx 2ε2 x + Γ Γ 2 12u + Γ 2 11(u 2 2 ). We note that the L estimate for the solutions L, M and Ñ to (2.8) with L > 0 is equialent to the L estimate for the solutions u and to (2.17) with > 0. Set f(u, ) = Γ ( Γ Γ 1 12)u + (2 Γ 2 12 Γ 1 11)u 2 + Γ Γ 2 11(u 2 2 )u, g(u, ) = Γ Γ 2 12u + Γ 2 11(u 2 2 ), then the system (2.17) becomes u y + (uu x x ) = f(u, ) + εu xx 2εu x x, y + (u x u x ) = g(u, ) + ε xx 2ε2 x. (2.18) (2.19)

6 6 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG The al existence of (2.8) is standard, and the global existence can be proed if the L boundedness of L, M and Ñ (or equialently the L boundedness of u and ) is established. The L uniform bound will be established in the next Section 3. The global existence of solution ( L, M) to (2.8) is equialent to the global existence of solution (u, ) to (2.19) which will be proed in Section L Uniform Estimate In order to establish the L bound of u and, we need to derie the equations of the Riemann inariants of (2.19). First we rewrite the system (2.19) in the following form: u y + u u x f(u, ) = + εu xx 2εuxx. (3.1) y u x g(u, ) ε xx 2ε2 x The eigenalues of (3.1) are and the Riemann inariants are λ 1 = u, λ 2 = u +, w = u +, z = u. Multiply (3.1) by (w u, w ) or (z u, z ) to obtain the equations satisfied by the Riemann inariants: w y + λ 1 w x = εw xx 2ε xw x z y + λ 2 z x = εz xx 2ε xz x + f(u, ) + g(u, ), + f(u, ) g(u, ). Then at the critical points of w, the first equation of (3.2) becomes (3.2) εw xx + f(u, ) + g(u, ) = 0, and at the critical points of z, the second equation of (3.2) becomes εz xx + f(u, ) g(u, ) = 0. Hence, by the parabolic maximum principle (see [24, 45]), we hae (a) w has no internal maximum when f(u, ) + g(u, ) < 0, (b) w has no internal minimum when f(u, ) + g(u, ) > 0, (c) z has no internal maximum when f(u, ) g(u, ) < 0, (d) z has no internal minimum when f(u, ) g(u, ) > 0. In order to find the inariant region of w, z, we need to analyze the source terms in (3.2), that is, the signs of f(u, ) + g(u, ) and f(u, ) g(u, ). We shall consider two types of surfaces that hae special metrics with F 0.

7 ISOMETRIC IMMERSIONS OF SURFACES Catenoid type surfaces: G(y) = ce(y), F = 0, c > 0. For the surfaces with metrics of the form: G(y) = ce(y), F = 0 (3.3) with constant c > 0, from the formulas of Γ k ij in (2.2) and (2.7), we can easily calculate that and thus If we assume where α is constant, then, by (2.3), Γ 2 22 = E 2E, Γ2 12 = 0, Γ 2 11 = E 2cE, Γ 1 22 = 0, Γ 1 12 = E 2E, Γ1 11 = 0, Γ 2 22 = E 2E + γ γ, Γ2 12 = 0, Γ2 11 = E 2cE, Γ 1 22 = 0, Γ1 12 = E 2E + γ 2γ, Γ1 11 = 0. γ γ = αe E, 2α E E = K K, that is K(y) = k 0 E(y) 2α, (3.4) with some constant k 0 > 0. From the expressions of f(u, ), g(u, ) in (2.18), one has and then Set f(u, ) = E 2E u E 2cE (u2 2 )u, g(u, ) = E E (α + 1 E ) 2 2cE (u2 2 ). f(u, ) + g(u, ) = E ( cu c(2α + 1) + (u 2 2 )(u + ) ), 2cE f(u, ) g(u, ) = E ( cu + c(2α + 1) + (u 2 2 )(u ) ). 2cE In particular, when α = 1, If we assume ϕ 1 (u, ) = cu c(2α + 1) + (u 2 2 )(u + ), ϕ 2 (u, ) = cu + c(2α + 1) + (u 2 2 )(u ). ϕ 1 (u, ) = (u c)(u + ), ϕ 2 (u, ) = (u c)(u ). E E (3.5) (3.6) (3.7) < 0, (3.8)

8 C 1 z=u C 2 8 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG w=u+ C 1 u C 2 Figure 1. Graphs of leel sets of w and z then the signs of f(u, ) + g(u, ), f(u, ) g(u, ) depend only on ϕ 1 (u, ), ϕ 2 (u, ) respectiely. We now derie and sketch the inariant regions. When α = 1, first we can easily sketch the leel sets of w, z in the u plane in Figure 1. To obtain the inariant region we need to sketch the graphs of ϕ 1 (u, ) = 0, ϕ 2 (u, ) = 0 in the u plane in Figure 2. Now let us find the inariant region in the upper half-plane. We draw a straight line parallel to u = 0 passing through the point C = (0, δ) with 0 < δ < c intersecting the hyperbola at the point and similarly, we get the point where A = (u 0, u 0 + δ), B = ( u 0, u 0 + δ), u 0 = u 0 ( 1) = c δ2. (3.9) 2δ We then draw a straight line perpendicular to u = 0 passing through the point A and another straight line perpendicular to u + = 0 through B. Then the two lines intersect the axis at the same point D = (0, c ) δ2 ; 2δ

9 ISOMETRIC IMMERSIONS OF SURFACES 9 u 2 2 +c=0 u+=0 c 1/2 u =0 u Figure 2. Graphs of ϕ 1 (u, ) = 0, ϕ 2 (u, ) = 0 when α = 1 B D A C u z= 2u 0 δ w=2u 0 +δ z= δ w=δ Figure 3. Inariant region when α = 1 see Figure 3. We see that the square ACBD in Figure 3 is an inariant region. Therefore we get the L estimate of (u, ), that is, c δ2 2δ u c δ2, δ c 2δ δ. (3.10)

10 10 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG ( 2cα c) 1/2 2 u 1 2 Figure 4. Graphs of ϕ 1 (u, ) = 0, ϕ 2 (u, ) = 0 when α < 1. When α < 1, the graphs of ϕ 1 (u, ) = 0, ϕ 2 (u, ) = 0 in the u plane look like the cures marked with 1 and 2 respectiely in Figure 4. Similarly to the case α = 1, we draw a straight line parallel to u = 0 passing through the point C = (0, δ) with 0 < δ < 2cα c intersecting the cure ϕ 1 (u, ) = 0 at the point A = (u 0, u 0 + δ). Then we draw a straight line parallel to u + = 0 passing through the point C = (0, δ) intersecting the cure ϕ 2 (u, ) = 0 at the point B = ( u 0, u 0 + δ). Moreoer, we draw a straight line perpendicular to u = 0 passing through point A and another straight line perpendicular to u + = 0 passing through point B. Then the two lines intersect the axis at the same point D = (0, 2u 0 + δ). Thus, the square ACBD in Figure 5 is an inariant region, and now the L estimate of (u, ) is u 0 u u 0, δ 2u 0 + δ, (3.11) where 0 < δ < 2cα c, u 0 = u 0 (α) = cα + 2δ2 c 2 α 2 4cαδ 2 4cδ 2. (3.12) 4δ

11 ISOMETRIC IMMERSIONS OF SURFACES 11 D B A z= 2u 0 δ δ C w=2u 0 +δ u z= δ w=δ Figure 5. Inariant region when α < 1 Example 3.1. For the following special catenoid type surfaces with ( ( y )) 2 β E(y) = c cosh 2 1 1, G(y) = c c 2 (β 2 1) 2 E(y), and K(y) = c 2 (β 2 1)E(y) β2, where c 0, β 2 are constants, one has E(y) > 0 wheneer y > 0, and E(y) < 0 wheneer y < 0. All the conditions (3.4) and (3.8) for the aboe inariant region are satisfied when y < 0. If we take Ω = {(x, y) : x R, y 0 y 0}, where y 0 > 0 is arbitrary, then in Ω the equations (2.19) are parabolic for y time like. Therefore, when the initial alue (u(x, y 0 ), (x, y 0 )) is in the square ACBD, the parabolic maximum/minimum principle ensures that the square ACBD is an inariant region, which yields the L estimate. We notice that the surface is just the classical catenoid when β = 2. Indeed, for the special catenoid type metric, the surface is gien by the following function: with r(x, y) = (r 1 (x, y), r 2 (x, y), r 3 (x, y)), r 1 (x, y) = r 2 (x, y) = r 3 (x, y) = ( ( y )) 1 β c cosh 2 1 sin(x), c ( ( y )) 1 β c cosh 2 1 cos(x), c y ( ( )) 4 2β 2 1 t β (β 2 1) 2 c cosh 2 1 dt. c

12 12 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG 3.2. Helicoid type surfaces: E(y) = B(y) 2, F = 0, G(y) = 1. For the surfaces with the metric of the form: we can also calculate that E(y) = B(y) 2, F = 0, G(y) = 1, B(y) > 0, (3.13) and Γ 2 22 = 0, Γ 2 12 = 0, Γ 2 11 = E 2E, Γ1 22 = 0, Γ 1 12 = E 2E, Γ1 11 = 0. Γ 2 22 = γ γ, Γ2 12 = 0, Γ2 11 = E 2, Γ 1 22 = 0, Γ1 12 = E 2E + γ 2γ, Γ1 11 = 0. Then Assuming and using E(y) = B(y) 2 and (2.3), we hae f(u, ) = E E u E 2 (u2 2 )u, g(u, ) = γ γ E 2 (u2 2 ). γ γ = ab B, that is, a E E = K K, K(y) = k 0 E(y) a, (3.14) with some constant k 0 > 0. Thus, and Now set f(u, ) = 2B B u BB (u 2 2 )u, g(u, ) = a B B BB (u 2 2 ), f(u, ) + g(u, ) = 2B B u + ab B BB (u 2 2 )(u + ), f(u, ) g(u, ) = 2B B u ab B BB (u 2 2 )(u ). ũ = Bu, ṽ = B, w = Bu + B = ũ + ṽ, z = Bu B = ũ ṽ. (3.15)

13 Since B depends only on y, we hae Note that then and similarly, ISOMETRIC IMMERSIONS OF SURFACES 13 w y + λ 1 w x = ε w xx 2ε x w x z y + λ 2 z x = ε z xx 2ε x z x + (f(u, ) + g(u, ))B + B B w, + (f(u, ) g(u, ))B + B B z. f(u, ) + g(u, ) = B B 2 ( 2Bu ab + ((Bu) 2 (B) 2 )(Bu + B) ), R 1 (u, ) :=(f(u, ) + g(u, ))B + B B w = B ( 2ũ aṽ + (ũ 2 ṽ 2 )(ũ + ṽ) ) + B (ũ + ṽ) B B = B (ũ (a + 1)ṽ + (ũ 2 ṽ 2 )(ũ + ṽ) ), B (3.16) R 2 (u, ) :=(f(u, ) g(u, ))B + B B z = B (ũ + (a + 1)ṽ + (ũ 2 ṽ 2 )(ũ ṽ) ). B Since B(y) > 0 is gien, it remains to find the inariant region of w, z. As in the catenoid case in Subsection 3.1, we need to analyze the signs of R 1 and R 2. If we assume or equialently, from E = B 2, and set B < 0, E < 0, (3.17) ψ 1 (ũ, ṽ) := ũ (a + 1)ṽ + (ũ 2 ṽ 2 )(ũ + ṽ), ψ 2 (ũ, ṽ) := ũ + (a + 1)ṽ + (ũ 2 ṽ 2 )(ũ ṽ), then the signs of R 1 (ũ, ṽ), R 2 (ũ, ṽ) depend on the signs of ψ 1 (ũ, ṽ), ψ 2 (ũ, ṽ) respectiely. Now we end up with a situation similar to the catenoid case in Subsection 3.1 with c = 1, a = 2α, and the corresponding ϕ i (i = 1, 2) in Subsection 3.1 are ϕ 1 (ũ, ṽ) = ψ 1 (ũ, ṽ), ϕ 2 (ũ, ṽ) = ψ 2 (ũ, ṽ). We can find the inariant regions just as in catenoid case in Subsection 3.1. Indeed, the inariant region of (ũ, ṽ) looks like the same as the inariant region ACBD in Figure 3 when a = 2 if we replace the u plane by the ũ ṽ plane and u 0 by ũ 0 = ũ 0 ( 2) defined below; and looks like the same as the inariant region ACBD in Figure 5 when a < 2 in the ũ ṽ plane, and thus we omit the sketch of the inariant regions. From the inariant region, when a = 2, 1 δ2 2δ ũ 1 δ2, δ ṽ 1 2δ δ ; (3.18)

14 14 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG and when a < 2, where ũ 0 ũ ũ 0, δ ṽ 2ũ 0 + δ, (3.19) 0 < δ < a 1, ũ 0 = ũ 0 (a) = a 4δ2 + a 2 8aδ 2 16δ 2. (3.20) 8δ Note that 0 < B(y) C 1,1 (Ω), we can easily obtain the L boundedness of u,. Example 3.2. For the helicoid surface with where c 0, we see that E(y) = c 2 + y 2, F (y) 0, G(y) 1, K(y) = (c 2 + y 2 ) 2, B(y) = c 2 + y 2, and B(y) > 0 for y > 0, and B(y) < 0 for y < 0. If we take Ω = {(x, y) : x R, y 0 y 0}, where y 0 > 0 is an arbitrary constant, then in Ω, the equations in (2.19) are parabolic for y time like. Therefore, when the initial alue (u(x, y 0 ), (x, y 0 )) is in the square ACBD, we hae the inariant region for the solutions. We note that the function for the helicoid in R 3 is r(x, y) = (y sin x, y cos x, cx). Remark 3.1. We can see from Figures 3 and 5 that the L estimates also hold for < 0 since we can also find the inariant regions in the lower half-plane of u, which are symmetric with the inariant regions in the upper half-plane around the u axis. c 2 4. H 1 Compactness In this section we shall proe the H 1 compactness of the approximate iscous solutions. For the strictly conex entropy η = M L, and entropy flux q = M 3 1, L 3 from the parabolic equations (2.8), one has ( M η y + q x = εη xx + Π( L, M) ε L2 L 3 x 2 M ) Lx + L M M 2 x x 2 L = εη xx + Π( L, M) ε L ( ) 2 2 x L + 1 L M L x 3 L M x, (4.1)

15 where From we hae and thus ISOMETRIC IMMERSIONS OF SURFACES 15 Π( L, M) = M L 2 + M L L = 1, L x = x 2, ) ( Γ2 22 L 2 Γ2 12 M + Γ2 11 Ñ ( Γ1 22 L 2 Γ1 12 M + Γ1 11 Ñ). M = u, Mx = u x 2 u x, ( ) η y + q x = εη xx + Π( L, M) 2 ε x + u2 x. (4.2) Due to the L uniform estimates for u and in Subsections 3.1 and 3.2, we hae 0 < b 1 (δ) L b 2 (δ), M b 3 (δ), (4.3) uniformly in ε in Ω, where b 1 (δ), b 2 (δ) and b 3 (δ) are positie constants depending on δ > 0. Therefore Π( L, M) is also uniformly bounded in ε. Let Ω = {(x, y) : x R, y 0 y 0}, where y 0 > 0 is arbitrary. Choose the test function φ Cc (Ω) satisfying φ K 1, 0 φ 1, where K is a compact set and K S = supp φ. From (4.2), we hae 0 ( ) 2 ε x y 0 + u2 x φdydx 0 ( εη xx η y q x + Π( L, M) ) φdydx y 0 (4.4) 0 ( = εηφ xx + ηφ y + qφ x + Π( L, M)φ ) dydx M(φ) y 0 for some positie constant M(φ) uniform in ε (0, 1). Since is uniformly bounded from below with positie lower bound, εx, 2 εu 2 x are bounded in L 1 (Ω). Since u is uniformly bounded and has uniform positie lower bound, one has ε L 2 x = ε 2 x 4 Cε2 x, ε M ( x 2 ux = ε 2 u ) x 2 Cε( 2 x + u 2 x), for some positie constant C uniform in ε, then we see that ε L 2 x and ε M x 2 are uniformly bounded in L 1 (Ω). Noting that ε L xx = ε( ε L x ) x,

16 16 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG and for arbitrary φ C c (Ω), 0 ε L xx φdydx = y 0 ( ε ε L 2 xdydx supp φ C ε 0 as ε 0, 0 ) 1 y 0 ε L x φ x dydx ( 2 (φ x ) 2 dydx Ω ) 1 2 (4.5) we see that ε L xx is compact in H 1 (Ω). From (2.8), we hae M x L y = Γ 2 L 22 2 Γ 2 M 12 + Γ 2 11Ñ ε L xx. (4.6) Since Γ 2 L 2 Γ 22 2 M 12 + Γ 2 11Ñ is uniformly bounded in Ω, it is uniformly bounded in L1 (Ω) and compact in W 1,p (Ω) with some 1 < p < 2 by the imbedding theorem and the Schauder theorem. Therefore M x L y is compact in W 1,p (Ω). Moreoer, we see that M x L y is uniformly bounded in W 1, (Ω) since M and L are uniformly bounded. Finally, by Lemma 4.1 below, we conclude that M x L y is compact in H 1 (Ω). Similarly, Ñ x M y is also compact in H 1 (Ω). Since γ is C1, we see that M x L y and N x M y are also compact in H 1 (Ω). We record the following useful lemma (see [7, 48]) here: Lemma 4.1. Let Ω R n be a open set, then (compact set of W 1,q of W 1,r (Ω) ) (compact set of H 1 (Ω) ) (bounded set (Ω)). where q and r are constants, 1 < q 2 < r. 5. Main Theorems and Proofs In this section, we shall state our main results and also gie the proof. In the preious sections, we hae established the L uniform estimate and H 1 compactness of the iscous approximate solutions to (2.8) for some special metrics of the form ds 2 = Edx 2 + Gdy 2. The corresponding Gauss curature K(x, y) = K has the following form (see [27]): ( ) K = 1 E 2 y + E x G x + G2 x + E y G y 2(E yy + G xx ). (5.1) 4EG E G y To proe the existence of isometric immersion, first let us recall the following compensated compactness framework in Theorem 4.1 of [8]: Lemma 5.1. Let a sequence of functions (L ε, M ε, N ε )(x, y), defined on an open subset Ω R 2, satisfy the following framework: (W.1) (L ε, M ε, N ε )(x, y) is uniformly bounded almost eerywhere in Ω R 2 with respect to ε; (W.2) M ε x L ε y and N ε x M ε y are compact in H 1 (Ω);

17 ISOMETRIC IMMERSIONS OF SURFACES 17 (W.3) There exist o ε j (1), j = 1, 2, 3, with oε j (1) 0 in the sense of distributions as ε 0 such that and M ε x L ε y = Γ 2 22L ε 2Γ 2 12M ε + Γ 2 11N ε + o ε 1(1), N ε x M ε y = Γ 1 22L ε + 2Γ 1 12M ε Γ 1 11N ε + o ε 2(1), (5.2) L ε N ε (M ε ) 2 = K + o ε 3(1). (5.3) Then there exists a subsequence (still labeled) (L ε, M ε, N ε ) conerging weak-star in L to (L, M, N)(x, y) as ε 0 such that (1) (L, M, N) is also bounded in Ω R 2 ; (2) the Gauss equation (5.3) is weakly continuous with respect to the subsequence (L ε, M ε, N ε ) conerging weak-star in L to (L, M, N)(x, y) as ε 0; (3) The Codazzi equations (5.2) as ε 0 hold for (L, M, N) in the sense of distribution. We now present the results on the existence of isometric immersion of surfaces with the two types of metrics studied in Section 3. Definition 5.1. A Riemannian metric on a two-dimensional manifold is called a catenoid metric if it is of the form ds 2 = E(y)dx 2 + ce(y)dy 2, with c > 0, E(y) > 0, E(y) < 0 for y < 0, and the corresponding Gauss curature is of the form with constants k 0 > 0 and β 2. K(y) = k 0 E(y) β2 From the Definition 5.1 and the formula in (5.1), we see that E(y) satisfies the following ordinary differential equation: (E(y) ) 2 EE(y) = 2k 0 E(y) 2 β2. (5.4) We can sole it through the following process utilizing the method of [42]. Set then and (5.4) becomes E(y) = e w(y), E(y) = e w(y) w(y), E(y) = e w(y) ( (w(y) ) 2 + w(y) ), w(y) = 2k 0 e β2 w(y). (5.5) Let f(w(y)) = w(y). After differentiating respect to y, we get i.e., f(w) w(y) = w(y), f(w)f(w) = 2k 0 e β2w,

18 18 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG therefore, noting that E < 0 implies w < 0, f(w) = dw dy = C 1 4k 0 β 2 e β2w. Then one has y = C 2 dw, C 1 4k 0 e β 2 β2 w Denote the right side of the aboe equation by h(w), then w(y) = h 1 (y) and E(y) = e h 1 (y), where h 1 (y) is the inerse function of h(w), C 1 and C 2 depend on the the alue of w(0) and w(0). Then the catenoid metric is of the form: ds 2 = e h 1 (y) dx 2 + ce h 1 (y) dy 2. (5.6) We note that the special catenoid type metric gien in Example 3.1: ( ( y )) 2 β E(y) = c cosh 2 1 1, G(y) = c c 2 (β 2 1) 2 E(y), with K(y) = c 2 (β 2 1)E(y) β2 is a catenoid metric in the sense of Definition 5.1. Definition 5.2. A Riemannian metric on a two-dimensional manifold is called a helicoid metric if it is of the form ds 2 = E(y)dx 2 + dy 2, with E(y) > 0, E(y) < 0 for y < 0, and the corresponding Gauss curature is of the form with constants k 0 > 0 and a 2. K(y) = k 0 E(y) a From Definition 5.2 and the formula (5.1), E(y) satisfies the following ordinary differential equation: (E(y) ) 2 2EE(y) = 4k 0 E(y) 2+a. (5.7) Set then (5.7) becomes E(y) = w(y) 2, w(y) = k 0 w(y) 2a+1. Letting g(w(y)) = w(y), and differentiating respect to y, one has i.e., Thus w(y) = u(w) w(y), g(w) g(w) = k 0 w 2a+1. g(w) = dw dy = C 1 + k 0 a + 1 w2a+2.

19 Then ISOMETRIC IMMERSIONS OF SURFACES 19 y = C 2 dw. C 1 + k 0 a+1 w2a+2 Denote the right side of the aboe equation by h(w), then w(y) = h 1 (y), and E(y) = (h 1 (y)) 2. Therefore, the helicoid metric is where h 1 (y) is the inerse function of h(w) = C 2 ds 2 = (h 1 (y)) 2 dx 2 + dy 2, dw. C 1 + k 0 a+1 w2a+2 Similar to the catenoid metric, C 1 and C 2 depend on the alue of w(0) and w(0). As an example, the helicoid surface with E(y) = c 2 + y 2, c 2 K(y) = (c 2 + y 2 ) 2 is a two-dimensional Riemannian manifold with the helicoid metric in the sense of Definition 5.2. Now we proe the existence of C 1,1 isometric immersion of surfaces with the aboe two types of metrics into R 3 by using Lemma 5.1. Theorem 5.1. For any gien y 0 > 0, let the initial data satisfy the following conditions: (u, ) y= y0 = (ū 0 (x), 0 (x)) := (u(x, y 0 ), (x, y 0 )) (5.8) ū and ū 0 0 are bounded, and inf (ū ) > 0, sup (ū 0 0 ) < 0. x R x R Then, for the catenoid metric in the sense of Definition 5.1, the Gauss-Codazzi system (2.1) has a weak solution in Ω = {(x, y) : x R, y 0 y 0} with the initial data (5.8). Remark 5.1. For the initial data (ū 0 (x), 0 (x)) satisfying the conditions in Theorem 5.1, there exists a constant δ > 0, such that (ū 0 (x), 0 (x)) ACBD := {(u, ) : δ w = u + 2u 0 + δ, (2u 0 + δ) z = u δ}, for the catenoid type metrics, where u 0 is defined in (3.12). That is, (ū 0 (x), 0 (x)) lies in the inariant region ACBD sketched in Figure 3 or Figure 5, i.e., ū 0 (x) is bounded, and 0 (x) has positie lower bound and upper bound. Proof. First for the initial data (5.8), the corresponding initial data for L 0 (x) = 1 0 (x), M0 (x) = ū 0(x) 0 (x), Ñ 0 (x) = ū0(x) 2 0 (x) 2. 0 (x) L, M, Ñ is We use system (2.19) to obtain the approximate iscous solutions and their L estimate, and then use (2.8) to obtain the H 1 compactness.

20 20 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG Step 1. We mollify the initial data (5.8) as ū ε 0(x) = ū 0 (x) j ε, ε 0(x) = 0 (x) j ε, where j ε is the standard mollifier and is for the conolution. From the aboe Remark 5.1, there exists a δ > 0, such that, ū ε 0(x) M(δ), 0 < δ ε 0(x) M(δ), where M(δ) is positie constant depending on δ, and (ū ε 0(x), ε 0(x)) (ū 0 (x), 0 (x)) as ε 0, a.e. Step 2. The al existence of (2.8) can be obtained by the standard theory, hence we hae the al existence of (2.19). From Section 3, we hae the L estimate for the approximate solution of (2.19), then we can obtain the global existence of the approximate solution as follows. We obsere that the first equation of (2.19) is of the diergence form, thus the estimate of u x can be handled in the standard way. Howeer, the second equation of (2.19) is not in diergence form. By differentiation of the second equation with respect to x, we get ( 2ε 2 ) ( x ) y + (u x u x ) x = g(u, ) x + ε( x ) xx x, x where we omit ε and still use u(x, y), (x, y) for the approximate solution of (2.19). Let G(x, y) is the heat kernel of h y = εh xx, then we can sole the aboe equation as the following: x = Therefore = x C y G(x ξ, y) ε 0x(ξ)dξ ( g(u, ) x (u x u x ) x + G(x ξ, y) ε 0x(ξ)dξ y y 0 ( 2ε 2 ) ) x G(x ξ, y η)dξdη x ( ) g(u, ) + (u x u x ) + 2ε2 x G(x ξ, y η) x dξdη. G(x ξ, y) ε 0x(ξ) dξ ( g(u, ) + (u x u x ) + 2ε 2 ) x G(x ξ, y η) x dξdη ε 0x C 0 + M(δ)( u x C 0 + x C 0 + ε x C 0) 0 y 0 G(x ξ, y η) x dξdη, where C 0 stands for the norm of continuous functions. The aboe process yields the C 0 estimate of x. Similarly, we can estimate C k (for any integer k 1) norms of u(x, y), (x, y), which can be bounded by the C k norms of ū ε 0 and ε 0. Thus, we obtain the global existence of smooth solutions to the system (2.19). Step 3. In Section 3 we hae proed L boundness of L, M, Ñ, and then L, M, N is in L for γ C 1. By the reerse process of Section 2, we can reformulate the equations

21 ISOMETRIC IMMERSIONS OF SURFACES 21 (2.19) of u and as the equations (2.8). Therefore, as in Section 4 we also obtain the H 1 compactness. So we hae proed that our approximate solutions satisfy (W.1) and (W.2) in the framework of Lemma 5.1. Furthermore, from Section 2, we hae ( M x L y ) ( Γ 2 22 L 2 Γ 2 12 M + Γ 2 11Ñ) = ε L xx, (Ñx M y ) ( Γ 1 22 L + 2 Γ 1 12 M Γ 1 11Ñ) = ε M xx. As in Section 4, from (4.5), we can get that, as ε 0, ε L xx 0, in the sense of distribution, and then M x L y = Γ 2 22 L 2 Γ 2 12 M + Γ 2 11Ñ + o 1(1) holds in the sense of distribution. Similarly, Ñ x M y = Γ 1 22 L + 2 Γ 1 12 M Γ 1 11Ñ + o 2(1) also holds in the sense of distribution. Here o 1 (1), o 2 (1) 0 as ε 0. We note that the Gauss equation holds exactly for the iscous approximate solutions. Therefore (W.3) is satisfied. Consequently, we complete the proof of the theorem and obtain the isometric immersion of the surface with the catenoid metric in R 3 using Lemma 5.1. Since we also obtained the L estimate and the H 1 compactness for the helicoid metric in the preious sections, we can hae the isometric immersion of surfaces with the helicoid metric in R 3 just as the case for the catenoid metric. Theorem 5.2. For any gien y 0 > 0, let the initial data satisfy the following conditions: and (u, ) y= y0 = (ū 0 (x), 0 (x)) := (u(x, y 0 ), (x, y 0 )) ū and ū 0 0 are bounded, inf (ū ) > 0, x R sup (ū 0 0 ) < 0. x R Then for the helicoid metric in the sense of Definition 5.2, the Gauss-Codazzi system (2.1) has a weak solution in Ω = {(x, y) : x R, y 0 y 0} with the initial data (5.8). Remark 5.2. Although the catenoid metric for β > 2 has also been studied by Chen- Slemrod-Wang in [8], their L estimate is different from ours, especially for L. In addition, we also proe the isometric immersion of catenoid for β = 2. Remark 5.3. By Remark 3.1, if we replace the second condition on the initial data in Theorem 5.1 and Theorem 5.2 by the following: sup (ū ) < 0, inf (ū 0 0 ) > 0, x R x R then both theorems still hold. Remark 5.4. By een symmetry from the weak solution in Ω = {(x, y) : x R, y 0 y 0} obtained in Theorem 5.1 and Theorem 5.2, we can obtain the weak solution in Ω = {(x, y) : x R, 0 y y 0 }.

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24 24 WENTAO CAO, FEIMIN HUANG, AND DEHUA WANG [50] S.-T. Yau, Reiew of geometry and analysis. In: Mathematics: Frontiers and Perspecties, pp , International Mathematics Union, Eds. V. Arnold, M. Atiyah, P. Lax, and B. Mazur, AMS: Proidence, W. Cao, Institute of Applied Mathematics, AMSS, CAS, Beijing , China. address: F. Huang, Institute of Applied Mathematics, AMSS, CAS, Beijing , China. address: D. Wang, Department of Mathematics, Uniersity of Pittsburgh, Pittsburgh, PA 15260, USA. address: