A Fluid Dynamic Formulation of the Isometric Embedding Problem in Differential Geometry
aa r X i v : . [ m a t h . A P ] A ug A FLUID DYNAMIC FORMULATION OF THEISOMETRIC EMBEDDING PROBLEM INDIFFERENTIAL GEOMETRY
GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG
Dedicated to Walter Strauss on the occasion of his 70th birthday
Abstract.
The isometric embedding problem is a fundamentalproblem in differential geometry. A longstanding problem is con-sidered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as iso-metric immersions into the three-dimensional Euclidean space. Aremarkable connection between gas dynamics and differential ge-ometry is discussed. It is shown how the fluid dynamics can be usedto formulate a geometry problem. The equations of gas dynamicsare first reviewed. Then the formulation using the fluid dynamicvariables in conservation laws of gas dynamics is presented for theisometric embedding problem in differential geometry. Introduction
We are concerned with isometric embeddings or immersions (i.e., re-alizations) of two-dimensional Riemannian manifolds in the Euclideanspace R . A classical question in differential geometry is whether onecan isometrically embed a Riemannian manifold ( M n , g ) into R N for N large enough. Nash [12] indicated that any smooth compact man-ifold ( M n , g ) can always be isometrically embedded into R s n +4 n for s n = n ( n − /
2. This important paper lays the foundation for thedevelopment of geometric analysis in the second half of the 20th cen-tury. Gromov [5] proved that one can embed any ( M n , g ) even into R s n +2 n +3 . Then a further natural question is to find the smallest di-mension N ( n ) for the Riemannian manifold ( M n , g ) to be isometricallyembeddable in R N ( n ) . In particular, a fundamental, longstanding open Date : August 5, 2008.2000
Mathematics Subject Classification.
Key words and phrases.
Isometric embedding, two-dimensional Riemannianmanifold, differential geometry, transonic flow, gas dynamics, viscosity method,compensated compactness. problem is to characterize intrinsic metrics on a two-dimensional Rie-mannian manifold M which can be realized as isometric immersionsinto R (cf. [6, 13, 14, 17] and the references cited therein). Importantresults have been achieved for the embedding of surfaces with positiveGauss curvature which can be formulated as an elliptic boundary valueproblem (cf. [6]). For the case of surfaces of negative Gauss curvaturewhere the underlying partial differential equations are hyperbolic, thecomplimentary problem would be an initial or initial-boundary valueproblem. When the Gauss curvature changes sign, the problem thenbecomes an initial-boundary value problem of mixed elliptic-hyperbolictype. Hong in [7] first proved that complete negatively curved surfacescan be isometrically immersed in R if the Gauss curvature decays atcertain rate in the time-like direction. In fact, a crucial lemma in Hong[7] (also see Lemma 10.2.9 in [6]) shows that, for such a decay rateof the negative Gauss curvature, there exists a unique global smooth,small solution forward in time for prescribed smooth, small initial data.We are interested in solving the corresponding problem for a class oflarge non-smooth initial data.In Chen-Slemrod-Wang [3], we have introduced a general approach,which combines a fluid dynamic formulation of balance laws with acompensated compactness framework, to deal with the isometric im-mersion problem in R (even when the Gauss curvature changes sign).In Chen-Slemrod-Wang [2], we have developed a vanishing viscositymethod to establish the existence of a weak entropy solution to thetransonic flow in gas dynamics past an obstacle such as an airfoil, viathe method of compensated compactness ([11, 15]). We have found in[3] that the idea of [2] for gas dynamics is useful for solving the isomet-ric embedding problem in differential geometry. In particular, in [3], wehave formulated the isometric immersion problem for two-dimensionalRiemannian manifolds in R via solvability of the Gauss-Codazzi sys-tem, and have introduced a fluid dynamic formulation of balance lawsfor the Gauss-Codazzi system. Then we have formed a compensatedcompactness framework and present one of our main observations thatthis framework is a natural formulation to ensure the weak continuity ofthe Gauss-Codazzi system for approximate solutions, which yields theisometric realization of two-dimensional surfaces in R . As a first ap-plication of this approach, we have focused on the isometric immersionproblem of two-dimensional Riemannian manifolds with strictly neg-ative Gauss curvature. Since the local existence of smooth solutionsfollows from the standard hyperbolic theory, we are concerned with theglobal existence of solutions of the initial value problem with large ini-tial data. The metrics ( g ij ) we study have special structures and forms LUID DYNAMIC FORMULATION FOR ISOMETRIC EMBEDDING 3 usually associated with the catenoid of revolution and the helicoid. Forthese cases, while Hong’s theorem [7] applies to obtain the existence ofa solution for small smooth initial data, our result yields a large-dataexistence theorem for a C , isometric immersion. To achieve this, wehave introduced a vanishing viscosity method depending on the fea-tures of the initial value problem for isometric immersions and havepresented a technique to make the apriori estimates including the L ∞ control and H − –compactness for the viscous approximate solutions.This yields the weak convergence of the vanishing viscosity approxi-mate solutions and the weak continuity of the Gauss-Codazzi systemfor the approximate solutions, hence the existence of a C , –isometricimmersion of the manifold into R with prescribed initial conditions.From Chen-Slemrod-Wang [2, 3], we have seen a remarkable connec-tion between the two distinct areas of gas dynamics and differentialgeometry. Here we present such a connection and show how the fluiddynamics can be used to formulate a geometry problem. Thus, wewill present first the equations in Chen-Slemrod-Wang [2] for the tran-sonic flow problem in gas dynamics, and then the formulation usingthe fluid dynamic variables in conservation laws of gas dynamics forthe isometric embedding problem in differential geometry.2. Equations of Gas Dynamics
In two space dimensions with variables ( x, y ), the steady transonicflow of isentropic case is governed by the following steady Euler equa-tions on conservations of mass and momentum in gas dynamics: ( ρu ) x + ( ρv ) y = 0 , ( ρu + p ) x + ( ρuv ) y = 0 , ( ρuv ) x + ( ρv + p ) y = 0 , (2.1)where ρ is the density, ( u, v ) is the velocity, and p = ρ γ γ ( γ ≥
1) is thepressure. If we assume that the flow is irrotational, then system (2.1)can be reduced to the following two equations of irrotationality andconservation of mass: ( v x − u y = 0 , ( ρu ) x + ( ρv ) y = 0 , (2.2)and, by scaling, the density ρ is determined by Bernoulli’s law: ρ = (cid:16) − γ − q (cid:17) γ − , (2.3) GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG where q is the flow speed defined by q = u + v . The sound speed c is defined as c = p ′ ( ρ ) = 1 − γ − q . (2.4)At the cavitation point ρ = 0, q = q cav := r γ − . At the stagnation point q = 0, the density reaches its maximum ρ = 1.Bernoulli’s law (2.3) is valid for 0 ≤ q ≤ q cav . At the sonic point q = c ,(2.4) implies q = γ +1 . Define the critical speed q cr as q cr := r γ + 1 . We rewrite Bernoulli’s law (2.3) in the form q − q cr = 2 γ + 1 (cid:0) q − c (cid:1) . (2.5)Thus the flow is subsonic when q < q cr , sonic when q = q cr , andsupersonic when q > q cr . For the isothermal flow ( γ = 1), p = c ρ where c > ρ is given byBernoulli’s law: ρ = ρ exp (cid:0) − u + v c (cid:1) (2.6)for some constant ρ >
0, and q cr = c .3. Isometric Embedding in Differential Geometry
In this section, we discuss the isometric embedding problem in dif-ferential geometry in R and its formulation of fluid dynamics.We first give the Gauss-Codazzi system of isometric embedding in R . Let g ij , i, j = 1 , , be the given metric of a two-dimensional Rie-mannian manifold M parameterized on an open set Ω ⊂ R . The firstfundamental form I for M on Ω is I := g ( dx ) + 2 g dxdy + g ( dy ) , and the isometric embedding problem is to seek a map r : Ω → R such that d r · d r = I, that is, ∂ x r · ∂ x r = g , ∂ x r · ∂ y r = g , ∂ y r · ∂ y r = g , so that { ∂ x r , ∂ y r } in R are linearly independent. The correspondingsecond fundamental form is II := h ( dx ) + 2 h dxdy + h ( dy ) . LUID DYNAMIC FORMULATION FOR ISOMETRIC EMBEDDING 5
The fundamental theorem of surface theory (cf. [4, 6]) indicates thatthere exists a surface in R whose first and second fundamental formsare I and II if the coefficients ( g ij ) and ( h ij ) of the two given quadraticforms I and II with ( g ij ) > h ij ) is only in L ∞ for given ( g ij ) in C , , for which theimmersion surface is C , . This shows that, for the realization of a two-dimensional Riemannian manifold in R with given metric ( g ij ) >
0, itsuffices to solve ( h ij ) ∈ L ∞ determined by the Gauss-Codazzi systemto recover r a posteriori. The Gauss-Codazzi system (cf. [4, 6]) can bewritten as ( ∂ x M − ∂ y L = Γ (2)22 L − (2)12 M + Γ (2)11 N,∂ x N − ∂ y M = − Γ (1)22 L + 2Γ (1)12 M − Γ (1)11 N, (3.1)with LN − M = κ, (3.2)where L = h p | g | , M = h p | g | , N = h p | g | , | g | = det ( g ij ) = g g − g ,κ ( x, y ) is the Gauss curvature that is determined by the relation: κ ( x, y ) = R | g | , R ijkl = g lm (cid:16) ∂ k Γ ( m ) ij − ∂ j Γ ( m ) ik + Γ ( n ) ij Γ ( m ) nk − Γ ( n ) ik Γ ( m ) nj (cid:17) ,R ijkl is the curvature tensor and depends on ( g ij ) and its first andsecond derivatives, andΓ ( k ) ij = 12 g kl ( ∂ j g il + ∂ i g jl − ∂ l g ij )is the Christoffel symbol and depends on the first derivatives of ( g ij ),where the summation convention is used, ( g kl ) denotes the inverse of( g ij ), and ( ∂ , ∂ ) = ( ∂ x , ∂ y ). Therefore, given a positive definite metric( g ij ) ∈ C , , the Gauss-Codazzi system gives us three equations for thethree unknowns ( L, M, N ) determining the second fundamental form II . Note that, although ( g ij ) is positive definite, R may change signand so does the Gauss curvature κ . Thus, the Gauss-Codazzi system(3.1)–(3.2) generically is of mixed hyperbolic-elliptic type, as in tran-sonic flow (cf. [2]). In Chen-Slemrod-Wang [3], we have introduced ageneral approach to deal with the isometric immersion problem involv-ing nonlinear partial differential equations of mixed hyperbolic-elliptictype by combining a fluid dynamic formulation of balance laws with acompensated compactness framework. As an example of direct applica-tions of this approach, we have shown how this approach can be applied GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG to establish an isometric immersion of a two-dimensional Riemannianmanifold with negative Gauss curvature in R .We now describe the fluid dynamic formulation of the Gauss-Codazzisystem (3.1)–(3.2) in detail. Although, from the viewpoint of geom-etry, the constraint condition (3.2) is a Monge-Amp`ere equation andthe equations in (3.1) are integrability relations, we can put the prob-lem into a fluid dynamic formulation so that the isometric immersionproblem may be solved via the approaches for transonic flows of fluiddynamics in Chen-Slemrod-Wang [2]. To do this, we set L = ρv + p, M = − ρuv, N = ρu + p, and set q = u + v as usual. Then the equations in (3.1) become thefamiliar balance laws of momentum: ( ∂ x ( ρuv ) + ∂ y ( ρv + p ) = − ( ρv + p )Γ (2)22 − ρuv Γ (2)12 − ( ρu + p )Γ (2)11 ,∂ x ( ρu + p ) + ∂ y ( ρuv ) = − ( ρv + p )Γ (1)22 − ρuv Γ (1)12 − ( ρu + p )Γ (1)11 , (3.3)and the Monge-Amp`ere constraint (3.2) becomes ρpq + p = κ. (3.4)We choose pressure p as for the Chaplygin-type gas: p = − ρ . Then, from (3.4), we have the “Bernoulli” relation: ρ = 1 p q + κ . (3.5)This yields p = − p q + κ, (3.6)and the formulas for u and v : u = p ( p − M ) , v = p ( p − L ) , M = ( N − p )( L − p ) . The last relation for M gives the relation for p in terms of ( L, M, N ),and then the first two give the relations for ( u, v ) in terms of (
L, M, N ).We rewrite (3.3) as ( ∂ x ( ρuv ) + ∂ y ( ρv + p ) = R ,∂ x ( ρu + p ) + ∂ y ( ρuv ) = R , (3.7) LUID DYNAMIC FORMULATION FOR ISOMETRIC EMBEDDING 7 where R and R denote the right-hand sides of (3.3). Then we canwrite down our “rotationality-continuity equations” as ( ∂ x v − ∂ y u = ρq (cid:16) u (cid:0) ρ∂ y κ + R (cid:1) − v (cid:0) ρ∂ x κ + R (cid:1)(cid:17) ,∂ x ( ρu ) + ∂ y ( ρv ) = ρuq ∂ x κ + ρvq ∂ y κ + vq R + uq R . (3.8)In summary, the Gauss-Codazzi system (3.1)–(3.2), the momentumequations (3.3)–(3.6), and the rotationality-continuity equations (3.5)and (3.8) are all formally equivalent. However, for weak solutions,we know from our experience with gas dynamics that this equivalencebreaks down. In Chen-Dafermos-Slemrod-Wang [1], the decision hasbeen made (as is standard in gas dynamics) to solve the rotationality-continuity equations and view the momentum equations as “entropy”equalities which may become inequalities for weak solutions. In ge-ometry, this situation is just the reverse. It is the Gauss-Codazzi sys-tem that must be solved exactly, and hence the rotationality-continuityequations will become “entropy” inequalities for weak solutions.We define the “sound” speed as: c = p ′ ( ρ ) = 1 ρ , (3.9)then from our “Bernoulli” relation (3.5), we see c = q + κ. (3.10)Hence, under this formulation,(i) when κ >
0, the “flow” is subsonic, i.e., q < c , and system (3.3)–(3.4) is elliptic;(ii) when κ <
0, the “flow” is supersonic, i.e., q > c , and system(3.3)–(3.4) is hyperbolic;(iii) when κ = 0, the “flow” is sonic, i.e., q = c , and system (3.3)–(3.4) is degenerate.In general, system (3.3)–(3.4) is of mixed hyperbolic-elliptic type.Thus, the isometric immersion problem involves the existence of so-lutions to nonlinear partial differential equations of mixed hyperbolic-elliptic type.In Chen-Slemrod-Wang [3], we have considered one of the spatialvariables x and y as time-like, have introduced a vanishing viscositymethod via parabolic regularization to obtain the uniform L ∞ esti-mate by identifying invariant regions for the approximate solutions,and have shown that the H − loc –compactness can be achieved for theviscous approximate solutions. Then, as in Chen-Slemrod-Wang [2],the compensated compactness framework yields a weak solution to theinitial value problem of system (3.3)–(3.4) when the initial data lies GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG in the diamond-shaped invariant region. This establishes a C , ( R )immersion of the Riemannian manifold into R . In particular, our ex-istence result asserts the existence of a C , -surface for the associatedmetric for a class of non-circular cross-sections prescribed at x = 0for catenoid. Our study in [3] also applies to the helicoid. See [3]for the details. Possible implication of our approach may be in exis-tence theorems for equilibrium configurations of a catenoidal shell asdetailed in Vaziri-Mahedevan [16]. However, the existence of isomet-ric embeddings/immersions of a general surface with negative Gausscurvature is still open. When the Gauss curvature κ changes sign, theproblem becomes transonic and thus mixed hyperbolic-elliptic type. Inthis mixed-type problem, only special local solutions are known to ex-ist for special data ([8, 6]), and the existence of global solutions is asignificantly difficult open problem. Acknowledgments.
Gui-Qiang Chen’s research was supported inpart by the National Science Foundation under Grants DMS-0807551,DMS-0720925, and DMS-0505473. Marshall Slemrod’s research wassupported in part by the National Science Foundation under GrantDMS-0647554. Dehua Wang’s research was supported in part by theNational Science Foundation under Grant DMS-0604362, and by theOffice of Naval Research under Grant N00014-07-1-0668.
References [1] G.-Q. Chen, C. Dafermos, M. Slemrod, and D. Wang,
On two-dimensionalsonic-subsonic flow , Commun. Math. Phys. 271 (2007), 635-647.[2] G.-Q. Chen, M. Slemrod, and D. Wang,
Vanishing viscosity method for tran-sonic flow , Arch. Rational Mech. Anal. 189 (2008), 159-188.[3] G.-Q. Chen, M. Slemrod, and D. Wang,
Isometric immersions and compen-sated compactness , submitted.[4] M. P. do Carmo,
Riemannian Geometry , Transl. by F. Flaherty, Birkh¨auser:Boston, MA, 1992.[5] M. Gromov,
Partial Differential Relations , Springer-Verlag: Berlin, 1986.[6] Q. Han, and J.-X. Hong,
Isometric Embedding of Riemannian Manifolds inEuclidean Spaces , AMS: Providence, RI, 2006.[7] J.-X. Hong,
Realization in R of complete Riemannian manifolds with nega-tive curvature , Comm. Anal. Geom. 1 (1993), no. 3-4, 487–514.[8] C.-S. Lin, The local isometric embedding in R of 2-dimensional Riemannianmanifolds with Gaussian curvature changing sign cleanly , Comm. Pure Appl.Math. 39 (1986), 867–887.[9] S. Maradare, The fundamental theorem of surface theory for surfaces withlittle regularity , J. Elasticity, 73 (2003), 251–290.[10] S. Maradare,
On Pfaff systems with L p coefficients and their applications indifferential geometry , J. Math. Pure Appl. 84 (2005), 1659–1692. LUID DYNAMIC FORMULATION FOR ISOMETRIC EMBEDDING 9 [11] F. Murat,
Compacite par compensation , Ann. Suola Norm. Pisa (4), 5 (1978),489-507.[12] J. Nash,
The imbedding problem for Riemannian manifolds , Ann. Math. (2),63, 20–63.[13] `E. G. Poznyak and E. V. Shikin,
Small parameters in the theory of isometricimbeddings of two-dimensional Riemannian manifolds in Euclidean spaces ,In:
Some Questions of Differential Geometry in the Large , Amer. Math. Soc.Transl. Ser. 2, (1996), 151–192, AMS: Providence, RI.[14] `E. R. Rozendorn,
Surfaces of negative curvature , In: Geometry, III, 87–178,251–256, Encyclopaedia Math. Sci. 48, Springer: Berlin, 1992.[15] L. Tartar,
Compensated compactness and applications to partial differentialequations. In, Nonlinear Analysis and Mechanics, Heriot-Watt SymposiumIV , Res. Notes in Math. 39, pp. 136-212, Pitman: Boston-London, 1979.[16] A. Vaziri, and L. Mahedevan,
Localized and extended deformations of elasticshells , Proc. National Acad. Sci, USA 105 (2008), 7913-7918.[17] S.-T. Yau,
Review of geometry and analysis , In:
Mathematics: Frontiers andPerspectives , pp. 353–401, International Mathematics Union, Eds. V. Arnold,M. Atiyah, P. Lax, and B. Mazur, AMS: Providence, 2000.
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