Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.


Introduction
C. L. Terng ([Te1]) defined the notion of a proper Fredholm submanifold in a (separable) Hilbert space as a submanifold of finite codimension satisfying certain conditions for the normal exponential map. Note that the shape operators of a proper Fredholm submanifold are compact operator. By using this fact, C. King and C. L. Terng ([KiTe]) defined the regularized trace of the shape operator for each unit normal vector of a proper Fredholm submanifold. Later, E. Heintze, C. Olmos and X. Liu ( [HLO]) defined another regularized trace of the shape operator for each unit normal vector of a proper Fredholm submanifold, which differs from one defined in [KiTe]. They called the regularized trace defined in [KiTe] ζ-reguralized trace. The regularized trace in [HLO] is easier to handle than one in [KiTe]. In almost all relevant cases, these regularized traces coincide. In this paper, we adopt the regularized trace defined in [HLO]. Let M be a proper Fredholm submanifold in V immersed by f . If, for each normal vector ξ of f , the regularized trace Tr r A ξ of the shape operator A ξ of f and the trace Tr A 2 ξ of A 2 ξ exist, then M (or f ) is said to be regularizable. See Section 2 about the definition of the regularized trace Tr r A ξ . Let M be a Hilbert manifold and {f t } t∈[0,T ) be a C ∞ -family of regularizable immersions of codimension one of M into V which admit a unit normal vector field ξ t . The regularized mean curvature vector H t is defined by H t := −Tr r ((A t ) −ξt ) · ξ t , where A t denotes the shape tensor of f t . Define a map F : M × [0, T ) → V by F (x, t) := f t (x) ((x, t) ∈ M × [0, T )). We call {f t } t∈[0,T ) the regularized mean curvature flow if the following evolution equation holds: This notion was introduced in [Koi2]. R. S. Hamilton ([Ha]) proved the existence and the uniqueness of (in short time) of solutions of a weakly parabolic equation for sections of a finite dimensional vector bundle. The evolution equation (1.1) is regarded as the evolution equation for sections of the infinite dimensional trivial vector bundle M × V over M . Since M is an infinite dimensional Hilbert manifold, we cannot apply the Hamilton's result to this evolution equation (1.1). Also, since M is of infinite dimension, M is not locally compact. Thus, we cannot show the existence and the uniqueness of solutions of (1.1) starting from f for a general regularizable C ∞ -immersion f of codimension one. Hence, at least, we must impose the finiteness of the cohomogeneity of f (M ) to the initial data f in order to use the compactness. Here "the finiteness of the cohomogeneity of f (M )" means that there exists a closed subgroup G of the (full) isometry group O(V ) ⋉ V (which is not a Banach Lie group) of V such that f (M ) is G-invariant and that f (M )/G is a finite dimensional manifold with singularity. Furthermore, in order to use the compactness, we must impose f the condition that f (M )/G is compact.
So, we ( [Koi2]) considered the following special case. Assume that an action of a Hilbert Lie group G on a Hilbert space V satisifies the following conditions: (I) The action G V is isometric and almost free, where "almost free" means that the isotropy group of the action at each point is finite; (II) All G-orbits are minimal regularizale submanifolds, that is, they are regularizable submanifold and their regularized mean curvature vectors vanish.
Note that V /G is a finite dimensional orbifold by the condition (II). Denote by N the orbit space V /G. Give N the Riemannian orbimetric g N such that the orbit map φ : V → N is a Riemannian orbisubmersion. Let M (⊂ V ) be a G-invariant hypersurface in V . Furthermore, we assume the following condition: Note that M is a compact hypersurface in N . Denote by f the inclusion maps of M into V and f that of M into N . We ( [Koi2]) showed that the regularized mean curvature flow starting from M exists uniquely in short time. However there were gaps in the statement and the proof. In this paper, we close the gaps (see the statement and the proof of Theorem 4.1 (also those of Theorem 3.1)). Here we note that the uniqueness of the flow is assured under the G-invariance of the flow.
In [Koi2], we mainly proved that the horizontally strongly convexity is preserved along the (G-invariant) regularized mean curvature flow in the case where M is a G-invariant hypersurface in V (see Thoerem 6.1 in [Koi2]), where "G-invariance" of a regularized mean curvature flow means that the regularized mean curvature flow consists of G-invariant regularizable hypersurfaces. In the statement of Theorem 6.1 of [Koi2], it is not specfied that the regularized mean curvature flow is G-invariant but the assumption of the G-invariance of the flow is needed in the statement.
In this paper, we prove that the G-invariant regularized mean curvature flow starting from a horizontally strongly convex G-invariant hypersurface in V collapses to a G-orbit in finite time under some additional conditions (see Theorem A). We shall state this collapsing theorem in detail. Let M be a hypersurface admitting a (global) unit normal vector field ξ. Denote by K the maximal sectional curvature of (N, g N ), which is nonnegative because V is flat. Set b := √ K. Let Σ be the singular set of (N, g N ) and {Σ 1 , · · · , Σ k } be the set of all connected components of Σ. Set B T r (x) := {v ∈ T x N | v ≤ r}. For x ∈ N and r > 0, denote by B r (x) the geodesic ball of radius r centered at x. Here we note that, even if x ∈ Σ, the exponential map exp x : T x N → N is defined in the same manner as the Riemannian manifold-case and B r (x) is defined by B r (x) := exp x (B T r (x)). We assume the following: ( * 1 ) M is included by B π b (x 0 ) for some x 0 ∈ N and exp x 0 | B T π b (x 0 ) is injective.
Furthermore, we assume the following: where α is a positive constant smaller than one and g denotes the induced metric on M and ω n denotes the volume of the unit ball in the Euclidean space R n . Denote by f the inclusion map of M to V and f that of M into N . Let {f t } t∈[0,T ) be the G-invariant regularized mean curvature flow starting from f . Denote by H t the regularized mean curvature vector of f t and set H s t := − H t , ξ t (= Tr r ((A t ) −ξt )), where ξ t is a unit normal vector field of f t such that ξ 0 = ξ and t → ξ t is continuous.
where H denotes the horizontal distribution of φ, ( H 1 ) u denotes the set of all unit horizontal vectors of φ at u and A φ denotes one of the O'Neill's tensors defined in [O'N] (see Section 4 about the definition of A φ ). Note that the restriction A φ | H× H of A φ to H × H is the tensor indicating the obstruction of the integrabilty of H. In this paper, we prove the following collapsing theorem. = 1" implies that f t (M ) converges to an infinitesimal constant tube over some G-orbit as t → T (or equivalently, φ(f t (M )) converges to a round point(=an infinitesimal round sphere) as t → T ) (see Figure 1.2).
In Section 2, we recall the definition of the the regularized mean curvature flow and, in Section 3, we discuss the existence and the uniqueness of mean curvature flows starting from a compact orbifold in a Riemannian orbifold. In the first-half part of Section 4, we give a new proof of the existence and the uniqueness of a Ginvariant regularized mean curvature flow starting from a G-invariant regularizable hypersurface satisfying the condition (III) in a Hilbert space V equipped with a Hilbert Lie group action G V satisfying the conditions (I) and (II). In the secondhalf part of the section, we prepare the evolution equations for some basic geometric quantities along the G-invariant regularized mean curvature flow. In Section 5, we prove the Sobolev inequality for Riemannian suborbifolds. Sections 6-8 is devoted to prove Theorem A. In Section 9, we we state a vision for applying the study of the regularized mean curvature flow to the gauge theory. In more detail, we state a vision for find and study interesting flows of hypersurfaces in the Yang-Milles (or self-dual) muduli space from regularlized mean curvature flows in a Hilbert space.  Let f be an immersion of an (infinite dimensional) Hilbert manifold M into a Hilbert space V and A the shape tensor of f . If codim M < ∞, if the differential of the normal exponential map exp ⊥ of f at each point of M is a Fredholm operator and if the restriction exp ⊥ to the unit normal ball bundle of f is proper, then M is called a proper Fredholm submanifold. In this paper, we then call f a proper Fredholm immersion. Then the shape operator A v is a compact operator for each normal vector v of M . Furthermore, if, for each normal vector v of M , the regularized trace In this paper, we then call f regularizable immersion. If Tr r A v = 0 holds for any v ∈ T ⊥ M , then f is said to be minimal. If f is a regulalizable immersion and if ρ u : v → Tr r (A u ) v (v ∈ T ⊥ u M ) is linear for any u ∈ M , then the regularized mean curvature vector H of f is defined as the normal vector field satisfying where , denotes the inner product of V and T ⊥ u M denotes the normal space of f at u.
Example 2.1. We consider the case where f is isoparametric. Then First we recall the notion of an isoparametric submanifold in a Hilbert space. If the normal connection of f is flat and if the principal curvatures of f for v are constant for any parallel normal vector field v, then it is called an isoparametric submanifold. Then, by analyzing the focal structure of f , we can show that the set Λ of all the principal curvatures of f is given by where λ a 's are parallel sections of the normal bundle T ⊥ M and b a 's are positive constants greater than one. See the first-half part of the proof of Theorem A in [Koi1] about the proof of this fact. Note that, even if Theorem A in [Koi1] is a result for isoparametric submanifolds in a Hilbert space arising from equifocal submanifolds in symmetric space of compact type, the first-half part of the proof is discussed for general isoparametric submanifolds in a Hilbert space. Hence the spectrum Spec (A u ) v of the shape operator (A u ) v for each normal vector v of f at u ∈ M is given by Hence the regularized trace Tr r (A u ) v is given by From this fact, it directly follows that ρ u is linear.
Example 2.2. We consider the case where f is a hypersurface. Then, since the normal space of M is of dimension one, ρ u is linear for each point u ∈ M .
We consider the case where f is a hypersurface and it admits a global unit normal vector field. Fix a global unit normal vector field ξ. Then we call Tr r A −ξ (= − H, ξ ) the regularized mean curvature of f and denote it by H s . Also, we call −A ξ the shape operator and denote it by the same symbol A.
Remark 2.1. In the research of the mean curvature flow starting from strictly convex hypersurfaces, it is general to take the outward unit normal vector field as the unit normal vector field ξ and −A ξ as the shape operator and − H, ξ as the mean curvature. Hence we take the shape operator A and the regularized mean curvature H s as above.
T ) the regularized mean curvature flow. We cannot show that there uniquely exists a regularized mean curvature flow satrting from f for any C ∞ -regularizable immersion f : M ֒→ V because M is not compact. However, for a G-invariant regularizable immersion f : M ֒→ V with the compact quotient f (M )/G in a Hilbert space V equipped with a special Hilbert Lie group action G V , it is shown that there uniquely exists a G-invariant regularized mean curvature flow starting from f (see Theorem 4.1).

The mean curvature flow in Riemannian orbifolds
The basic notions for a Riemannian orbifold and a suborbifold were defined in [AK,BB,GKP,Sa,Sh,Th]. In [Koi2], we introduced the notion of the mean curvaure flow starting from a suborbifold in a Riemannian orbifold. We shall recall this notion shortly. Let M be a paracompact Hausdorff space and O := {(U λ , ϕ λ , U λ /Γ λ ) | λ ∈ Λ} an n-dimensional C k -orbifold atlas of M , that is, a family satisfying the following condition (i)-(iv): where π Γ λ , π Γµ and π Γν are the orbit maps of Γ λ , Γ µ and Γ ν , respectively.
The pair (M, O) is called an n-dimensional C k -orbifold. and each (U λ Denote by (Γ λ ) x the conjugate class of this group (Γ λ ) x , This conjugate class is called the local group at x. If the local group at x is not trivial, then x is called a singular point of (M, O). Denote by Sing (M, O) (or Sing(M )) the set of all singular points of (M, O). This set Sing (M, O) is called the singular set of (M, O).
Let (M, O M ) and (N, O N ) be orbifolds, and f a map from M to N . If, for each x ∈ M and each pair of an orbifold chart (U λ In the sequel, we assume that r = ∞. If a (0, 2)-orbitensor field g of class C k on (M, O M ) is positive definite and symmetric, then we call g a C k -Riemannian orbimetric and (M, O M , g) a C k -Riemannian orbifold. See Section 3 of [Koi2] about the definition of (0, 2)-orbitensor field of class C k . Let ) such that they give the mean curvature flow in ( V µ , g ∧ µ ), where g ∧ µ is the local lift of g to V µ . Then we call f t (0 ≤ t < T ) the mean curvature flow in (N, O N , g).
In [Koi2], we proved the existence and the uniqueness theorem of a mean curvature flow starting from a C ∞ -orbiimmersion f of a compact C ∞ -orbifold into a C ∞ -Riemannian orbifold (see Theorem 3.1 in [Koi2]). However, there was a gap in the proof. Hence we shall close the gap. The mean curvature flow equation is the same kind of partial differential equation as the Ricci flow equation. For the existence and the uniqueness of solutions of the Ricci flow equation in a compact orbifold, the following fact is known. According to Subsection 5.2 of [KL], it is shown that, for any C ∞ -orbimetric g on a compact orbifold M , there uniquely exists a Ricci flow starting from g in short time. The method of the proof is as follows. The existence and the uniqueness of solutions of the Ricci flow equation in short time is reduced to those of a standard quasi-linear parabolic partial differential equation called the Ricci-de Turck equation by the de Turck trick. Since the Ricci-de Turck equation is a standard quasi-linear partial differential equation, it is shown that, in the case where M is a compact manifold (without boundary), there uniquely exists a solution of the Ricci-de Turck equation having g as the initial data in short time for any C ∞ -Riemannian metric g. Hence, in this case, it is shown that, for any C ∞ -Riemannian metric g, there uniquely exists a Ricci flow starting from g in short time. As stated in Subsection 5.2 of [KL], even if M is a compact orbifold (not manifold), it is shown similary that, for any C ∞ -Riemannian orbimetric g on M , there uniquely exists a Ricci flow starting from g in short time.
We prove the following statement by applying this method of the proof to the case of the mean curvature flow starting from a C ∞ -orbiimmersion of a compact orbifold M into a Riemannian orbifold (N, g). Proof. First we consider the case where M and N are manifolds. Then the existence and the uniqueness of solutions of the mean curvature flow equation on a compact manifold (without boundary) in short time is reduced to those of a standard quasilinear parabolic partial differential equation called the mean curvature-de Turck equation by the de Turck trick (see [Z], [CY] and [Koi3] etc. for example). Let's recall the definition of the mean curvature-de Turck equation. Let {f t } t∈[0,T ) be a C ∞ -family of immersions of M into (N, g). Set g t := f * t g and denote by ∇ t the Riemannian connection of g t . Fix a torsion-free connection∇ on M . Define a (1, 2)-tensor field S t on M by S t := ∇ t −∇ and a vector field V (f t ) on M by V (f t ) := Tr gt S t , where Tr gt S t is the trace of S t with respect to g t . The mean curvature-de Turck equation is defined by We give the local expressions of the mean curvature flow equation and the mean curvature-de Turck equation. Let n := dim M and n + r := dim N . Take a local coordinate (U, (x 1 , · · · , x n )) of M and a local coordinate (W, (y 1 , · · · , y n+r )) of N with f (U ) ⊂ W . The local expression of the mean curvature flow equation is given by where (f t ) γ 's are the components of f t with respect to (U, (x 1 , · · · , x n )) and (W, (y 1 , · · · , y n+r )), ((g t ) ij ) is the inverse matrix of the matrix ((g t ) ij ) consisting of the components (g t ) ij 's of the induced metric g t with respect to (U, (x 1 , · · · , x n )), Γ γ αβ 's are the Christoffel's symbols of g with respect to (W, (y 1 , · · · , y n+r )) and (Γ t ) k ij is the Christoffel's symbols of g t with repspec to (U, (x 1 , · · · , x n )). Here we note that (g t ) ij is given by and (Γ t ) k ij is given by By the existence of the final term of the right-hand side of (3.2), the equation (3.2) is a quasi-linear parabolic partial differential equation but not strongly parabolic.
On the other hand, the local expression of the mean curvature-de Turck equation is given by where (f t ) γ 's, ((g t ) ij ) and Γ γ αβ are as above, andΓ k ij 's are the Christoffel's symbol of a fixed connection∇ of M . Since the final term of the right-hand side of (3.2) is changed by n i,j,k=1 ij , the equation (3.3) (hence (3.1)) is a strongly parabolic quasi-linear parabolic partial differential equation. Hence, if M is a compact manifold (without boundary), then it is shown that there uniquely exists a solution {f t } t∈[0,T ) of (3.1) with f 0 = f in short time. As B. Kleiner and J. Lott state in Subsection 5.2 of [KL] in the case of the Ricci flow on a compact orbifold, even if M is a compact orbifold (not manifold), it is shown similarly that there uniquely exists a solution {f t } t∈[0,T ) of (3.1) with f 0 = f in short time. For the solution {f t } t∈[0,T ) , we consider the ordinary differential equation Let {ψ t } t∈[0,T ′ ) (T ′ < T ) be the solution of this ordinary differential equation with the initial condition ψ 0 = id M , where we note that each ψ t is a C ∞ -diffeomorphism of M onto oneself. Then it is shown that {f t • ψ t } t∈[0,T ′ ) is a mean curvature flow starting from f . Conversely, it is easy to show that any mean curvature flow starting from f is given like this. This completes the proof.
Remark 3.1. Let f be a C ∞ -immersion of a manifold M into a complete Riemannian manifold (N, g) and D a relative compact domain of M . The existence of a mean curvature flow starting from f | D is shown but its uniqueness is not shown. In fact, it is shown that there exist infinitely many mean curvature flows starting from f | D as follows. Let f | D : M ֒→ N be an immersion satisfying the following conditions: is an immersed complete Riemannian submanifold of bounded second fundamental form in (N, g), where we note that the condition of "bounded second fundamental form" controls the behavior of the submanifold f | D (M ) near the infinity.
It is clear that there exists infinitely many complete exrensions f | D satisfying the condition (ii). Assume that the norms of the curvature tensor of (N, g), its first derivative and its second derivative are bounded. Then there uniquely exists a mean curvature flow {( f | D ) t } t∈[0,T ) of bounded second fundamental form starting from f | D by the uniqueness theorem in [CY]. It is clear that gives a mean curvature flow starting from f | D and that this flow depends on the choice of the complete extension f | D . Thus we see that there exist infinitely many mean curvature flows starting from f | D .

Evolution equations
Let G V be an isometric almost free action with minimal regularizable orbit of a Hilbert Lie group G on a Hilbert space V equipped with an inner product , . The orbit space V /G is a (finite dimensional) C ∞ -orbifold. Let φ : V → V /G be the orbit map and set N := V /G. Here we give an example of such an isometric almost free action of a Hilbert Lie group.
Example 4.1. Let G be a compact semi-simple Lie group, K a closed subgroup of G and Γ a discrete subgroup of G. Denote by g and k the Lie algebras of G and K, respectively. Assume that a reductive decomposition g = k + p exists. Let B be the Killing form of g and g the bi-invariant metric of G induced from −B. Also, let H 0 ([0, 1], g) be the Hilbert space of all paths in the Lie algebra g of G which are L 2 -integrable with respect to −B, and H 1 ([0, 1], G) the Hilbert Lie group of all paths in G which are of class H 1 with respect to g. This group H 1 ([0, 1], G) acts on H 0 ([0, 1], g) isometrically and transitively as a gauge action: Set P (G, Γ×K) := {g ∈ H 1 ([0, 1], G) | (g(0), g(1)) ∈ Γ×K}. The group P (G, Γ×K) acts on H 0 ([0, 1], g) almost freely and isometrically, and the orbit space of this action is diffeomorphic to the orbifold Γ \ G / K. Furthermore, each orbit of this action is regularizable and minimal (see [HLO], [PiTh], [Te1], [Te2], [TeTh], [Koi2]). In particular, in the case of K = Γ = {e}, φ is a Riemannian submersion of H 0 ([0, 1].g) onto (G, g) and is called the parallel transport map for G.
Let g N be the Riemannian orbimetric on N such that φ is a Riemannian orbisubmersion of (V, , ) onto (N, g N ). By using Theorem 3.1, we prove the following unique existence theorem for a G-invariant regularized mean curvature flow starting from a G-invariant regularizable hypersurface with compact quotient.
. Let H t be the regularized mean curvature vector of f t and H t the mean curvature Thus the existence of a G-invariant regularized mean curvature flow starting from f is shown.
Next we shall show the uniqueness of a G-invariant regularized mean curvature is the flow constructed as above. Thus the uniqueness of a G-invariant regularized mean curvature flow starting from f also is shown.
Remark 4.1. We cannot conclude whether there uniquely exists a (not necessarily G-invariant) regularized mean curvature flow starting from a G-invariant reguralized immersion f as in the statement of Theorem 4.1.
In the sequel, we consider the case where the codimension of M is equal to one. Denote by H (resp. V) the horizontal (resp. vertical) distribution of φ. Denote by pr H (resp. pr V ) the orthogonal projection of T V onto H (resp. V). For simplicity, for X ∈ T V , we denote pr H (X) (resp. pr V (X)) by . Fix a unit normal vector field ξ t of f t . Denote by g t , h t , A t , H t and H s t the induced metric, the second fundamental form (for −ξ t ), the shape operator (for −ξ t ) and the regularized mean curvature vector and the regularized mean curvature (for −ξ t ), respectively. The group G acts on M through f t . Since φ : V → V /G is a G-orbibundle and H is a connection of this orbibundle, it follows from Proposition 4.1 in [Koi2] , where the right-hand side of this relation is .
The restriction of B H to H × · · · × H (s-times) is regarded as a section of the (r, s)-tensor bundle H (r,s) of H. This restriction also is denoted by the same sym- Now we shall recall the evolution equations for some geometric quantities given in [Koi2]. By the same calculation as the proof of Lemma 4.2 of [Koi2] (where we replace H = H ξ in the proof to H = H, ξ ξ = −H s ξ), we can derive the following evolution equation.
According to the proof of Lemma 4.3 in [Koi2], we obtain the following evolution equation.
Lemma 4.3. The unit normal vector fields ξ t 's satisfy the following evolution equation: where ∇ is the connection of π * M (T (r,s) M ) (or π * M (T (r,s+1) M )) induced from ∇ and (e 1 , · · · , e n ) is an orthonormal base of H (u,t) with respect to (g H ) (u,t) . Also, we define a section△ H S H of H (r,s) by where ∇ H is the connection of H (r,s) (or H (r,s+1) ) induced from ∇ H and {e 1 , · · · , e n } is as above. Let A φ be the section of means that e i is entried into the j-th component and the k-th component of S (u,t) .
In [Koi2], we derived the following relation.
Lemma 4.4. Let S be a section of π * M (T (0,2) M ) which is symmetric with respect to g. Then we have According to the proof of Lemma 4.5 in [Koi2], we obtain the following Simonstype identity.
Lemma 4.5. We have Note. In the sequel, we omit the notation F * for simplicity.
Define a section R of π * M (H (0,2) ) by According to Theorem 4.6 in [Koi2], we obtain the following evolution equation from from Lemmas 4.3, 4.4 and 4.5.
Lemma 4.6. The sections (h H ) t 's of π * M (T (0,2) M ) satisfies the following evolution equation: According to the proof of Lemma 4.8 of [Koi2], we obatin the following relation from Lemma 4.2.
Lemma 4.7. Let X and Y be local sections of H such that g(X, Y ) is constant.
According to Lemmas 4.8 and 4.10 in [Koi2], we obtain the following relation .
where we omit F * . In particular, we have Simple proof of the third relation. We give a simple proof of Tr • g H R(•, •) = 0. Take any (u, t) ∈ M × [0, T ) and an orthonormal base (e 1 , · · · , e n ) of H (u,t) with respect to g (u,t) . According to Lemma 4.4 and the definiton of R, we have h(e i , e i ) (which holds because the fibres of φ is regularized minimal).
According to the proof of Corollary 4.11 in [Koi2], we obatin the following evolution equation.
Lemma 4.9. The norms H s t 's of H t satisfy the following evolution equation: According to the proof of Corollary 4.12 in [Koi2], we obtain the following evolution equation.
Lemma 4.10. The quantities ||(A H ) t || 2 's satisfy the following evolution equation: From Lemmas 4.9 and 4.10, we obtain the following evolution equation.
n 's satisfy the following evolution equation: where gradH s is the gradient vector field of H s with respect to g and ||gradH s || is the norm of gradH s with respect to g.
Set n := dim H = dim M and denote by n H * the exterior product bundle of degree n of H * . Let dµ g H be the section of π * M ( n H * ) such that (dµ g H ) (u,t) is the volume element of (g H ) (u,t) for any (u, t) ∈ M × [0, T ). Then we can derive the following evolution equation for Proof. Let (e 1 , · · · , e n ) be a local orthonormal base of H (u 0 ,t 0 ) with respect to (g H ) (u 0 ,t 0 ) and (E 1 , · · · , E n ) a local frame field of H t 0 | U (U : an open set of M ) with (E i ) u 0 = e i (i = 1, · · · , n) and (ω 1 , · · · , ω n ) the dual frame field of (( in the second equality. By using this relation, we can derive On the other hand, by using ∇ ∂ ∂t Therefore we obtain the desired evolution equation.

Sobolev inequality for Riemannian suborbifolds
In this section, we prove the divergence theorem for a compact Riemannian orbifold and Sobolev inequality for a compact Riemannian suborbifold, which may have the boundary. Let (M , g) be an n-dimensional compact Riemannian orbifold, Σ the singular set of (M , g) and {Σ 1 , · · · , Σ k } be the set of all connected components of Σ. Theorem 5.1. For any C 1 -orbitangent vector field X on (M , g), the relation M div g X dv g = 0 holds.
Proof. Let U i (i = 1, · · · , k) be a sufficiently small tubular neighborhood of Σ i with gives a division of cl(U i ) (i = 1, · · · , k). Denote by π ij the projection π Γ ij : U ij → U ij /Γ ij and l i the cardinal number of Γ ij , which depends only on i. Let ξ i be the outward unit normal vector field of ∂U i and ι i is the inclusion map of ∂U i into M . Also, let ξ ij be the outward unit normal vector field of ∂U ij Denote by ι ij the inclusion map of ∂ U ij into R n and g ij the local lift of g with respect to (U ij , ϕ ij , U ij /Γ ij ). Then, by using the divergence theorem (for a compact Riemannian manifold with boundary), we have Also, by using the divergence theorem, we can show and hence From (5.1) and (5.2), we obtain M div g X dv g = 0.
in fact In 1974, D. Hoffman and J. Spruck ( [HoSp]) proved the same Sobolev inequality in a general Riemannian manifold, where we note that the integrand must vanishes on the boundary of the submanifold and furthemore, the volume of the support of the integrand must satisfy some estimate from above related to the curvature and the injective radius of the ambient space. We shall show the following Sobolev inequality for compact Riemannian suborbifolds.
Theorem 5.2. Let (M , g) be a compact Riemannian suborbifold isometrically immersed into (N, g N ) by f . Assume that the sectional curvature K of a complete Riemannian orbifold (N, g N ) satisfies K ≤ b 2 (b is a non-negative real number or the purely imaginary number), M satisfies the following condition ( * 1 ): Let ρ be any non-negative C 1 -function on M satisfying where g denotes the induced metric on M , ω n denotes the volume of the unit ball in the Euclidean space R n , l denotes the cardinality of the local group at x 0 and α is any fixed positive constant smaller than one. Then the following inequality for ρ holds: where H denotes the mean curvature vector of f and C(n, α) is the positive constant depending only on n and α.
Remark 5.1. In the case where (M , g) is a compact Riemannian manifold, the statement of this theorem follows from the Sobolev's inequality in [HoSp] because the condition ( * 1 ) assures the condition (2.3) in Theorem 2.1 of [HoSp].
We shall prepare some lemmas to prove this theorem. Let Γ be the local group at x 0 and set and define a map f : and that f is a C ∞ -immersion. Also, it is clear that π M is a C ∞ -orbisubmersion and that exp x 0 •π • f = f • π M holds. Let g be the Riemannian metric on M such that π M : ( M , g) → (M , g) is a Riemannian orbisubmersion. Let ∇ N be the Riemannian connection of g N and ∇ that of g. Let X ∈ Γ ∞ ( f * T ( B π b ( x 0 ))). Let X T (resp. X ⊥ ) be the tangential (resp. the normal) component of X, that is, .
where (e 1 , · · · , e n ) is an orthonormal base of T x M with respect to g x and ( ∇ N ) f denotes the induced connection of ∇ N by f . π y 6 x 0  First we prepare the following lemma.
where grad g (•) denotes the gradient vector field of (•) with respect to g.
See the proof of Lemma 3.2 in [HoSp] about the proof of this lemma. Let where grad g N (•) denotes the gradient vector field of • with respect to g N . Also, define a C ∞ -vector field P over the tangent space T x 0 R n+1 by By using the discussion in the proof Lemmas 3.5 and 3.6 in [HoSp], we can show the following fact.
Lemma 5.4. (i) For any unit vector v of B π b ( x 0 ) at any x ∈ B π b ( x 0 ), the following inequality holds: (ii) For any (x, y) ∈ M , the following inequality holds: Let λ be a C 1 -function over R satisfying the following condition: denotes the geodesic ball of radius s centered at x 0 . Let ρ be a C 1 -function as in the statement of Theorem 5.2 and set According to the proof of Lemma 4.1 in [HoSp], we can derive the following fact by using Lemmas 5.3 and 5.4.
Lemma 5.5. For all s ∈ [0, π b ), the following inequality hold: According to the proof of Lemma 4.2 in [HoSp], we can show the following result by using Lemma 5.5.
By using Lemma 5.6, we prove Theorem 5.2.
Proof of Theorem 5.2. We shall prove the statement in the case where b is real (similar also the case where b is purely imaginary). Let α,α be constants with 0 < α < 1 ≤α and λ ε (ε > 0) be C 1 -funnction over R satisfying (5.4) and (5.5). Define a function λ ε (ε > 0) over R by λ ε (s) := λ ε (s + ε) and define a function Since ρ satisfies the condition ( * 2 ) in Theorem 5.2, ρ ε,t satisfies the conditions (ii) and (iii) in Lemma 5.6. By using Lemma 5.6 and discussing as in the proof of Theorem 2.1 in [HoSp], we can derive Clearly we have M ρ n n−1 dv g = l · M ρ n n−1 dv g and M grad g ρ + ρ · H s dv g = l · M grad g ρ + ρ · H s dv g .
Hence we obtain 6 Approach to horizontally totally umbilicity In this section, we recall the preservability of horizontally strongly convexity along the mean curvature flow. Let G V be an isometric almost free action with minimal regularizable orbit of a Hilbert Lie group G on a Hilbert space V equipped with an inner product , and φ : V → N := V /G the orbit map. Denote by ∇ the Riemannian connection of V . Set n : is compact. Let f be an inclusion map of M into V and f t (0 ≤ t < T ) the G-invariant regularized mean curvature flow starting from f . We use the notations in Sections 4. In the sequel, we omit the notation f t * for simplicity. As stated in Introduction, set where H 1 := {X ∈ H | ||X|| = 1}. Assume that L < ∞. Note that L < ∞ in the case where N is compact. In [Koi2], we proved the following horizontally strongly convexity preservability theorem by using evolution equations stated in Section 4 and the discussion in the proof of Theorem 5.1.
In this section, we shall prove the following result for the approach to the horizontally totally umbilicity of f t as t → T .
Proposition 6.2. Under the hypothesis of Theorem 6.1, there exist positive constants δ and C 0 depending on only f, L, K and the injective radius i(N ) of N such that We prepare some lemmas to show this proposition. In the sequel, we denote the fibre metric of H (r,s) induced from g H by the same symbol g H , and set ||S|| := g H (S, S) for S ∈ Γ(H (r,s) ). Define a function ψ δ over M by Lemma 6.2.1. Set α := 2 − δ. Then we have Proof. By using Lemmas 4.9 and 4.11, we have (6.2) Also we have (6.3) From (6.2) and (6.3), we obtain the desired relation.
Then we have the following inequalities.
By using the Codazzi equation, we can derive the following relation.

Then we can show that
holds for all t ∈ [0, T ). Without loss of generality, we may assume that ε ≤ 1. Then we have the following inequalities.
Lemma 6.2.3. Let ε be as above. Then we have the following inequalities: Proof. First we shall show the inequality (6.4). Fix (u, t) ∈ M × [0, T ). Take an orthonormal base {e 1 , · · · , e n } of H (u,t) with respect to g (u,t) consisting of the eigenvectors of (A H ) (u,t) . Let (A H ) (u,t) (e i ) = λ i e i (i = 1, · · · , n). Note that λ i > εH s (> 0) (i = 1, · · · , n). Then we have On the other hand, we have From these inequalities, we can derive the inequality (6.4). Next we shall show the inequality (6.5). By using Lemma 6.2.2, we can show For simplicity, we set It is clear that (6.5) holds at (u, t) if (dH s ) (u,t) = 0. Assume that (dH s ) (u,t) = 0. Take an orthonormal base (e 1 , · · · , e n ) of H (u,t) with respect to (g H ) (u,t) with e 1 = (dH s ) (u,t) ||(dH s ) (u,t) || . Then we have where we use ||A H e|| ≤ H s holds for any unit vector e of H. Thus we see that (6.5) holds at (u, t). This completes the proof.
From Lemma 6.2.1 and (6.5), we obtain the following lemma.
Lemma 6.2.4. Assume that δ < 1. Then we have the following inequality: On the other hand, we can show the following fact for ψ δ .
Lemma 6.2.5. We have Proof. According to (4.16) in [Koi2], we have (6.6) Also we have By using Lemmas 4.4, 4.5 and these relations, we can derive (6.8) By substituting this relation into (6.3), we obtain From this relation, we can derive the desired relation.
From this lemma, we can derive the following inequality for ψ δ directly.
Lemma 6.2.6. We have We call this function the function over M × [0, T ) associated with ρ. Denote by g N the Riemannian orbimetric of N and setḡ t :=f * t g N . Also, denote by dv t the orbivolume element ofḡ t . Define a sectionḡ of π * Let ρ and ρ B be as above. According to Theorem 5.1, we have (6.9) From the inequlaity in Lemma 6.2.6 and (6.9), we can derive the following integral inequality.
Lemma 6.2.7. Assume that 0 ≤ δ ≤ 1 2 . Then, for any β ≥ 2, we have where C i (i = 1, 2) are positive constants depending only on K and L (L is the constant defined in the previous section).
Also, we can derive the following inequality.
By using Lemmas 6.2.7 and 6.2.8, we can derive the fact.
Lemma 6.2.10. Take any positive constant k. Assume that Then the following inequality holds: where C is as in Lemma 6.2.9.
Proof of Proposition 6.2. (Step I) First we shall show T < ∞. According to Lemma 4.10, we have ∂H s ∂t Let ρ be the solution of the ordinary differential equation dy ∂t = 1 n y 3 with the initial condition y(0) = min M H s 0 . This solution ρ is given by We regard ρ as a function over M × [0, T ). Then we have Furthermore, by the maximum principle, we can derive that H s ≥ ρ holds over M × [0, T ). Therefore we obtain This implies that T ≤ Step II) Take positive constants δ and β satisfying (6.22) and (6.23). Define a function ψ δ,k by ψ δ,k := max{0, ψ δ (·, t) − k}, where k is any positive number with which is finite because of T < ∞. For a functionρ over M × [0, T ), denote by A(k)ρ dv the function over [0, T ) defined by assigning At(k)ρ (·, t) dv t to each t ∈ [0, T ). By multiplying the inequality in Lemma 6.2.4 by βψ β−1 δ,k , we can show that the inequality in Lemma 6.2.8 holds for ψ δ,k instead of ψ δ . From the inequality, the following inequality is derived directly: By integrating both sides of this inequality from 0 to any t 0 (∈ [0, T )), we have where we use k ≥ sup M ψ δ (·, 0). By the arbitrariness of t 0 , we have From k ≥ sup M ψ δ (·, 0), we have A 0 (k) = ∅. Since f satisfies the conditions ( * 1 ) and ( * 2 ), so is also f t (0 ≤ t < T ) because Volḡ t (M ) decreases with respect to t by Lemma 4.12. Hence we can apply the Sobolev's inequality in Theorem 5.2 to f t (0 ≤ t < T ). By using the Sobolev's inequality in Theorem 5.2 and the Hölder's inequality, we can derive where C is as in Lemma 6.2.9. Hence we obtain Assume that k ≥ k 1 . Then we have From (6.24) and (6.25), we obtain any positive number (n = 2) and q 0 := 2 − 1/q and By using the interpolation inequality, we can derive By using this inequality and the Young inequality, we can derive We may assume thatĈ(n, k) < 1 holds by replacing C(n) to a bigger positive number and furthemore k to a positive number bigger such that 1 − C(n) · C k β/n > 0 holds for the replaced number C(n). Then, from (6.26) and (6.27), we obtain On the other hand, by using the Hölder's inequality, we obtain where r is any positive constant with r > 1. From (6.28) and this inequality, we obtain On the other hand, according to Lemma 6.2.10, we have for some positive constant C (depending only on K, L and f ) by replacing r to a bigger positive number if necessary. Also, by using the Hölder inequality, we obtain From (6.29), (6.30) and this inequality, we obtain We may assume that 2 − 1/q 0 − 1/r > 1 holds by replacing r to a bigger positive number if necessary. Take any positive constants h and k with h > k ≥ k 1 . Then From this inequality and (6.31), we obtain Since • → ||A t (•)|| T is a non-increasing and non-negative function and (6.32) holds for any h > k ≥ k 1 , it follows from the Stambaccha's iteration lemma that ||A t (k 1 + d)|| T = 0, where d is a positive constant depending only on β, δ, q 0 , r, C, C (n, k) and ||A t (k 1 )|| T . This implies that sup This completes the proof.
7 Estimate of the gradient of the mean curvature from above In this section, we shall derive the following estimate of gradH s from above by using Proposition 6.2.
We prepare some lemmas to prove this proposition.
Lemma 7.1.1. The family {||grad t H s t || 2 } t∈[0,T ) satisfies the following equation: Furthermore, according to the Young's inequality: (where p and q are any positive constants with 1 p + 1 q = 1 and ε is any positive constant), we have whereC(n, C 0 , δ) is a positive constant only on n, C 0 and δ. Also, we have From (7.4) and these inequalities, we can derive the desired evolution inequality.
Proof of Proposition 7.1. Define a function ρ over M × [0, T ) by where b is any positive constant and C 1 is a positive constant which is sufficiently big compared to n and b. By using Lemmas 4.10, 7.1.2, 7.1.3 and 7.1.4, we can derive Also, in similar to (7.5), we obtain This implies together with (7.7) that Denote by T 1 V the unit tangent bundle of V . Define a function Ψ over T 1 V by It is clear that Ψ is continuous. Set we have (7.10) for some positive constant K 2 because of the homogeneity of N . By using (7.7), (7.9), (7.10), ||A H || ≤ H s , 1 n ||gradH s || 2 ≤ ||∇ H A H || 2 and Proposition 6.2, we can derive (7.11) Furthermore, by using the Young's inequality (7.6) and the fact that C 1 is sufficiently big compared to n and b, we can derive that holds for some positive constant C 3 (n, C 0 , C 1 , b, δ, K 1 , K 2 ) only on n, C 0 , C 1 , b, δ, K 1 and K 2 . This together with T < ∞ implies that Furthermore, by using the Young inequality (7.6), we obtain ||grad H s || 2 ≤ 2b(H s ) 4 + C 4 (n, C 0 , C 1 , b, δ, K 1 , K 2 , T ) holds for some positive constant C 4 (n, C 0 , C 1 , b, δ, K 1 , K 2 , T ) only on n, C 0 , C 1 , b, δ, K 1 K 2 and T . Since b is any positive constant and C 4 (n, C 0 , C 1 , b, δ, K 1 , K 2 , T ) essentially depends only on n and f 0 , we obtain the statement of Proposition 7.1.

Proof of Theorem A.
In this section, we shall prove Theorem A. G. Huisken ([Hu]) obtained the evolution inequality for the squared norm of all iterated covariant derivatives of the shape operators of the mean curvature flow in a complete Riemannian manifold satisfying curvature-pinching conditions in Theorem 1.1 of [Hu]. See the proof of Lemma 7.2 (Page 478) of [Hu] about this evolution inequality. In similar to this evolution inequality, we obtain the following evolution inequality.
Lemma 8.1. For any positive integer m, the family {||(∇ H ) m A H || 2 } t∈[0,T ) satisfies the following evolution inequality: where C 4 (n, m) is a positive constant depending only on n, m and C i (m) (i = 5, 6) are positive constants depending only on m.
In similar to Corollary 12.6 of [Ha], we can derive the following interpolation inequality.
Lemma 8.2. Let S be an element of Γ(π * M (T (1,1) M )) such that, for any t ∈ [0, T ), S t is a G-invariant (1, 1)-tensor field on M . For any positive integer m, the following inequality holds: where C(n, m) is a positive constant depending only on n and m.
From these lemmas, we can derive the following inequality.
Lemma 8.3. For any positive integer m, the following inequality holds: This fact together with the arbitrariness of t 0 implies that Φ t is uniform bounded. Thus, we see that According to this lemma, we see that such a case as in Figure 8.1 does not happen. Set Λ min (t) := min x∈M λ min (x, t). Let x min (t) be a point of M with λ min (x min (t), t) = Λ min (t) and setx min (t) := φ M (x min (t)). Denote by γf t(xmin(t)) the normal geodesic off t (M ) starting fromf t (x min (t)). Set p t := γf t(xmin(t)) (1/Λ min (t)). Since N is of non-negative curvature, the focal radii of M t along any normal geodesic are smaller than or equal to 1 Λ min (t) . This implies thatf t (M ) is included by the geodesic sphere of radius 1 Λ min (t) centered at p t in N . Hence, since lim t→T 1 Λ min (t) = 0, we see that, as t → T , M t collapses to a one-point set, that is, M t collapses to a G-orbit.
Denote by (Ric M ) t the Ricci tensor of g t and let Ric M be the element of Γ(π * M (T (0,2) M )) defined by (Ric M ) t 's. To show the statement (ii) of Theorem A, we prepare the following some lemmas.
Lemma 8.6. (i) For the section Ric M , the following relation holds: (ii) Let λ 1 be the smallest eigenvalue of A (x,t) . Then we have Proof. Denote by Ric the Ricci tensor of N . By the Gauss equation, we have Also, by a simple calculation, we have . From these relations, we obtain the relation (8.3).

Next we show the inequality in the statement (ii). Since
Hence, from the relation in (i), we can derive the inequality (8.4).
By remarking the behavior of geodesic rays reaching the singular set of a compact Riemannian orbifold (see Figure 8.2) and using the discussion in the proof of Myers's theorem ( [M]), we can show the following Myers-type theorem for Riemannian orbifolds.
Theorem 8.7. Let (N, g) be an n-dimensional compact (connected) Riemannian orbifold. If its Ricci curvature Ric of (N, g) satisfies Ric ≥ (n − 1)K for some positive constant K, then the first conjugate radius along any geodesic in (N, g) is smaller than or equal to π √ K and hence so is also the diameter of (N, g).
By using Propositions 7.1, 8.4, Lemmas 8.6 and Theorem 8.7, we prove the statement (ii) of Theorem A.
The case where g| Ui is a flat metric . Let x t 0 be a maximal point of ||H t 0 ||. Take any geodesic γ of length 1 √ 2||Ht 0 ||max·b 1/4 starting from x t 0 . According to (8.5), we have ||H t 0 || ≥ (1 − b 1/4 )||H t 0 || max along γ. From the arbitrariness of t 0 , this fact holds for any t ∈ [t(b), T ). ( Step II) For any x ∈ M , denote by γ f t (x) the normal geodesic of f t (M ) starting from f t (x). Set p t := γ f t (x) (1/λ min (x, t)) and q t (s) := γ f t (x) (s/λ max (x, t)). Since N is of non-negative curvature, the focal radii of f t (M ) at x are smaller than or equal to 1/λ min (x, t). Denote by G 2 (T N ) the Grassmann bundle of N of 2-planes and Sec : G 2 (T N ) → R the function defined by assigning the sectional curvature of Π to each element Π of G 2 (T N ). Since ∪ (M ). Denote by κ max this maximum. It is easy to show that the focal radii of f t (M ) at x are bigger than or equal to c/λ max (x, t) for some positive constant c depending only on κ max . Hence a sufficiently small neighborhood of f t (x) in f t (M ) is included by the closed domain surrounded by the geodesic spheres of radius 1/λ min (x, t) centered at p t and that of radius c/λ max (x, t) centered at q t ( c). On the other hand, according to Lemma 8.5, we have By using these facts, we can show for any (x, t) ∈ M × [0, T ) and any v ∈ T x M . Hence, according to Theorem 8.7, the first conjugate radius along any geodesic γ in (M , g t ) is smaller than or equal to π ε 0 (H s t ) min for any t ∈ [0, T ). This implies that holds for any t ∈ [0, T ), where exp ft(x) denotes the exponential map of (M , g t ) at denotes the closed ball of radius π ε 0 (H s t ) min in T f t (x) M centered at the zero vector 0. By the arbitrariness of b (in (Step I)), we may assume that b ≤ ε 4 0 4π 4 C 4 0 . Then we have . Let t 0 be as in Step I. Then it follows from the above facts that

Towards application to the Guage theory
In this section, we shall state the vision for applying the study of regularized mean curvature flows to the Gauge theory. In the future, we plan to find interesting Riemannian submanifolds and interesting flows (of Riemannian submanifolds) in the Yang-Milles moduli space or the self-dual moduli space. To state the strategy of this plan, we first recall some basic notions in the theory of the connections of the principal bundles. Let π : P → B be a principal bundle over a compact manifold B having a compact semi-simple Lie group G as the structure group. Fix an Ad(G)invariant inner product , g (for example, the (−1)-multiple of the Killing form) of the Lie algebra g of G, where Ad denotes the adjoint representation of G. Denote by g G the bi-invariant metric of G induced from , g . Set where V denotes the vertical distribution of the bundle P . Each element of Ω ∞ T ,1 (P, g) is called a g-valued tensorial 1-form of class C ∞ on P . Also, let Ω ∞ 1 (B, Ad(P ))(= Γ ∞ (T * B ⊗ Ad(P ))) be the space of all Ad(P )-valued 1-forms of class C ∞ over B, where Ad(P ) denotes the adjoint bundle P × Ad g. The space A ∞ P is the affine space having Ω ∞ T ,1 (P, g) as the associated vector space. Furthermore, Ω ∞ T ,1 (P, g) is identified with Ω ∞ 1 (B, Ad(P )) under the correspondence A ↔ A defined by u · A u (X) = A π(u) (π * X) (u ∈ P, X ∈ T u P ).
Denote by A w,s P the space of all s-times weak differentiable connections of P and Ω w,s i (B, Ad(P )) the space of all s-times weak differentiable Ad(P )-valued i-form on P . Fix a C ∞ -connection ω 0 of P . Define an operator ω 0 : Ω w,s i (B, Ad(P )) → Ω w,s−2 i (B, Ad(P )) by where d ω 0 denotes the covariant exterior derivative with respect to ω 0 and d * ω 0 deontes the adjoint operator of d ω 0 with respect to the L 0 -inner products of Ω w,j i (B, Ad(P )) (j ≥ 0). The H s -inner product , ω 0 s of T ω A w,s P (≈ Ω w,s 1 (B, Ad(P )) ≈ Γ w,s (T * B ⊗ Ad(P ))) is defined by where s ω 0 (A 2 ) denotes the element of Ω w,0 1 (B, Ad(P )) corresponding to s ( A 2 ), , B,g denotes the fibre metric of T * B ⊗ Ad(P ) defined by the the Riemannian metric of B and , g and dv B denotes the volume element of the Riemannian metric of B. Let Ω H s 1 (B, Ad(P )) be the completion of Ω ∞ 1 (B, Ad(P )) with respect to , ω 0 s , that is, Ω H s 1 (B, Ad(P )) := {A ∈ Ω w,s 1 (B, Ad(P )) | A, A ω 0 s < ∞} and A H s P the completion of A ∞ P with respect to , ω 0 s , that is, Let Ω H s T ,1 (P, g) be the completion of Ω ∞ T ,1 (P, g) corresponding to Ω H s 1 (B, Ad(P )).
Let G ∞ P be the group of all C ∞ -gauge transformations g's of P with π • g = π. For each g ∈ G ∞ P , g ∈ C ∞ (P, G) is defined by g(u) = u g(u) (u ∈ P ). This element g satisfies g(ug) = Ad(g −1 )( g(u)) (∀u ∈ P, ∀g ∈ G), where Ad denotes the homomorphism of G to Aut(G) defined by Ad(g 1 )(g 2 ) := g 1 · g 2 · g −1 1 (g 1 , g 2 ∈ G). Under the correspondence g ↔ g, G ∞ P is identified with For g ∈ G ∞ P , the C ∞ -sectiong of the associated G-bundle P × Ad G is defined by g(x) := u · g(u) (x ∈ M ), where u is any element of π −1 (x). Under the correspondence g ↔g, G ∞ P (= G ∞ P ) is identified with the space Γ ∞ (P × Ad G) of all C ∞ -sections of P × Ad G. The H s+1 -completion of Γ ∞ (P × Ad G) was defined by Groisser and Parker (see Section 1 (P668) of [GP1]). Denote by Γ H s+1 (P × Ad G) this completion. Also, denote by G H s+1 P (resp. G H s+1 P ) the completion of G ∞ P (resp. G ∞ P ) corresponding to Γ H s+1 (P × Ad G). Assume that s > 1 2 dim M − 1. Then, according to Lemma 1.2 of [U], this H s+1 -completion G H s+1 P is a C ∞ -Hilbert Lie group and the gauge action G H s+1 P A H s P is of class C ∞ . However, this action does not act isometrically on the Hilbert space (A H s P , , ω 0 s ). Define a Riemannian metric g s on A H s P by (g s ) ω := ω s (ω ∈ A H s P ). This Riemannian metric g s is non-flat and translationinvariant. The gauge action G H s+1 P A H s P acts isometrically on the Riemannian Hilbert manifold (A H s P , g s ). Note that the Hilbert space (A H s P , , ω 0 s ) is regarded as the tangent space of (A H s P , g s ) at ω 0 . Give the moduli space M H s P := A H s P /G s+1 P the Riemannian orbimetric g s such that the orbit map π M P : (A H s P , g s ) → (M H s P , g s ) is a Riemannian orbisubmersion. (t ∈ [0, 1]). Fix x 0 ∈ B and u 0 ∈ π −1 (x 0 ). Take c : S 1 → B be a C ∞ -loop with c(1) = x 0 . Denote by π c : c * P → S 1 the induced bundle of P by c, which is identified with the trivial G-bundle P o := S 1 × G over S 1 by σ. Define an immersion ι c of the induced bundle c * P into P by ι c (z(t), u) = u ((z(t), u) ∈ c * P ).
Definition 9.1. Define a map hol c : A H s P → G by where P ω c•z (resp. P ω 0 c•z ) denotes the parallel translation along c • z with respect to ω (resp. ω 0 ). We call this map hol c the holonomy map along c.
In particular, in the case where P is the trivial G-bundle π o : S 1 × G → S 1 , A H s P is identified with the Hilbert space H s ([0, 1], g) of all H s -curves in the Lie algebra g of G and the holonomy map hol c along c(t) = t (t ∈ [0, 1]) coincides with the parallel transport map φ : H s ([0, 1], g) → G for G stated in Example 4.1, where s may be any non-negative integer because [0, 1] is of one-dimension.
The based gauge group (G H s+1 P ) x at x ∈ M is defined by where e denotes the identity element of G. For a finite subgroup Γ of G and u ∈ P , we define a closed subgroup (G H  where g is the element of H s+1 (P, G) corresponding to g.
Example 9.1. Let G be a compact semi-simple Lie group and consider the trivial Gbundle P o := S 1 ×G over S 1 . Also let s be any non-negative integer. Since Ad(P o ) is identified with the trivial g-bundle P ′ o := S 1 × g over S 1 , A H s Po (≈ Ω H s 1 ([0, 1], Ad(P o ))) is identified with H s ([0, 1], g). Also, G H s+1 In particular, denote by Λ H s+1 e (G) the loop group P (G, {e} × {e}) at the idenitity element e of G. Let Γ be a finite subgroup of G and K a closed subgroup of G such that (G, K) is a reductive pair, that is, there exists a subspace p of g satisfying g = k ⊕ p and [k, p] ⊂ p, where g and k denote the Lie algebras of G and K, respectively. Let B := A H 0 Po and G 1 := P (G, Γ × K). Then the pair (B, G 1 ) satisfies the conditions (I) and (II) stated in Introduction, where we note that the moduli space B/G 1 is orbi-diffeomorphic to the orbifold Γ \ G / K.
Denote by C ∞ ([0, 1], M ) the space of all C ∞ -paths in M . Take c ∈ C ∞ ([0, 1], M ). Denote by π c : c * P → [0, 1] the induced bundle of P by c, which is isomophic to the trivial G-bundle P o (= [0, 1] × G) → [0, 1]. Define an immersion ι c of the induced bundle c * P into P by ι c (t, u) = u ((t, u) ∈ c * P ). Fix a base point ω 0 of A ∞ P . We shall define a map linking A H s P to H 0 ([0, 1], g). Let x 0 , c, u 0 be as above and σ the horizontal lift of c • z starting from u 0 with respect to ω 0 .
Also, it is easy to show that the following facts holds: (iii) The operator norm (dµ c ) ω op of the differential (dµ c ) ω of µ c at any point ω is smaller than or equal to one.
Denote by π H s M P the orbit map of the action G H s+1 P A H s P . Also, let exp ω 0 be the exponential map of the Riemannian Hilbert manifold (A H s P , g s ) at ω 0 . Then, for an orbisubmanifold S in H 0 ([0, 1], g)/P (G, Γ ω 0 ), we can costruct an orbisubmanifolds π H s M P (exp ω ((π Γ • hol c ) −1 (S))) in the muduli space (M H s P , g s ) (see Figure 9.2). Let YM H s P be the Hilbert space of all Yang-Mills H s -connections of the G-bundle π : P → M . Also, in the case of dim M = 4, denote by SD H 2 P the Hilbert space of all self-dual H 2 -connections of the G-bundle π : P → M , where we note that s ≥ 2 in this case. Note that a self-dual connection means "instanton" because M is compact. The Yang-Mills moduli space M YM P := YM H s P /G H s+1 P and the self-dual moduli space M SD P := SD H s P /G H s+1 P are finite dimensional manifolds with singularity in general. Give these moduli spaces the singular Riemannian metrics induced naturally from the (non-flat) Riemannian metric g s on A H s P . Denote by g s these singular Riemannian metrics.
(A H s P , g s ) (M H s P , g s ) H 0 ([0, 1], g)/P (G, Γ ω0 ) π H s MP (exp ω0 ((π Γ • hol c ) −1 (S))) (A H s P , , ω0 s )(= (T ω0 A H s P , (g s ) ω0 )) (π Γ • hol c ) −1 (S) Figure 9.2 : Submanifolds in the moduli space defined by hol c Strategy (i) We plan to find a pair (S, c) of an orbisubmanifold S in the Riemannian orbifold H 0 ([0, 1], g)/P (G, Γ) and a C ∞ -loop c in M such that π H s M P (exp ω ((π Γ • hol c ) −1 (S))) ∩ YM H s P (resp. π H s M P (exp ω ((π Γ • hol c ) −1 (S))) ∩ SD H s P ) gives an interesting submanifold in YM H s P (resp. SD H s P ). gives an interesting flow in YM H s P (resp. SD H s P ) (for example, a good flow collapsing to a singular point of YM H s P (resp. SD H s P )). We will use Theorem A to find a good flow collapsing to a singular point of YM H s P or SD H s P .