A Lie Transformation Group Attached to a Second Order Elliptic Operator

New York Journal of Mathematics

New York J. Math. ⁵(1999) 143{146.

A Lie Transformation Group Attached to a Second Order Elliptic Operator

Atallah Aane

Abstract. Given a second order elliptic dierential operator ^Lon a com- pact^C1manifold, we prove that the group of transformations which preserve the sheaf of functions annuled by^Lis a Lie transformation group under the compact-open topology.

Contents

1. Introduction 143

2. Proofs 144

References 146

1. Introduction

It is well known that for a given manifold, the group of dieomorophisms which preserve some geometric structure is often a Lie transformation group. For instance, in 3], the authors deduce from a famous theorem of Palais a large list of results of this kind. In this note, where all objects are assumed to be of class C¹, we consider, on a compact manifoldM, a partial dierential operator which has in any local coordinates the form

L=a^ij @²

@xⁱ@x^{j +}b^k @

@x^{k +}c with the following conditions:

the coecientsa^ij b^kandc areC¹ functions, the matrix (a^ij) is symmetric and positive denite, the functionc has negative values.

Using the Bochner-Montgomery Theorem 2] on the group of dierentiable trans- formations, we study the case where the geometric structure considered is the sheaf KerL. More precisely, fori= 1, 2, letMi be anmi-manifold provided with a par- tial dierential operatorLisatisfying the three conditions above. We introduce the

Received October 15, 1999.

Mathematics Subject Classication. 58D05, 58G03.

Key words and phrases. Lie transformation groups, elliptic operators.

c

1999StateUniversityofNewYork

ISSN1076-9803/99

143

144

Atallah Aane

subsetEL(M¹M²) of all mapsf ²C⁰(M¹M²) such that for any open subsetU² ofM²and'²C¹(U²) satisfyingL²'= 0 onU²the composite function'f also satises L¹('f) = 0 onf^;1(U²): In the case M¹ =M² =M, we shall consider the group EL(M) of all homeomorphisms of M such that for any pair (U²') as above,L'= 0 onU²if and only ifL('f) = 0 onf^;1(U²):In the next section we shall prove the following results:

Proposition 1.1.

^WhenM1is compact,EL(M¹M²) provided with the compact- open topology is locally compact and contained in C¹(M¹M²):

Proposition 1.2.

EL(M) is a Lie transformation group under the compact-open topology.

2. Proofs

First, we give two technical lemmas.

Lemma 2.1.

For any point pthere exists a coordinate system x^im_i=1 dened on a neighborhood U such that

Lxⁱ0 fori= 1:::m.

Proof.

By the imposition of a suitable coordinate system, we may assume thatp= 0,M is a neighborhood ofpin

R

^mandL=a^kl @²

@x^k@x^l+b^j @

@x^j+cwitha^ll(x)⁶= 0 for all x in M and l = 1:::m. For i = 1:::m, we prove the existence of a function uⁱ such that Luⁱ = 0 uⁱ(0) = 0 and d⁰uⁱ = dxⁱ. To do this, we apply the theorem of Hormander given in the appendix of 4] and sinceLis elliptic, the solutionsuⁱ areC¹and give a coordinate system.

The next lemma seems classical under other forms.

Lemma 2.2.

LetL=a^kl @²

@x^k@x^l+b^j @

@x^j+cbe an elliptic dierential operator on an open subsetW of

R

^m with C¹ coecients, and suppose that the functionc is negative. Let ^ff_n^g_n^2N be a sequence of continuous functions onW satisfying

(a) Lf_n= 0 for all n²

N

.

(b) There existsC >0 such that^jf_n(x)^jC for allx²W andn²

N

. Then(i) fn:²C¹(W) for all n²

N

.

(ii) One can extract a subsequence^ff_n^kg_k¹ which converges inC¹(W):

Proof.

Assertion (i) follows from the ellipticity ofL. Let us prove assertion (ii).

Let K be a compact subset of W and " > 0 such that the closed ball B(p") =

fjx^;p^j"^gis contained inW wheneverp²K:Given a pointp²K, we consider the linear mapSfromC⁰(@B(p")) intoC¹(B(p")) which sends²C⁰(@B(p")) on the unique solution of the Dirichlet problem:

Lu= 0 onB(p") u=on@B(p"):

In fact S is continuous. Indeed, if ^fn^g converges to in C⁰(@B(p")) then, by the maximum principle ^fSn^gconverges to S uniformly on B(p") this makes sure that S is closed and one can apply the closed graph theorem. Now, by our

A Lie Transformation Group

145 hypothesis the sequence n=fn^j@B⁽p"⁾ is bounded and fn = Sn thus, the sequence^ffn^gis bounded in the Montel spaceC¹(B(p")) and one can extract a converging subsequence. Since K is compact, one can nd a neighborhood U of K and a subsequence which converges inC¹(U). But W is a countable union of compact subsets, and so the classical diagonal method gives the conclusion.

Proofs of Propositions 1.1 and 1.2.

Since the two manifolds are metrizable, the compact-open topology is equivalent to the one of the uniform convergence onM¹. By Lemma 2.1, we have an open covering^fU^g2ofM²where theUare relatively compact, such that on any U there is a coordinate system y = ⁿy^o

1m²

satisfying

L²y= 0 on U and there existsC>0 with yC on U: By classical topology, there exists a second open covering^fU^0g2 of the para- compact manifold M² with the inclusions U⁰ U. Forf ² E(M¹M²) we put V=f^;1(U⁰) from the compactness ofM¹we deduce a nite part ⁰ of and an open covering^fV^0g20 ofM¹such thatV⁰ V for all²⁰. Clearly, the subset

=ⁿf ²EL(M¹M²)^jf(V⁰)U⁰ for all^20o

is a neighborhood off inEL(M¹M²) in the compact-open topology. Let^fhn^gn¹

be a sequence in . Let d be a metric on M¹ and " > 0 such that for any p ² M¹ the ball B(p") =^fd(xp)"^gis contained in the intersection of some chart domain with someV⁰⁽_p⁾((p)²⁰): Given a pointp²M¹, we apply Lemma 2.2 to each of sequences ⁿy⁽_p⁾hn

o

n¹ and we obtain a subsequence y⁽p⁾hn^k

which converges uniformly on B(p"). As constructed, the limit has its values in y⁽p⁾(U⁽p⁾) and by composition we get a subsequence of^fhn^gwhich converges uniformly onB(p"):The rst part of the proposition results from the compactness ofM¹and the obvious fact thatEL(M¹M²) is closed inC⁰(M¹M²) provided with the compact-open topology. The ellipticity ofL¹ gives the second part.

For Proposition 1.2,E_L(M) is obviously a group. Furthermore, we know from a result of Arens 1] that it is a topological group under the compact-open topology.

Moreover, we can deduce easily from Proposition 1.1 that it is locally compact and the Bochner-Montgomery Theorem 2] gives the conclusion.

Corollary 2.3.

Suppose that the manifold M is compact. Then the groupE_L⁰(M) of all homeomorphisms f ²C⁰(MM) such that for any open subsetU of M and '²C¹(U) we have:

(L')f =L('f) on f^;1(U) is a Lie transformations group under the compact-open topology.

Proof.

Since the linear dierential operators are continous on the distributions, one can verify that E_L⁰(M) is a closed subgroup in EL(M) and use the Cartan Theorem.

146

Atallah Aane

Remark 2.4.

1. In Proposition 1.2, the compactness ofM is not necessary.

2. E_L⁰(M) may be a proper subgroup ofEL(M)

Firstly, if L is the Laplace-Beltrami operator of a Riemannian manifold M, EL(M) is the group of conformal transformations which is a Lie transformation group when m 3 (see 5, p. 310]). Secondly, whenM is the euclidian space

R

^m

E_L⁰(M) is the group of isometries.

References

1] R. Arens, Topologies for homeomorphism groups, Amer. J. Math. ⁶⁸ (1946), 593{610, MR 8,479i, Zbl 061.24306.

2] S. Bochner, D. Montgomery,Locally compact groups of dierentiable transformations, Ann.

of Math.⁴⁷(1946), 639{653, MR 8,253c Zbl 061.04407.

3] H. Chu, S. Kobayashi,The automorphism group of a geometric structure, Trans. Amer. Math.

Soc.¹¹³, (1964), 141{150, MR 29 #1596, Zbl 131.19704.

4] B. Fuglede,Harmonic morphisms between semi-riemannian manifolds, Ann. Sci. Fenn. Math.

21(1996), 31{50, MR 97i:58035, Zbl 847.53013.

5] S. Kobayashi, K. Nomizu,Foundations of Dierential Geometry, I. John Wiley & Sons, New York, 1963, MR 27 #2945.

Institut de Mathematiques, U.S.T.H.B,, El-Alia, B.P. 32 Bab-Ezzouar, 16111 Alger, Algerie.

[email protected]

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