The Bao-Ratiu Equations on SurfacesAuthor(s): Bennett PalmerSource: Proceedings: Mathematical and Physical Sciences, Vol. 449, No. 1937 (Jun. 8, 1995), pp.623-627Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52683 .

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The Bao-Ratiu equations on surfaces

BY BENNETT PALMER

Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, U.K.

We study the existence of asymptotic directions for the volume preserving dif- feomorphism group of a compact surface E, considered as a submanifold of the full diffeomorphism group. We show that when the curvature of Z is positive, there are no asymptotic directions. Related results are obtained for domains with boundary.

1. Main result

Let 7 be a smooth compact oriented d-dimensional Riemannian manifold with volume form dV. For s > 1 + d/2, let

Diff8 = Diff'(S) := {: S -> S p is an HS diffeomorphism},

Diffv - DiffS () := { e Difff Io* dV = dV} be the Hs diffeomorphism and Hs volume preserving diffeomorphism groups, respectively. For (o E Diff5,

T, Diff (S) (F~?p*TS), where rF denotes the space of Hs sections. Right translation defines an isomor- phism

p, : rr(TE) - rS((*TE), o =: For < C Diffs,

T? Diff = p({( e rFS(TE) I div 0}). The metric on Diff5 is defined by

(eW, )O:= '-p ({, r/)'(p) dV(p), i,r/ eC T, Diff ,

which defines a weak Riemannian structure on Diff5 which induces a right invari- ant weak Riemannian structure on Diff>. This metric may also be used to define a connection D and a curvature tensor on Diff5 in a natural way. We refer the reader to Bao et al. (1993) for a description of these constructions.

Using the connection D, the second fundamental form of Diff' can be defined as

IH(, Tr) - (Dr/l), ,I rE TDiffv, where L denotes projection orthogonal to Diffv. This projection can be defined

using the Hodge decomposition.

Proc. R. Soc. Lond. A (1995) 449, 623-627 ) 1995 The Royal Society Printed in Great Britain 623 T1X Paper

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It is well known that the intrinsic geometry of Diffz plays an important role in the study of fluid mechanics on S. Arnold (1966) showed that the Euler equations of a perfect fluid on S could be interpreted as the geodesic equations on Diffv. As in finite-dimensional submanifold theory, the intrinsic and extrinsic geometries of Diffv are closely related. For example, it is straightforward to show that the Gauss, Codazzi and Ricci equations hold in this infinite-dimensional setting.

In Bao & Ratiu (1993) and Bao et al. (1993), the authors initiated the in- vestigation of the asymptotic directions of DiffV. By definition, an asymptotic direction is a non-zero tangent vector to Diffv, satisfying

J7(^)-0.

As shown in Bao & Ratiu (1993), an H8 vector field ~ on E defines an asymptotic direction on Diffv if, and only if, ( solves

div O = 0, div V(= O. (*)

Here, V denotes the metric connection on S. We refer to (*) as the Bao-Ratiu equations.

Theorem 1.1. Let E be a two-dimensional compact oriented surface with Gauss curvature K > O. Then, for s > 4, Diffv admits no asymptotic directions.

In contrast, it is shown in Bao et al. (1993) that every surface admits a metric such that Diffv possesses asymptotic directions for all s > 0. It is also shown that even when E has strictly positive curvature, the curvature of Diff8 is only non-negative. Also, the following example will indicate that the theorem has no direct generalization to higher dimensions. Let G be a compact Lie group. Then G may be endowed with a bi-invariant metric. With respect to this metric, the covariant derivative for left-invariant vector fields X and Y is given by

VxY= [X,Y].

Note that since the metric is bi-invariant, each left-invariant vector field is di- vergence free. It then follows that every left-invariant vector field solves (*). In particular, note that this is true for G = SU(2) = S3.

Proof. Suppose, to the contrary, that ~ exists. By the Sobolev imbedding the- orem (Wells 1993, pp. 114), ( c C2. Let J denote the almost-complex structure of E. By the Gauss-Bonnet theorem, E is topologically a 2-sphere and, hence, any C2 divergence-free vector field is of the form

&= JVu, u C3.

As in Bao & Ratiu (1993), one sees that ~ solves (*) if, and only if, u solves the Monge-Ampere-type equation

F[u] :=-2det(uij)+ K Vul2 = 0, (1.1)

where uij := (VVu, ej) are the components of the Hessian of u with respect to an orthonormal frame {el, e2}. Equation (1.1) can be rewritten as

IVVUl2-(A)2 + KIVu2 = 0, (1.2)

using

VVul2 (Uij)2', Au= suii.

Proc. R. Soc. Lond. A (1995)

624 B. Palmer

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The Bao-Ratiu equations on surfaces

Since

jAu-0,

and since u is not constant, the set

:= {xlAu > 0}

is non-empty. Let Qo be any component of Q, and let

Q ':= {zx e Qo\Au > 6}.

By Sard's theorem and the assumptions on u, almost every e is a regular value of Au and, for these values of E, OQg is a smooth compact curve. Hence, by Stokes's theorem,

:= / Au= - Vunds< (maxl7Vul) 02,1

where 10Q&j denotes the length of (9Q. Using (1.2), we obtain, on 0Q,,

= Au = vVVlu2 + KIVu u> v/KolVl, (1.3)

where Ko is the minimum of K in E. Therefore,

I K la1 1 (1.4) V/o

holds for almost all E. By the assumptions on u, and the co-area formula (Chavel 1984, pp. 85),

oo > IVAu - ]01, I de. 0o 0

Since, for every t > 0,

oo - dE/ ln(1/c), /o

there exists a sequence Ej of regular values of Au such that Ej \ 0 and

lafk I < I

j-= 1, 2, 3.. Ejln(1/ j)'

From (1.4), we obtain

Kj <

/K0(/) O, as j -+ .

However, Ie is positive and increases with j, yielding a contradiction. 1

2. Domains with boundary In this section, we consider asymptotic directions for Diffv when S is a rela-

tively compact domain in a Riemannian manifold with smooth immersed bound- ary 0Z. In this case, asymptotic directions are defined by the following boundary value problem (see, for example, Bao & Ratiu 1993):

div = , in X, (2.1) Proc. R. Soc. Lond. A (1995)

625

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divV - 0, in X, (2.2)

(,n) - 0, on 01 , (2.3)

(V, n) = 0, on 0E. (2.4)

Theorem 2.1. Let E be a smooth compact two-dimensional manifold with smooth immersed boundary. Assume that the Gaussian curvature of S is positive and that the geodesic curvature kg of OE vanishes, at most, at finitely many points. Then there is no non-trivial ~ C C2(E) which solves (2.1)-(2.4).

Proof. Suppose, to the contrary, that ~ exists. We first show that the boundary conditions, together with the assumption on the geodesic curvature, imply that ( vanishes on the boundary. Let p E 80 and suppose that <(p) / 0. Then, from

(2.3), one sees that ~ is tangent to C0 and so

0 -(V,n)< - 12kg(p)

shows that kg(p) = 0. Hence, J can only be non-zero at finitely many boundary points so, by continuity, ~ is identically zero on the boundary.

By Stokes's theorem, we have

div J() = JJ() nds= 0. (2.5)

Using (2.2), and the fact that any divergence free vector field on a surface is locally Hamiltonian, we see from (1.2) that S satisfies

IV2 -(div J)2 + K l2 = 0, (2.6)

from which we see that if div Jd vanishes identically, then so does (. It follows from (2.5) that div JI must take on positive values somewhere in S.

We now proceed as in the proof of theorem 1.1. Define Q, QO, f,, etc., as before, but with Au replaced by div JJ. The only difference is that now Q, may not be compactly contained in the interior of E. However, if p c 0Q,, then either p is in the interior of X, in which case we obtain e >, /K1oll from (2.6) as before, or p c 8Q whence 0[ = 0. We therefore obtain

h ( sup )110QA\OI. a00 \aO

The co-area formula may be applied as follows. Let E -> E' be any embedding of S into a proper open subset of a surface 5'; for example, the double of S. Let f be any C1 function which agrees with div Jc on S and vanishes off a neighborhood of S. Then we obtain

oo > / ifl {- f s}) d j I6,\os d?.

The rest of the proof follows as in theorem 1.1. ?

Corollary 2.2. Under the assumptions of the previous theorem, Diff' admits no asymptotic directions for sufficiently large s.

Proof. It is well known that the restriction map f - flor, f C C(ES), extends to a continuous map

T : H () -, Hs-1/2 (O),

Proc. R. Soc. Lond. A (1995)

B. Palmer 626

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The Bao-Ratiu equations on surfaces

called the trace. Using T, theorem 2.1 and the Sobolev imbedding theorem, one sees that for s > 0, an HS solution of (2.1)-(2.4) must, in fact, be in C2(E). U

We conclude with a simple non-existence result for domains in arbitrary dimen- sions. Recall that a C2 function u is called strictly convex if its Hessian (V Vu, ) is positive definite.

Theorem 2.3. Let N be a complete Riemannian manifold and let S be a relatively compact subdomain with smooth immersed boundary. Assume that there exists a function w C C2(() which is almost everywhere strictly convex. Then there is no non-trivial ~ C C2(S) which solves (2.1)-(2.4).

Proof. Assume, to the contrary, that S exists. Using (2.2), we obtain

div(uVW ) = (Vu, Ve) + u div V~ = (Vu, Ve).

Therefore, using (2.4), we obtain

0 - u(Ve, n) ds= j div(uVz)- (Vu, V ). (2.7)

Note that by (2.1),

div((Vu, ) = (V(Vu, +), ) + (Vu, ) div

= (V(Vu, ), J)

= (Vu, ) - (VVu, ) + (Vu, Vi).

Using this, (2.2), (2.3) and (2.7), we obtain

0- = (Vu, ) ,n) ds= jdiv((Vu, )e)

= j(vcu, e), which gives a contradiction. ?

For every n, the n-dimensional Euclidean and n-dimensional hyperbolic space admit global convex functions. Consequently, no subdomains of these manifolds admit C2 solutions of (2.1)-(2.4). If S is the upper hemisphere of Sn(1), embedded in Rn+l in the standard way, then minus the height function defines an almost- everywhere convex function on S and the same conclusion holds.

References

Arnold, V. I. 1966 Sur la geometrie differentielle des groupes de Lie de dimension infinite et ses applications a la l'hydrodynamique des fluides parfait. Ann. Inst. Fourier 16, 1319-1361.

Bao, D. & Ratiu, T. 1993 On the geometric origin and the solvability of a degenerate Monge- Ampere equation. Proc. Symp. Pure Math. 54, part 1, 55-69.

Bao, D., Lafontaine, J. & Ratiu, T. 1993 On a nonlinear equation related to the geometry of the diffeomorphism group. Pacific J. Math. 158, 223-242.

Chavel, I. 1984 Eigenvalues in Riemannian geometry. New York: Academic.

Wells, R. 0. 1973 Differential analysis on complex manifolds. Englewood Cliffs, NJ: Prentice Hall.

Received 24 May 1994; accepted 25 October 1994

Proc. R. Soc. Lond. A (1995)

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