minimal submanifolds in path space

MINIMAL SUBMANIFOLDS IN PATH SPACE

C. King, C.-L. Terng*

December 1992

1. Introduction

The first fundamental form, shape operators, and the induced normal connectionof a submanifold in Rn are related by the structure equations of Rn, and they de-termine the submanifold up to rigid motion. These local invariants can be definedsimilarly for submanifolds in a Hilbert space. But in order to apply infinite dimen-sional differential topology and Morse theory to submanifolds in Hilbert space, werestrict ourselves to the class of proper Fredholm submanifolds (cf. [T1]). Recallthat a submanifold M of a Hilbert space V is proper Fredholm (PF) if the restric-tion of the endpoint map η : ν(M) → V to νr(M) is proper Fredholm, where ν(M)is the normal bundle, 0 < r < ∞, and

νr(M) = v ∈ ν(M)| ||v|| ≤ r.

Although there is a good theory relating the local and global geometry of PFsubmanifolds, it is not clear how one should define the induced volume and themean curvature vector. The main purpose of this paper is to provide definitions ofthese quantities for a class of PF submanifolds and prove an infinite dimensionalanalogue of the Hsiang-Lawson theorem [HL] for invariant minimal submanifoldsin Hilbert spaces.

Let G be a compact, simply connected Lie group, g its Lie algebra, ( , ) anAd-invariant inner product on g, and ds2 the bi-invariant metric on G defined by( , ). Let V = H0([0, 1], g) denote the Hilbert space of square integrable mapsu : [0, 1] → g with the L2-norm defined by

| u |2=∫ 1

0

(u(t), u(t)) dt.

A natural family of examples of PF submanifolds is provided by the principal orbitsof the gauge group L(G) = g ∈ H1([0, 1], G) | g(0) = g(1) acting on the pathspace (or space of connections) V = H0([0, 1], g). It was shown in [T1] that theseorbits are isoparametric, meaning that the normal bundle is flat and the principal

1991 Mathematics Subject Classification. Primary 53C40, 53C42, 58G32, 60J65.

Key words and phrases. minimal submanifolds, regularized mean curvature vector, Wiener

measure.

*research supported in part by DMS 9103221

Typeset by AMS-TEX

MINIMAL SUBMANIFOLDS IN PATH SPACE 3

curvatures along any parallel normal field are constant. Furthermore the spectrumof the shape operator on each orbit can be computed explicitly.

A finite dimensional submanifold of Euclidean space inherits a volume form fromthe first fundamental form of the embedding. If the embedding is varied along anormal vector field v, then to first order the variation of the volume is the meancurvature along v. A submanifold is called minimal if its mean curvature vectorfield vanishes; this condition implies that the volume of the submanifold (if it isfinite) is stationary under variations of the embedding. This observation provides animportant analytical tool for the investigation of submanifolds of extremal volume.

It is an interesting problem to find an appropriate definition of minimal subman-ifold in the infinite dimensional case. For a PF submanifold of Hilbert space theanalogue of the mean curvature along a normal vector field is the trace of the corre-sponding shape operator. Unfortunately, in the interesting case of the gauge grouporbits in path space, the shape operators are not trace class. Furthermore sincethere is no infinite dimensional version of Lebesgue measure on Hilbert space, the“volume form” for the induced metric on the orbit is not meaningful. So neither themean curvature vector field nor the local variation of volume of these submanifoldsis well-defined.

In some cases it is possible to make sense of the trace of a non trace-class compactoperator by using zeta-function regularization. In section 5 we apply this techniqueto the shape operator of the principal gauge orbits in path space for the groupSU(2). We also use the zeta-function regularization technique to compute the“variation of the regularized induced volume form” as the embedding varies overthe gauge orbits. Both calculations give sensible answers, and both predict thesame (unique) minimal orbit.

Motivated by the SU(2) case just described, we present in section 4 the definitionof what we call a regularizable submanifold, and its regularized mean curvaturevector. Both these definitions rely on the zeta-function regularization of the traceof the shape operator making sense. Furthermore, we define a submanifold to beminimal if it is PF, regularizable, and its regularized mean curvature vector is zero.As an application we prove that the orbits of the based loop group Ωe(G) = g ∈H1([0, 1], G)|g(0) = g(1) = e in V are regularizable and minimal. Let H1

e ([0, 1], G)be the Hilbert Lie group of H1-paths g with g(0) = e, equipped with the right-invariant H1-metric. Let

E : H0([0, 1], g) → H1e ([0, 1], G)

denote the parallel transport map, i.e., E(u) is obtained as the solution of thedifferential equation for parallel transport along [0, 1] for the connection u(t)dtstarting at e. Let ϕ : H0([0, 1], g) → G be the parallel transport from t = 0 tot = 1, i.e., ϕ(u) = E(u)(1). It is known that the orbits of Ωe(G) in V are thefibers of the parallel transport map ϕ : V → G. One of our main results is thatfor any closed submanifold M of G, the corresponding set M = ϕ−1(M) is PF andregularizable. Furthermore if M is minimal in the Riemannian manifold (G, ds2)then M is minimal in V . We also consider a family of submanifolds of V , which wecall cylinder sets; these are defined by putting conditions on the parallel transportmap at a sequence of times between 0 and 1 (see section 3). To be more precise,let t denote a sequence of times 0 = t0 < t1 < . . . < tn = 1, and M a closedsubmanifold of

∏ni=1 Gi, where Gi is the group G equipped with the bi-invariant

4 C. KING & C.-L. TERNG

metric 1ti−ti−1

ds2. We prove that the cylinder set

Mt = u ∈ V | (E(u)(t1), E(u)(t1)−1E(u)(t2), . . . , E(u)(tn−1)−1E(u)(1)) ∈ Mis PF and regularizable. Moreover, if M is minimal in

∏ni=1 Gi then Mt is minimal

in V .Our result stated above can be viewed as an infinite dimensional analogue of

a theorem of Hsiang and Lawson on invariant minimal submanifolds [HL], andcan also be viewed as an example for which the “symmetric criticality principle”(cf. Palais [P]) holds. Recall that the Hsiang-Lawson theorem can be stated asfollows: Let G be a compact Lie group, N a Riemannian manifold, and G acton N isometrically. Let M be a submanifold of N invariant under the actionof G, I(M, N) and IG(M, N) denote the space of immersions and G-equivariantimmersions of M into N , and A : I(M, N) → R the area functional. If f ∈IG(M, N) is a critical point for A | IG(M, N) then f is a critical point for A, i.e.,f is a minimal immersion. When M is a G-orbit in N , this is an earlier theorem ofHsiang [H], which states that for each orbit type of G that occurs in N , there is aminimal orbit of that type. It is known that if N has only one orbit type then thereis a unique Riemannian metric ds2 on the orbit space N/G making the projectionπ : N → N/G a Riemannian submersion. Hsiang and Lawson also proved thatπ−1(B) is minimal in N if and only if B is minimal in N/G with respect to themetric v(x)2/kds2(x), where k is the dimension of N/G and v(x) is the volume ofthe fiber π−1(x). If all G-orbits are congruent under Iso(N), then all G-orbits areminimal, π−1(B) is minimal in N if and only if B is minimal in N/G, and by aformula in [PT1] we also have

H = H∗. (1.1),

where H∗ be the horizontal lifting of the mean curvature vector H of B to π−1(B)and H is the mean curvature vector of π−1(B). The symmetric criticality principle(SCP) is a principle often used in the calculus of variations to find symmetricsolutions. To state the principle, we let M be a Hilbert manifold, G a HilbertLie group acting on M, and f : M → R be a smooth G-invariant function. LetS = x ∈ M | g · x = x ∀ g ∈ G be the fixed point set of G. Then SCP saysthat a critical point s ∈ S for f | S is a critical point for f . In [P], Palais gaveexamples for which SCP fails (even when both M and G are of finite dimension), andproved that SCP is valid if the action of G on M is isometric. The Hsiang-Lawsontheorem is an application of SCP. Although the space I(M, V ) of PF immersionsfrom M to V does not have a Hilbert manifold structure and the area functionalA : I(M, V ) → R is not well-defined, we prove that SCP is valid when we usethe action of the base loop group Ωe(G) on V (or more generally the action of∏n

i=1 Ωe([ti−1, ti], G) on V ). Moreover, we also prove the formula (1.1) holds forregularized mean curvature vectors.

In section 6 we approach the question of assigning a meaning to the volume (orvariation of volume) of PF submanifolds. As a guide to what to expect, it is knownthat the orbits of Ωe(G) on V are congruent under isometries of V , suggesting thatthese orbits should be assigned equal volume. In particular let M1 and M2 besubmanifolds of G and define as before Mi = ϕ−1(Mi) for i = 1, 2. Then for thisreason we expect that any definition of volume will imply the following result:

vol(M1)vol(M2)

=vol(M1)vol(M2)

. (1.2)


There is no translation invariant measure on Hilbert space to play the role ofthe infinite dimensional Lebesgue measure. The closest thing is Wiener measure(or Brownian motion) µT on the space of continuous paths C0([0, T ], g) startingat 0. We would like to use this to provide an analogue of Lebesque measure onH0([0, 1], g). To do this, we identify H1

0 ([0, T ], g) with H0([0, 1], g) by the mapψ(w)(s) = sw′(sT ) for s ∈ [0, 1], where w ∈ H1

0 ([0, T ], g), and note that H10 ([0, T ], g)

is a subspace of C0([0, T ], g). This would seem to provide a one parameter familyof measures on H0([0, 1], g); however the space H1

0 ([0, T ], g) has measure zero withrespect to µT . In section 6, we expound and elaborate on these ideas and use themto invent a plausible definition of the volume of a subset ϕ−1(M), where M is asubmanifold of G and ϕ is the parallel transport map as above.

This paper is organized as follows: we review local geometry of submanifolds insection 2, and study the geometry of the parallel transport map ϕ : H0([0, 1], g) →G in section 3. In section 4, we define regularizable submanifolds and regular-ized mean curvature vectors and prove that if M is a minimal submanifold of Gthen M = ϕ−1(M) is regularizable and its regularized mean curvature is zeroin H0([0, 1], g). In section 5, we give explicit formulas for the regularized meancurvature and volume ratio when G = SU(2) or when M is an isoparametric hy-persurface. Finally, in section 6 we give a definition of the volume of M = ϕ−1(M)and show that the equality (1.2) is true under this definition.

2. Local geometry of submanifolds in Hilbert space

Let M be a codimension p submanifold of a Hilbert space (V, 〈 , 〉), X : M → Vthe inclusion map, ∇ the Levi-Civita connection for the inner product on V , andlet d denote the usual differentiation. A smooth vector field on V can be identifiedas a smooth map v : V → V , and ∇uv = dv(u). We will review below the localtheory of submanifolds in Hilbert space ([T1]). The first fundamental form I of Mis the induced metric on M , i.e., the inner product Ix on TMx is the restrictionof 〈 , 〉 to TMx. Let ν(M) denote the normal bundle of M in V , and πT and πν

the orthogonal projection from TN onto TM and ν(M) respectively. For a smoothnormal field v,

Av(xo)(u) = −πT (∇uv)(xo)

defines a self-adjoint linear operator on TMxo, which only depends on the value of v

at xo. The operator Av is called the shape operator of M with respect to the normalvector v, and the eigenvalues of Av are called the principal curvatures of M alongv. The induced normal connection ∇ν on ν(M) is defined by ∇ν(v) = πν(∇v).

The above local invariants of submanifolds can be expressed in terms of movingframes. A local orthonormal frame field e1, . . . , en, . . . defined on an open subsetU of V is called adapted if for all i > p, ei restricted to M is tangent to M .Henceforth we will use the following index ranges:

1 ≤ A, B, C, 1 ≤ α, β, γ ≤ p, i, j, k > p.

Let ωA be the coframe defined on U , and ωAB the Levi-Civita connection 1-formfor the inner product 〈 , 〉, i..e., ωAB is defined by

dωA =∑B

ωAB ∧ ωB , ωAB + ωBA = 0.


Restricting to M , we have ωα = 0, and

dX =∑

i

ωi ⊗ ei,

I = 〈dX, dX〉 =∑

i

ωi ⊗ ωi,

Aeα=

∑i,α

ωiα ⊗ ei =∑i,j,α

hαijωj ⊗ ei,

∇νeα =∑

β

ωαβ ⊗ eβ ,

where ωiα =∑

j hαijωj .

In the following we will derive the normal variation of the induced metric onthe space I(M, V ) of immersions. We may assume the deformation vector field ξis equal to feβ for some smooth function f and some β, and consider the normaldeformation of M along ξ:

Xt = X + tfeβ .

We obtain

dXt =∑

i

ωiei + tfdeβ + tdfeβ

=∑

i

ωiei − tf∑ij

hβijωjei + tf

∑α

ωβαeα + tdfeβ ,

and the induced metric It of the immersion Xt is

It = 〈dXt, dXt〉

=∑ij

(δij − 2tfhβ

ij + t2f2hβkih

βkj + t2(fifj + f2

∑α

ΓiβαΓj

βα))wiwj ,

wheredf = Σfiwi, ωβα =

∑i

Γiβαωi.

Let C and Gt be the linear operators defined by

〈C(ei), ej〉 = f2∑

k

hβikhβ

jk + fifj + f2∑α

ΓiβαΓj

βα,

Gt = Id −2tfAeβ+ t2C.

ThenIt(u1, u2) = I0(Gt(u1), u2〉.

So informally the induced volume form for the immersion Xt is

dvt =√

det(Gt)ωp+1 ∧ ωp+2 ∧ · · · =√

det(Gt) dv0,

and the variation of volume gives

d

dt

t=0

vol(Xt(M)) = −∫

M

f∑

i

hβiidv = −

∫M

〈ξ, H〉dv.


So the first variation of the volume functional is zero when H = 0. This calculationis rigorous if dim(V ) < ∞, and is informal otherwise.

Next we want to express the volume form of the induced metric of a submani-fold Mn in a Riemannian manifold (Nn+k, g) as the measure “induced” from theRiemannian measure of N . Let dv and dv0 denote the Riemannian volume formson N and M respectively. Let

η : ν(M) → N, v ∈ ν(M)p → expp(v)

be the end point map of M in N . Given a coordinate system (U, x) of M andan orthonormal normal frame field eα | n + 1 ≤ α ≤ n + k defined on U , thenthere exists ε > 0 such that η maps νε(U) diffeomorphically onto a neighborhoodNε(U) of N , and (x, y) → η(

∑α yαeα(x)) is a local coordinate system on Nε(U).

Let πα : Nε(U) → Rk be the yα coordinate. Since eα is orthonormal and d exp0 =id, the volume form dv on Nε(U) has the following expression

dv(x, y) = (1 + O(y))dv0 ∧ dyn+1 ∧ · · · ∧ dyn+k.

Now let δm be a sequence of integrable functions on I = [−ε, ε] converging weaklyto the delta function δ at 0, i.e., for any continuous function f : I → R, we have

limm→∞

∫I

δm(t)f(t) dt = f(0).

So if f : Nε(U) → R is a continuous bounded function, then

limm→∞

∫Nε(U)

n+k∏α=n+1

δm(πα(y))f(x, y) dv(x, y) =∫

U

f(x) dv0(x).

3. Parallel transport map

Let G be a connected, compact Lie group, g its Lie algebra, ( , )o a fixed Ad-invariant inner product on g, and ds2

o the corresponding bi-invariant metric on G.Let V = H0([a, b], g) denote the space of L2-paths with inner product defined by

〈u, v〉 =∫ b

a

(u(t), v(t))o dt.

Let H1e ([a, b], G) denote the H1 Sobelov space of paths g : [a, b] → G with g(a) = e,

which is a Hilbert manifold. Henceforth we will adopt the following notations forg ∈ G and v ∈ g:

vg = (Rg)∗(v), gvg−1 = Ad(g)∗(v),

where Rg : G → G is the right multiplication by g. Note that

T (H1e ([a, b], G))g = vg | v ∈ H1([0, 1], g), v(0) = 0, and

〈v1g, v2g〉g =∫ 1

0

(v′1(t), v′2(t))o dt

defines a right-invariant Riemannian metric on H1e ([0, 1], G). Recall that an action

of a Lie group H on a manifold M is called proper if the map

H ×M → M ×M, (h, x) → (hx, x)

is proper, i.e., if hnxn → y and xn → x then hn has a convergent subsequence inH. According to the Sobolev embedding theorem, H1([0, 1], G) ⊂ C0([0, 1], G) andthe inclusion map is compact. Using this we obtain:


3.1 Proposition ([T1]). The action of the group H1([a, b], G) on H0([a, b], g)by the gauge transformation

g ∗ u = gug−1 − g′g−1

is proper.

3.2 Proposition. The orbit map ψ : H1e ([0, 1], G) → H0([0, 1], g) defined by

ψ(g) = g−1 ∗ 0 = g−1g′ is an isometric diffeomorphism.

Proof. Since the action of H1e ([0, 1], G) on H0([0, 1], g) is proper, the image of ψ

is closed. It follows from the existence and uniqueness theorem for linear ordinarydifferential equations, that the space C0([0, 1], g) of continuous paths is containedin the image of ψ. Since Image(ψ) is closed and C0 is dense in H0, ψ is onto. Toprove ψ is one to one, we assume that ψ(g1) = ψ(g2) = u, i.e., g−1

1 ∗0 = g−12 ∗0 = u.

Let h = g2g−11 . Then h ∗ 0 = −h′h−1 = 0, But h(0) = e, so h(t) = e for all t,

i.e., g1 = g2. A direct computation gives dψg(vg) = g−1v′g, which implies thatdψg is an isometry. It follows from the inverse function theorem that ψ is a localdiffeomorphism. Since ψ is a bijection, ψ is a diffeomorphism. 3.3 Corollary. The action of H1

e ([0, 1], G) on H0([0, 1], g) is isometric, free andtransitive.

Let E : H0([0, 1], g) → H1e ([0, 1], G) be the inverse of the isometric diffeomor-

phism ψ above. In other words, E is the parallel translation in the trivial bundle[0, 1]×G over [0, 1] defined by the connection 1-form u(t)dt: E = E(u) is the uniquesolution of

E−1E′ = u, E(0) = e.

Let u ∈ H0([0, 1], g), [a, b] ⊂ [0, 1], and ϕba : H0([a, b], g) → G denote the parallel

transport map from a to b, i.e., ϕba(u) = g(b), where g the unique solution to the

initial value problemg−1g′ = u | [a, b], g(a) = e.

The uniqueness of solutions of ordinary differential equation implies that

ϕba(u | [a, b]) = E(u)(a)−1E(u)(b).

Next we collect some known geometric properties for the parallel transport map.

3.4 Theorem ([T2]). Let ϕ = ϕ10 : H0([0, 1], g) → G be the parallel transport

map from t = 0 to t = 1, i.e., ϕ(u) = E(u)(1). Then(1) ϕ(g ∗ u) = g(0)ϕ(u)g(1)−1 for any g ∈ H1([0, 1], G),(2) if ϕ(v) = h1ϕ(u)h−1

2 for some (h1, h2) ∈ G×G then there exists g ∈ H1([0, 1], G)with g(0) = h1, g(1) = h2 such that g ∗ u = v.

3.5 Corollary. Let Ωe([0, 1], G) = g ∈ H1([0, 1], G) | g(0) = g(1) = e denotethe based loop group. Then the action of Ωe([0, 1], G) on H0([0, 1], g) by the gaugetransformation is free, the orbit space is G, and the map ϕ in 3.4 is the principalΩe([0, 1], G)-bundle associated to the action. Moreover, any two Ωe(G)-orbits arecongruent with respect to isometries of H0([0, 1], g).

3.6 Theorem ([TT]). Let ϕ = ϕ10 : H0([0, 1], g) → G be as in 3.4. Then ϕ is a

Riemannian submersion.


3.7 Corollary. Let G[a,b] be the Lie group G equipped with the bi-invariant

metric 1b−a ds2

0, and ϕba : H0([a, b], g) → G[a,b] the parallel transport map from

a to b. Let Ωe([a, b], G) denote the group of H1-paths g : [a, b] → G such thatg(a) = g(b) = e. Then

(1) ϕba is a principal Ωe([a, b], G)-bundle, and the fibers of ϕb

a are the Ωe([a, b], G)-orbits under the gauge transformations,

(2) ϕba is a Riemannian submersion.

In the following we consider an analogue of the cylinder sets occuring in theWiener measure for submanifolds in the path space. Given any partition s: 0 = s0 <s1 < · · · < sn = 1 of [0, 1], it is obvious that H0([0, 1], g) is naturally isomorphic to⊕n

1H0([si−1, si], g) via the isomorphism

u → (u | [0, s1], . . . , u | [sn−1, 1]),

so the following is a direct consequence of 3.7.

3.8 Corollary. Let s : 0 = s0 < s1 < · · · < sn = 1 be a partition of [0, 1], andΦs : H0([0, 1], g) →

∏n1 Gi the map defined by

Φs(u) = (ϕs10 (u | [o, s1]), ϕs2

s1(u | [s1, s2]), . . . , ϕ1

sn−1(u | [sn−1, 1])),

where Gi is the Lie group G equipped with the bi-invariant metric 1si−si−1

ds20.

Then(1) Φs is a Ω-principal bundle, where Ω =

∏n1 Ωe([si−1, si], G),

(2) Φs is a Riemannian submersion,(3) all fibers of Φs are congruent; in particular they are isometric,(4) if M is a submanifold of

∏n1 Gi, then Φ−1

s (M) is PF and isometric to the fol-

lowing submanifold of H1e ([0, 1], G)

g ∈ H1e ([0, 1], G) | (g(s1), g(s1)−1g(s2), . . . , g(sn−1)−1g(1)) ∈ M.

3.9 Remark. Since the fibers of the Riemannian submersion Φs as in 3.8 arecongruent, they should have the same induced volume, if the latter are defined.Let Mt be a family of deformation of submanifolds in

∏n1 Gi, and dvt, dvt the

induced volume form for Mt and Mt = Φ−1s (Mt) respectively. Suppose dvt(x) =

f(x, t)dv0(x). Then informally we have

dvt(y) = f(Φ(y), t)dv0(y),vol(Mt)vol(M0)

=vol(Mt)vol(M0)

.

4. Regularized mean curvature vector

Since the shape operators of a PF submanifold in a Hilbert space V are in generalnot of trace class, in order to make sense of the mean curvature vector we considerthe class of operators with regularized trace (defined below).

Given a compact, self-adjoint operator T : V → V , we arrange its non-zeroeigenvalues counted with multiplicities as follows:

−µ1(T ) ≤ −µ2(T ) ≤ · · · < 0 < · · · ≤ λ2(T ) ≤ λ1(T ).


Let Ls(V ) denote the space of all compact, self-adjoint operators T on V such that

| T |s=(∑

n

λsn +

∑n

µsn

)1/s

< ∞.

4.1. Definition. A compact, self adjoint operator T on V is called regularizableif T ∈ Ls(V ) for all s > 1 and the regularized trace

trζ(T ) = lims1

(∑n

λsn −

∑n

µsn

)

exists.The following theorem is needed later for proving our results on minimal sub-

manifolds.

4.2 Theorem. Suppose B is a regularizable operator on V , C is a self-adjointoperator of finite rank on V , and A = B + C. Then

(1) A is regularizable,(2) trζ(A) = trζ(B) + tr(C).

By induction, we may assume that the operator C has rank 1. With this assump-tion, we divide the proof of 4.2 into several lemmas. The first two are well-known(cf. p. 908, p.1089, p.1091 and p. 1097 of [DS]):

4.3 Lemma. Let T, T1 and T2 be compact, self-adjoint operators on V . Then

λ1(T ) = max|x|=1

(Tx, x),

λn+1(T ) = miny1,... ,yn

max|x|=1,

(x,y1)=...=(x,yn)=0

(Tx, x)

λn+m+1(T1 + T2) ≤ λn+1(T1) + λm+1(T2).

Applying 4.3 to −T and −Ti, we get a similar result for the µn’s.

4.4 Lemma. If Tn and T are compact, self-adjoint operators on V , and Tn → Tin the norm topology, then λj(Tn) → λj(T ), and µj(Tn) → µj(T ).

4.5 Lemma. λn+1(B) ≤ λn(A) ≤ λn−1(B), and µn+1(B) ≤ µn(A) ≤ µn−1(B).Thus A ∈ Ls(V ) for s > 1.

Proof. Since C has rank one, λj(C) = λj(−C) = 0 for all j ≥ 2. Then by 4.3,

λn+1(B) = λn+1(A + (−C)) ≤ λn(A) + λ2(−C) = λn(A)

= λn(B + C) ≤ λn−1(B) + λ2(C) = λn−1(B),

and a similar argument will give the inequality for the µ’s. That A ∈ Ls(V ) fors > 1 now follows from the comparison test. 4.6 Lemma. Let

∑n Mn be a convergent series with non-negative terms.

(1) Suppose xn(s) is a sequence of functions defined on the interval I = [1, 2] suchthat | xn(s) |≤ Mn and for all n and all s ∈ I, and for each n let lims1 xn(s) = yn.Then lims1

∑n xn(s) =

∑n yn.

(2) Suppose for each positive integer N , aNk is a sequence of complex numbers

with | aNk |≤ Mk for all N and k, and for each k let limN→∞ aN

k = ak. ThenlimN→∞

∑k aN

k =∑

k ak.


Proof. By the classic “Weierstrass M -Test”, the series∑

n xn(s) converges uni-formly in s, and similarly the series

∑k aN

k converges uniformly in N . The inter-change of limits and summation, lims1

∑n =

∑n lims1, and limN→∞

∑k =∑

k limN→∞ is then in each case a trivial consequence of the uniformity.

In applying 4.6 below we will take Mn = λn−1(B)− λn(B) ≥ 0. Note that sinceλn(B) → 0, the telescoping series

∑∞n=2 Mn is clearly convergent to λ1.

4.7 Lemma. A is regularizable and

trζ(A)− trζ(B) =∑

n

(λn(A)− λn(B))−∑

n

(µn(A)− µn(B)).

Proof. Let xn(s) = λsn(A)− λs

n(B) for s ≥ 1. Note that f(x) = xs is an increasingfunction on [0,∞). Applying 4.5, we have

λn+1(B)s − λn(B)s ≤ xn(s) ≤ λsn−1(B)− λs

n(B).

This implies that| xn(s) |≤ bn = λx

n−1(B)− λsn+1(B).

But∑

n bn is obviously convergent, and lims1 xn(s) = λn(A)− λn(B). By 4.6,

lims1

∑n

(λsn(A)− λs

n(B)) =∑

n

(λn(A)− λn(B)).

Mutatis mutandis, we get a similar identity with µ replacing λ. Since both A andB are in Ls(V ) if s > 1, all the four series in the following expression convergeabsolutely:

X = lims1

((∑

n

λsn(A)−

∑n

µsn(A))− (

∑n

λsn(B)−

∑n

µsn(B))

),

hence in computing the limit we may combine terms in a different order and obtain

X = lims1

(∑n

(λsn(A)− λs

n(B))−∑

n

(µsn(A)− µs

n(B)))

=∑

n

(λn(A)− λn(B))−∑

n

(µn(A)− µn(B)).

Since, by assumption, the limit trζ(B) = lims1

∑n(λs

n(B) − µsn(B)) exists, it

follows by adding these two limits that the limit trζ(A) = lims1

∑n(λs

n(A) −µs

n(A)) also exists and equals X + trζ(B).

4.8. Proof of 4.2. Since C is self-adjoint and of rank one, there exist a unit vectoru ∈ V and a ∈ R such that C(x) = a(x, u)u. Let vn be an orthonormal eigenbasisof B and let u =

∑n cnvn. Let VN be the linear span of v1, . . . , v

N, so u

N=∑N

n=1 cnvn is the orthogonal projection of u on VN , and let CN

(x) = a(x, uN )uN

,and A

N= B + C

N. Clearly C

Nis 0 on V ⊥N , so A

N= B on V ⊥N , and it follows that


trζ(AN) = trζ(B) + tr(C

N). Clearly, tr(C

N) = a‖u

N‖2 → a‖u‖2 = tr(C). Now

from 4.7,

tr(CN ) = trζ(AN )− trζ(B) =∑

n

(λn(AN )− λn(B))−∑

n

(µn(AN )− µn(B)),

so

tr(C) = limN→∞

∑n

(λn(A)− λn(B))−∑

n

(µn(A)− µn(B)).

Now, by 4.4, limN→∞ λn(AN ) = λn(A), and limN→∞ µn(AN ) = µn(A). Also, using4.5, λn(AN ) ≤ λn−1(B), which implies that λn(AN )−λn(B) ≤ λn−1(B)−λn(B) =Mn, (with a corresponding esimate for the µ’s), so by (2) of 4.6

tr(C) =∑

n

(λn(A)− λn(B))−∑

n

(µn(A)− µn(B)),

and 4.2 now follows from 4.7.

4.9. Definition. A PF submanifold M of a Hilbert space V is called regularizableif for every unit normal vector v the shape operator Av is regularizable. For a reg-ularizable submanifold M , the normal vector Hx ∈ ν(M)x defined by the condition〈H, v〉 = trζ(Av) for all v ∈ ν(M)x is called the regularized mean curvature vectorof M in V at x.

It is obvious that if e1, . . . , ep is an orthonormal basis for ν(M)x, then theregularized mean curvature vector of a regularizable submanifold M at x is

H =∑α

trζ(Aeα)eα.

4.10. Definition. A PF submanifold M of V is called minimal if M is regular-izable and its regularized mean curvature vector field H is zero.

There are many known examples of PF submanifolds. In fact, it is proved in [TT]that if M is a closed submanifold of G, then M = ϕ−1(M) is a PF submanifold inH0([0, 1], g). Or more generally, if M is a closed submanifold of

∏i Gi as in 3.8, then

Φ−1s (M) is PF. Since all fibers of the Riemannian submersion ϕb

a are congruent, theyshould be minimal according to the intuition from the finite dimensional theory.The following theorem says they are indeed minimal.

4.11 Theorem. Let ϕba be the parallel transport map from a to b as in 3.4. Then

all the fibers of ϕba are regularizable and minimal in H0([a, b], g).

Proof. We will only give a proof of the Theorem for ϕ = ϕ10, because the proof

for ϕba is the same. Since all fibers of ϕ are congruent, it suffices to prove that

F = ϕ−1(e) is minimal. For x ∈ g, let x denote the constant path with value x. By3.6, Ωe = Ωe([0, 1], G) acts on H0([0, 1], g) freely, F = Ωe ∗ 0, ν(F )0 = x | x ∈ g,and

TF0 = u′ | u ∈ H1([0, 1], g), u(1) = u(0) = 0

= ξ ∈ H0([0, 1], g) |∫ 1

0

ξ(t)dt = 0.


Let a ∈ g, and a : F → V be the map defined by a(g ∗ 0) = gag−1. Then a is anormal vector field on F . So the shape operator Aa of F is given by

Aa(u′) = [u, a]T ,

the orthogonal projection of [u, a] to TF0. Let t be a maximal abelian subalgebraof g containing a. Then there is an orthonormal basis

a1, . . . , ar ∪ xα, yα | α ∈ +

of g such that ai’s form a basis for t and for all a ∈ t

[a, xα] = α(a)yα, [a, yα] = −α(a)xα, [a, ai] = 0,

where + is a set of positive roots. Let zα = xα + iyα, and zα,n = zαe2πint.Then [a, zα] = −iα(a)zα. Set u = zα,n − zα. Then we have u(1) = u(0) = 0,u′ = 2πnzα,n, and

Aa(u′) = Aa(2πinzα,n) = 2πinAa(zα,n)

= [zα,n − zα, a]T = iα(a)zα,n.

This implies that

Aa(zα,n) =α(a)2πn

zα,n.

Leteα,n = (zαe2πint), fα,n = (zαe2πint),

rα,n = (aje2πint), sα,n = (aje

2πint).

Theneα,n, fα,n, rα,n, sα,n | α ∈ +, n = 0

forms an eigenbasis for Aa with eigenvalues

α(a)2nπ

,α(a)2nπ

, 0, 0

respectively. This proves that Aa is regularizable. To compute trζ(Aa), we notethat since α(a)/2πn converges absolutely for s > 1, we can pair positive andnegative terms:

trζ(Aa) = lims1

∑α∈+

∞∑n=1

2( | α(a) |

2πn

)s( 1ns− 1

ns

)= 0.

Hence trζ(Aa) = 0 and H(0) = 0. Since F is an orbit of the isometric action of Ωe

on V , H = 0 everywhere, and F is minimal. 4.12 Theorem. Let M be a closed submanifold of G, and M the submanifoldϕ−1(M) of V = H0([0, 1], g). Then

(a) M = ϕ−1(M) is regularizable,

(b) the regularized mean curvature vector H of M in V is the horizontal lifting ofthe mean curvature vector H of M in G,

(c) if M is minimal in G, then M is minimal in V .


Proof. By right translation, we may assume that e ∈ M . Let a ∈ ν(M)e, and a theconstant path with value a as before. Then a ∈ ν(M)0 is the horizontal lifting ofa at 0. Let F = ϕ−1(e). Identifying TMe with a linear subspace of ν(F )a via thehorizontal lifting, we have

TM0 = TF0 ⊕ TMe.

Let A and A denote the shape operator of M and F in the direction of a respectively.A direct computation implies that A = B +C, where B and C are linear operatorson TM0 defined as follows: for v = v1 + v2 with v1 ∈ TF0 and v2 ∈ TMe

B(v1) = A(v1), B(v2) = 0

C(v1) = π2(A(v1)), C(v2) = A(v2),

where π2 : TM0 → TMe is the orthogonal projection. By 4.11, B is regularizableand trζ(B) = 0. Since

C(TM0) ⊂ TMe + A(TMe),

C has finite rank. So 4.2 implies that A is regularizable and

trζ(A) = trζ(B) + tr(C) = tr(C).

This completes the proof of our theorem. 4.13. Remark. Let M be a hypersurface in G. Although we do not have anexplicit formula for the principal curvatures of the hypersurface M = ϕ−1(M) inV = H0([0, 1], g), by 4.12 the regularized mean curvature of M in V at y is equalto the mean curvature of M in G at ϕ(y). In particular, a constant mean curvaturehypersurface in G gives rises to a constant mean curvature hypersurface in V .

4.14. Examples. If M is a minimal surface in S3 = SU(2), then 4.11 impliesthat ϕ−1(M) is a minimal hypersurface in H0([0, 1], su(2)). Moreover, there aremany minimal surfaces in S3. In fact, given any p, Lawson constructed in [L1] aclosed minimal surface of genus p in S3; and Hsiang and Lawson constructed in[HL] infinitely many minimal tori in S3.

4.15. Examples. Let H be a closed subgroup of G ×G. Then the action of Hon G defined by (h1, h2) · g = h1gh−1

2 is isometric. Hsiang’s theorem implies thatfor each orbit type occurs of H occurs in G, there is a minimal orbit of that type.Now suppose M = H · ea is a minimal orbit in G. Then ϕ−1(M) is a homogeneousminimal submanifold in V = H0([0, 1], g). In fact, let

P (G, H) = g ∈ H1([0, 1], G) | (g(0), g(1)) ∈ H

act on V by gauge transformations, then (c.f. [T2])

ϕ−1(M) = P (G, H) ∗ a, P (G, H)a = Hea .

As a consequence of Hsiang’s theorem and 4.12 we conclude that for each orbit typeof P (G, H) that occurs in V , there is a minimal orbit of that type. Note that theaction of the gauge group L(G) = g ∈ H1([0, 1], G) | g(0) = g(1) on the space Vof connections is the P (G, H)-action with H = (g, g) | g ∈ G.


4.16. Remark. Maeda, Rosenberg, and Tondeur use a different approach todefine the mean curvature vector for gauge orbits of the space of L2-connections ofa principle G-bundle over a Riemannian n-manifold in [MRT].

An argument similar to the proof of 4.12 gives the following more general result:

4.17 Theorem. Let Φs : V = H0([0, 1], g) →∏

i Gi be as in 3.8, M a closed

submanifold of∏

i Gi, and M = Φ−1s (M). Then

(a) M is regularizable,

(b) the regularized mean curvature vector H of M in V is the horizontal lifting ofthe mean curvature vector H of M in

∏i Gi,

(c) if M is minimal in∏

i Gi, then M is minimal in V .

5. Examples

Let G = SU(2), (x, y) = −1/2 tr(xy) the inner product on g = su(2), and ds2

the bi-invariant metric on G defined by ( , ). Then G is the unit 3-sphere with thestandard metric. Let V = H0([0, 1], g), and ϕ = ϕ1

0 : V → G the parallel transportmap as before. Let a = diag(iθ,−iθ) with 0 ≤ θ ≤ π, a the constant path inV with constant value a, Mθ the conjugacy class of G containing ϕ(a) = ea =diag(eiθ, e−iθ), and Mθ = ϕ−1(Mθ). Note that Mθ = e if θ = 0, Mθ = −eif θ = π, and Mθ is a two-sphere of radius sin θ otherwise. In the following, weassume 0 < θ < π. We will compute explicitly the regularized mean curvature ofthe hypersurface Mθ in V , and the informal expression for the ratio of “the inducedvolumes” vol(Mθ) : vol(M1).

As a consequence of 3.4, Mθ is the orbit of the isometric action of the loop groupL(G) = g ∈ H1([0, 1], G) | g(0) = g(1) on V through a. So the regularized meancurvature of Mθ is constant. Next we compute the regularized mean curvature ata. Note that b = diag(i,−i) is a unit normal vector of Mθ at ea, and

b∗ : Mθ → V, defined by b∗(g ∗ a) = gbg−1

is a well-defined unit normal vector field on Mθ. Let A denote the shape operatorof Mθ in the direction of b∗. Using the definition of shape operators and a directcomputation, we see that A([u, a]− u′) = [u, b], and

un = x cos 2nπt− y sin 2nπt, vn = x sin 2nπt + y cos 2nπt, n ∈ Z

rn = h cos 2nπt, sn = h sin 2nπt, n ∈ Z, n = 0

is an orthonormal eigenbasis for A with eigenvalues (nπ − θ)−1, (nπ − θ)−1, 0, 0respectively, where

h = i(e11 − e22), x = e12 − e21, y = i(e12 + e21),

and eij is the matrix with all entries zero except the ij-th entry equal to 1. So theprincipal curvatures of the hypersurface Mθ are λn = (nπ − θ)−1 (n ∈ Z) and 0with multiplicity 2 and ∞ respectively. In particular, using a standard formula forinfinite series (cf. [GR]) the regularized mean curvature Hθ of Mθ is

Hθ =∑n∈Z

2nπ − θ

= 2/π∑n∈Z

1n− θ/π

= −2 cot θ.


So Mθ is minimal if and only if θ = π/2, or equivalently Mθ is minimal.

Note thatf : M1 → Mθ, f(g ∗ b) = g ∗ a

is a well-defined L(G)-equivariant diffeomorphism, and T (M1)b = T (Mθ)a, whichis equal to the hyperplane V0 = u ∈ V | 〈b, u〉 = 0. Next we want to relate theinduced metrics on M1 and Mθ. Let P be the positive, self-adjoint operator on V0

such that 〈dfb(u1), dfb(u2)〉 = 〈Pu1, u2〉. Note that

dfb([u, b]− u′) = [u, a]− u′

and P = df∗bdfb. It is easy to see that the eigenbasis of the shape operator of Mθ at

a is the eigenbasis of P and the eigenvalues of P are ( θ−nπ1−nπ )2 (n ∈ Z) and 1 with

multiplicity 2 and ∞ respectively. Now let Iθ denote the induced metric on Mθ,and Pu the positive operator on T (M1)u such that for v1, v2 ∈ T (M1)u

Iθ(dfu(v1), dfu(v2)) = I1(Pu(v1), v2).

Since Mθ is an orbit, if u = g ∗ b then Pu = gPg−1. In particular, this implies that

dvθ =√

det(P )dv1 =∞∏

n=−∞

(θ − nπ

1− nπ

)2 =∞∏

n=−∞

(1− θ − 1

nπ − 1)2

.

Using the standard ζ-function regularization technique, we can compute the infiniteproduct by grouping the terms with n and −n together then taking the productover all n > 0, i.e.,

dvθ

dv1= θ

∞∏n=1

(1−

θ−1π

n− 1π

)2(1 +θ−1

π

n + 1π

)2.

Using the following formula (c.f. [GR])

sinπ(x + a)sinπa

=x + a

a

∏k>0

(1− x

k − a)(1 +

x

k + a).

we getdvθ

dv1=

sin(θ)2

sin(1)2,

which is equal to the ratio vol(Mθ)/ vol(M1).

5.1. Remarks. The above hypersurface Mθ is isoparametric in H0([0, 1], su(2)),and in fact, the mean curvature and volume ratio can be computed explicitly for anyisoparametric hypersurface of a Hilbert space. We refer to [TT] for the constructionof infinitely many topologically distinct families of isoparametric hypersurfaces inHilbert space. Now let M be an isoparametric hypersurface of a Hilbert space V ,and v a unit normal vector field on M . Then it is known [T1] that there exist B > 0and 0 < a < B such that the principal curvatures of M are

1nB− a

, n ∈ Z


with multiplicity m0 for even n, and m1 for odd n. Set

Mt = xt = x + (t− a)v(x) | x ∈ M.

Then we have(1) if 0 < t < B, then Mt is isoparametric and vt(xt) = v(x) is normal to Mt,(2) the principal curvatures of Mt are (nB− t)−1 (n ∈ Z) with multiplicity m0 for n

even and m1 for n odd,(3) the regularized mean curvature Ht of Mt is

Ht = − π

2B

(m0 cot

πt

2B−m1 tan

πt

2B

);

in particular, Mt is minimal if t = 2(π tan−1

√m0m1

,

(4) let dvt denote the informal volume form for Mt, then

dvt

dv 2

= 2m0+m1 sin(πt

2B)m0 cos(

πt

2B)m1 ,

and this implies thatd

dtlog(

dvt

dv 2

) = −Ht.

(5) M0 and M( are minimal submanifolds of V with codimension m0 +1 and m1 +1respectively.

5.2. Remark. Let P (G, H) be as in 4.15. Suppose the H-action on G ishyperpolar , i.e., there is a closed, flat submanifold S of G that meets every H-orbit orthogonally. We refer to [HPTT] for a general theory and many examples ofhyperpolar actions on G. Then principle P (G, H)-orbits are isoparametric. Nowlet M be an isoparametric submanifold of a Hilbert space V , eβ a parallel normalfield on M , and Xt = X + teβ . With the same notation as in section 1, we obtain

It(u1, u2) = I0(Gt(u1), u2),

whereGt = I − 2tAeβ

+ t2A2eβ

= (I − tAeβ)2.

Moreover, (Gt)x and (Gt)y are conjugate for x, y ∈ M . Using the same zeta functionregularization, we obtain

f(t) =dvt

dv0=

√det(Gt) = det(I − tAeβ

),

d

dtlog(f(t)) = −〈Ht, eβ〉.


6. Measures on Path Spaces

In section 3 we defined the parallel transport map ϕ : H0([0, 1], g) → G, whichturned out to be a Riemannian submersion. This led to consideration of subman-ifolds of Hilbert space of the form ϕ−1(M), where M is a submanifold of G. Inparticular, Theorem 4.12 showed that ϕ−1(M) is minimal if M is minimal in G. Itis natural to expect that a minimal submanifold has minimal volume. This leadsto the problem of computing volumes of submanifolds in H0([0, 1], g), and hence tothe problem of defining a measure on this Hilbert space.

The problem of defining measures on path spaces has a long history ([S]). Theprototype is Wiener measure on the space C([0, T ], Rn), consisting of continuousvector valued functions on [0, T ] satisfying f(0) = 0. This well known Gaussianmeasure is also called Brownian motion on Rn. Since the vector space g has aninner product, it can be identified with Rr, where r = dim (g). So Wiener measureis also defined on the following space of continuous paths:

W0T (g) := C([0, T ], g) = w : [0, T ] → g |w(0) = 0, w(t) continuous

Let H01([0, T ], g) denote the Sobolev H1-space of paths w in g with w(0) = 0.

Clearly this is a subspace of W0T (g). Furthermore, there is a one to one mapping

ψ of H01([0, T ], g) into H0([0, 1], g) , given by

ψ(w)(s) =d

dsw(sT ), 0 ≤ s ≤ 1

from which it follows that∫ T

0

||w′||2dt =1T

∫ 1

0

||ψ(w)||2ds

At first sight this seems to answer the problem of defining a measure on theHilbert space H0([0, 1], g), since it is isomorphic to a subspace of the measure spaceW0

T (g). Unfortunately H01([0, T ], g) has measure zero with respect to Wiener

measure, so this does not work.We now present an alternative approach to this problem. We will define a map

Φ : W0T (g) → G which can be thought of as an extension of ϕ ψ. Using Wiener

measure we can then compute the “volume” of the submanifold Φ−1(M). Despitethe fact that Φ−1(M) consists of infinite energy paths which lie outside the Hilbertspace H0

1([0, T ], g), our results will be in pleasant agreement with Theorem 4.12.

Let Cx0([0, T ], M) denote the space of continuous maps f : [0, T ] → M , satisfyingf(0) = x0, where M is a Riemannian manifold. Let be the Laplace-Beltramioperator on M , and let pt : M ×M → R be the heat kernel of . This satisfies theconvolution equation

(etF )(x) =∫

M

pt(x, y)F (y) dy

where dy is Riemannian measure on M .A cylinder set is a set of the form f ∈ C([0, T ], M) | f(t1) ∈ B1, . . . , f(tm) ∈

Bm where B1, . . . , Bm are arbitrary Borel subsets of M , and 0 ≤ t1 < · · · < tm ≤ 1


is a partition of [0, 1]. Let F denote the σ-algebra generated by all cylinder sets. Inorder to define a measure on (Cx0([0, T ], M),F), we first define a positive, additivefunction PT on the algebra of cylinder sets:

PT f ∈ Cx0([0, T ], M)|f(t1) ∈ B1, . . . , f(tm) ∈ Bm =∫B1

· · ·∫

Bm

pt1(x0, y1)pt2−t1(y1, y2) . . . ptm−tm−1(ym−1, ym)dy1 . . . dym

It is a classic theorem of Kolmogorov [W] that PT can be extended to a proba-bility measure on the σ-algebra generated by the cylinder sets.

We mention in particular two examples which will be important. The first isM = g with the Euclidean metric, in which case

pt(X, Y ) = (2πt)−r2 exp

(−1

2||X − Y ||2

t

)

The corresponding measure is Wiener measure, which we will denote by µT .The second example is M = G, with the metric induced by the Ad-invariant

inner product on g. The corresponding measure is called left invariant Brownianmotion on G (cf. [E]), and we will denote it by νT . The heat kernel in this examplecan be re-expressed in terms of a map Qt : G → R as

(etF )(g) =∫

G

Qt(h−1g)F (h) dh (6.1)

Our goal is to define a map Φ : W0T (g) → G which is the analog of ϕψ. We will

do this using the solution of a stochastic differential equation. It will be necessaryto consider both G and g as matrix subgroups of U(N), for some N. Let Ta bean orthonormal basis of g, and let C denote the quadratic Casimir operator:

C =r∑

a=1

Ta2

6.1 Proposition. Consider the following Ito stochastic differential equation onthe interval [0, T ], where w(t) is Brownian motion in g:

dg(t) = g(t) dw(t) +12Cg(t) dt, g(0) = e

Then for each t ∈ [0, T ], there is a unique solution g(t) ∈ L2(W0T (g),F , µt), which

lies in the group G. Furthermore, g(t) is a left invariant Brownian motion on G ,so g∗µT = νT .

The proof of existence and uniqueness of solutions of Ito stochastic differentialequations is well known. The fact that the solution of this equation generatesBrownian motion on G is also standard; see for example [E].


6.2. Definition. The stochastic parallel transport along the interval [0,T] is themap Φ : W0

T (g) → G given by Φ = g(T ).

Notice that Φ is defined almost everywhere on W0T (g); this is sufficient for our

purposes. In order to motivate later definitions in this section, let us write thefollowing well known informal expression for the measure dµT :

dµT (w) =1

ZTexp(−1

2

∫ T

0

||w′||2 dt)Dw

where Dw is an infinite dimensional “Lebesgue measure” on W0T (g). The factor

ZT is meant to represent the normalization which makes µT a probability measure.Proceeding informally, we can push forward µT to get a measure on H0([0, 1], g),namely

dµT (u) =1

ZT

exp(− 12T

∫ 1

0

||u||2 dt)Du

Our goal is to assign meaning to the “Lebesgue measure” Du on H0([0, 1], g).Informally, we see that in the limit T → ∞, the quantity ZT dµT becomes Du.Therefore if F1, F2 are functions on H0([0, 1], g), we have an informal expression forthe ratio of their integrals:

∫F1(u)Du∫F2(u)Du

∼ limT→∞

∫F1 dµT∫F2 dµT

(6.2)

The normalization constant ZT cancels in this ratio. Since the map ψ is linear,its Jacobian is constant, so

∫F1 dµT∫F2 dµT

=∫

F1 ψ dµT∫F2 ψ dµT

Combining these two results leads to the following informal expression for theratio of the integrals of F1 and F2, which will serve as motivation for our laterdefinitions: ∫

F1Du∫F2Du

∼ limT→∞

∫F1 ψ dµT∫F2 ψ dµT

(6.3)

Now suppose M is a submanifold of G, and let M = ϕ−1(M) be the correspond-ing submanifold of H0([0, 1], g). Let πα , α = 1, . . . , p be normal coordinatesdefined on an open neighborhood of M , as in section 2. Here p is the codimensionof M . Then since ϕ is a Riemannian submersion, the maps πα ϕ are normalcoordinates on an open neighborhood of M . Therefore by analogy with the resultsof section 2 we propose the following definition.

6.3. Definition. The regularized volume of M is

vT (M) = limm→∞

∫ p∏α=1

δm(πα(Φ(w))) dµT

We use the word regularized because the Wiener measure depends on the pa-rameter T (which we will eventually send to infinity). Also we associate vT withthe submanifold M , even though the integral is computed in W0

T (g).


6.4 Lemma.vT (M) =

∫M

QT (g) dg

where dg is the measure on M induced by the Riemannian metric on G.

Proof. Since g(t) is a Brownian motion on G, its distribution is given by (6.1).Therefore

∫ p∏α=1

δm(πα(Φ(w))) dµT =∫

G

p∏α=1

δm(πα(g))QT (g)dg (6.4)

Furthermore, for all T > 0 the heat kernel QT (g) is bounded and continuous.Therefore (6.4) has a limit as m →∞, and from the results of section 2 we get thestated result.

Our final definition is motivated by the heuristic discussion leading up to (6.2),where it was noted that Du represents “Lebesgue measure” on the path spaceH0([0, 1], g). Let M and N be submanifolds of G, and denote by M and N theirpreimages in H0([0, 1], g). We expect that these submanifolds of H0([0, 1], g) shouldhave infinite volume. However the perturbative computation in section 5 suggeststhat the ratio of their volumes may be finite. Pursuing this idea, we can representthe ratio of their volumes using our infinite dimensional version of section 2. Letπ1, . . . , πp and ρ1, . . . , ρq be normal coordinates in open neighborhoods of Mand N , respectively. Then using our informal result (6.3), we have

vol(M)vol(N)

∼ limm→∞

limT→∞

∫ ∏pα=1 δm(πα(ϕ ψ)) dµT∫ ∏qβ=1 δm(ρβ(ϕ ψ)) dµT

(6.5)

As indicated before, the map Φ is the appropriate extension to W0T (g) of the

function ϕ ψ on H01([0, T ], g). Therefore (6.5) is a informal expression for the

limit as T → ∞ of the ratio of the regularized volumes of M and N . This is themotivation for our next definition.

6.5. Definition. Let M and N be submanifolds of G, and let M and N be theirpreimages in H0([0, 1], g). Then we define the ratio of volumes of M and N to be

v(M, N) = limT→∞

vT (M)vT (N)

It is quite straightforward to compute this ratio of volumes. Since G is compact,we have the result that for all g ∈ G,

limT→∞

QT (g) = 1

Together with Lemma 6.4 this implies the following result.

6.6 Proposition.

v(M, N) =vol(M)vol(N)

where vol(M) is the volume of the submanifold M in G.


We remark that Proposition 6.6 allows the interpretation that the fibers of thebundle ϕ : H0([0, 1], g) → G all have “equal volume” . This is consistent withCorollary 3.5, which asserts that the fibers are isometric.

These results extend to more general subsets of H0([0, 1], g), in the following way.Let 0 = s0 < s1 < s2 < · · · < sn = 1 be a partition of [0, 1], and write ϕj(u) ∈ G forthe parallel transport from sj−1 to sj , which is the solution E(sj) of the differentialequation E−1E′ = u, E(sj−1) = e. So (ϕ1, . . . , ϕn) is the map Φs defined inCorollary 3.8. Let K be a submanifold of the direct product

∏ni Gi, where Gi is

the Lie group G equipped with the bi-invariant metric 1si−si−1

ds20. Then we define

K(s) = K(s1, s2, . . . , sn) = u ∈ V | (ϕ1(u), . . . , ϕn(u)) ∈ K

Following the same methods as above we can define the regularized volumeof K(s) using the Wiener measure on path space. For each j = 1, . . . , n theparallel transport ϕj on H0([0, 1], g) corresponds to the random variable hj :=g(sj−1T )−1

g(sjT ). Then we define

vT (K(s)) = limm→∞

∫ ∏α

δm(πα(h1, h2, . . . , hn))dµT

where πα are again normal coordinates in an open neighborhood of K. Using thefinite dimensional distributions of the Brownian motion on G, we get

vT (K(s)) = limm→∞

∫G

· · ·∫

G

∏α

δm(πα(g1, g−11 g2, . . . , g

−1n−1gn))

×n∏

i=1

Q(si−si−1)T (g−1i−1gi) dg1 . . . dgn

(6.6)

where g0 := e, and dgi is the Riemannian measure on Gi. By making a change ofvariables, using the translation invariance of Haar measure, and letting m → ∞,(6.6) becomes

vT (K(s)) =∫

K

n∏i=1

Q(si−si−1)T (gi) dνK(g1, . . . , gn)

Here νK is the induced Riemannian measure on K. By letting T →∞ we obtainresults analogous to Proposition 6.6.


References

[DS] Dunford, N. and Schwartz J.T., Linear Operators, Part II, Interscience Publishers Inc.

New York, 1958.

[E] Elworthy, K. D., Stochastic Differential Equations on Manifolds, London Mathematical

Society Lecture Note Series vol. 70, Cambridge University Press, 1982.

[GR] Gradshteyn, I.S. and Ryzhik, I.M., Tables of integrals, series, and products, Academic

Press, 1980.

[HPTT] Heintze, E., Palais, R.S., Terng, C.L. and Thorbergsson, G., Hyperpolar actions on

symmetric spaces, in preparation.

[H] Hsiang, W.Y., On the compact homogeneous minimal submanifolds 56 (1966), Proc.

Nat. Acad. Sci. USA, 5–6.

[HL] Hsiang, W.Y. and Lawson, B.H.Jr, Minimal submanifolds of low cohomogeneity 5 (1971),

J. Differential Geometry, 1–38.

[L1] Lawson, B.H.Jr, Complete minimal surfaces in S3 90 (1970), Ann. of Math., 335–374.

[L2] Lawson, B.H.Jr, Lectures on Minimal submanifolds, Publish or Perish, Berkeley, 1980.

[MRT] Maeda, Y., Rosenberg, S. and Tondeur, P., The mean curvature of gauge orbits, preprint.

[P] Palais, R.S., The principle of symmetric criticality 69 (1979), Comm. Math. Phys.,

19–30.

[PT1] Palais, R.S. and Terng, C.L., Reduction of variables for minimal submanifolds 98 (1986),

Proceedings Amer. Math. Soc., 480–484.

[PT2] Palais R.S., and Terng C.L., Critical Point Theory and Submanifold Geometry, Lecture

Notes in Math., vol. 1353, Springer-Verlag, Berlin and New York, 1988.

[PS] Pressley, A. and Segal, G. B., Loop Groups, Oxford Science Publ., Clarendon Press,

Oxford, 1986.

[S] Simon, B., Functional Integration and Quantum Physics, Pure and Applied Mathemat-

ics, vol. 86, Academic Press, 1979.

[T1] Terng, C.L., Proper Fredholm submanifolds of Hilbert spaces 29 (1989), J. Differential

Geom., 9–47.

[T2] Terng, C.L., Polar actions on Hilbert spaces, to appear in Jour. of Geometric analysis.

[TT] Terng, C.L. and Thorbergsson, G., Submanifolds with constant focal structure, preprint.

[W] Wentzell, A. D., A Course in the Theory of Stochastic Processes, McGraw-Hill, 1981.

Department of Mathematics, Northeastern University, Boston, MA 02115

E-mail address: [email protected],[email protected]

https://www.researchgate.net/publication/234300518_Tables_of_Integrals_Series_and_Products?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==

https://www.researchgate.net/publication/234300518_Tables_of_Integrals_Series_and_Products?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==

https://www.researchgate.net/publication/258505616_Complete_Minimal_Surfaces_in_S_3?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==



https://www.researchgate.net/publication/240478798_Proper_Fredholm_submanifolds_of_Hilbert_space?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==

https://www.researchgate.net/publication/240478798_Proper_Fredholm_submanifolds_of_Hilbert_space?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==

https://www.researchgate.net/publication/243062829_Reduction_of_Variables_for_Minimal_Submanifolds?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==

https://www.researchgate.net/publication/243062829_Reduction_of_Variables_for_Minimal_Submanifolds?el=1_x_8&enrichId=rgreq-c3cdcec4a154f47b78f6c11b9336ad99-XXX&enrichSource=Y292ZXJQYWdlOzIyODc2OTYxNztBUzoyNTAxMzk5ODYxMDAyMjVAMTQzNjY0OTQyODU0Nw==

minimal submanifolds in path space

Documents