A modified mean curvature flow in Euclidean space and soap bubbles in symmetric spaces
aa r X i v : . [ m a t h . DG ] M a y A modfied mean curvature flowin Euclidean space and soap bubblesin symmetric spaces
Naoyuki Koike
Abstract
In this paper, we show that small spherical soap bubbles in irreducible simplyconnected symmetric spaces of rank greater than one are constructed from thelimits of a certain kind of modified mean curvature flows starting from smallspheres in the Euclidean space of dimension equal to the rank of the symmetricspace, where we note that the small spherical soap bubbles are invariant underthe isotropy subgroup action of the isometry group of the symmetric space.Furthermore, we investigate the shape and the mean curvature of the smallspherical soap bubbles.
Let f be an immersion of an n -dimensional compact oriented manifold M into an( n + 1)-dimensional oriented Riemannian manifold ( f M , e g ). If f is of constant meancurvature, then f ( M ) is called a soap bubble . Soap bubbles in the Euclidean spacehave studied by many geometers. In 1989, W.T. Hsiang and W.Y. Hsiang ([HH])studied isoperimetric soap bubbles in the product space H n ( c ) × · · · × H n k ( c k ) ofhyperbolic spaces or H n ( c ) × · · · × H n k ( c k ) × R n k +1 , where “isoperimetric” meansthat the soap bubble is a solution of the isopermetric problem. They proved thatall isoperimetric soap bubbles in the product spaces are invariant under the isotropysubgroup action of the isometry group of the product space. Also, they ([HH])proved that isoperimetric soap bubbles with the same constant mean curvature in H n ( c ) × R are congruent. Furthermore, they found the lower bound of the constantmean curvatures of isoperimetric soap bubbles in H n ( c ) × H n ( c ) or H n ( c ) × R n .In 1992, W.Y. Hsiang ([Hs]) found the lower bound of the constant mean curvaturesof (not necessarily isoperimetric) soap bubbles in a rank l ( ≥
2) symmetric space
G/K of non-compact type and, in the case where
G/K is irreducible, he gave the explicit1escription of the lower bound by using the root system of
G/K . Furthermore, heproved that, for each real number b greater than the lower bound, there exists a K -invariant spherical soap bubble of constant mean curvature b in G/K .In this paper, we introduce the notion of a weighted root system and define themodified mean curvature flow in a Euclidean space associated to the system. Weshow that the flows starting from small spheres exist in infinite time and convergeto an embedded hypersurfaces in C ∞ -topology. Furthermore, in the case where thesystem is one associated to an irreducible simply connected symmetric space of rankgreater than one, we show that spherical soap bubbles in the symmetric spaces areconstructed from the limit hypersurfaces of the flows starting from small spheresand investigate the shape and the mean curvature of the spherical soap bubbles.First we shall introduce the notion of a weighted root system. Let S = ( V, ( △ , { m α | α ∈ △} , ε )) be a system satisfying the following four conditions:(i) V is a l ( ≥ h , i ,(ii) △ is a subset of the dual space V ∗ of V and it is a root system of rank l inthe sense of [He] (i.e., it is of type a l , b l , c l , d l , bd l , e (in case of l = 6), e (incase of l = 7), e (in case of l = 8), k (in case of l = 4), g (in case of l = 2)),(iii) m α ( α ∈ △ ) are positive integers and ε is equal to 1 or − S a weighted root system and l the rank of the system. Denoteby rank S the rank of S . Let W be the group generated by the reflections withrespect to α − (0)’s ( α ∈ △ ). We call W the reflection group associated with S . Let S i = ( V i , ( △ i , { m α | α ∈ △ i } , ε i )) ( i = 1 ,
2) be weighted root systems. If ε = ε andif there exists a linear isometry Φ of V onto V satisfying(i) { α ◦ Φ | α ∈ △ } = △ ,(ii) m α ◦ Φ = m α ( α ∈ △ ),then we say that S is isomorphic to S and call Φ an isomorphism of S onto S .Let S = ( V, ( △ , { m α | α ∈ △} , ε )) be a weighted root systems. Let S V ( r ) is thesphere of radius r centered at the origin in V . The reflection group W preserves S V ( r ) invariantly and hence it acts on S V ( r ). Let △ + ( ⊂ △ ) be the positive rootsystem under a lexicographic ordering of V ∗ and δ ( ∈ △ + ) the highest root. See[He] about the definitions of the positive root system and the highest root. Definea positive number r S by r S := ( π || δ || (in case of ε = 1) ∞ (in case of ε = − , where || δ || is the norm of δ with respect to the inner product of V ∗ induced from h , i .2et B V ( r S ) be the open ball of radius r S centered at the origin in V and set B S := B V ( r S ) \ { } . Fix r ∈ (0 , r S ). Denote by C ∞ W ( S V ( r ) , V ) (resp. C ∞ W ( S V ( r ) , B S ))the space of all W -equivariant C ∞ -maps from S V ( r ) to V (resp. B S ) and byImm ∞ W ( S V ( r ) , V ) (resp. Imm ∞ W ( S V ( r ) , B S )) the space of all W -equivariant C ∞ -immersions of S V ( r ) into V (resp. B S ). For φ ∈ Imm ∞ W ( S V ( r ) , B S ), define a W -invariant function ρ S ,φ over S V ( r ) by(1 . ρ S ,φ ( Z ) := X α ∈△ + m α √ εα ( ν ( Z ))tan( √ εα ( φ ( Z )) ( Z ∈ S V ( r )) , where ν (: S V ( r ) → S V (1)) is the Gauss map of φ (defined by assigning the outwardunit normal vector of φ at Z to each point Z of S V ( r )). Note that, if α ( φ ( Z )) = 0,then we have ν ( Z ) = φ ( Z ) || φ ( Z ) || and hence √ εα ( ν ( Z ))tan( √ εα ( φ ( Z ))) implies || φ ( Z ) || . Define a map D S from Imm ∞ W ( S V ( r ) , B S ) to C ∞ W ( S V ( r ) , V ) by(1 . D S ( φ ) := R S V ( r ) (cid:0) || ∆ b g φ || + ρ S ,φ (cid:1) dv b g R S V ( r ) dv b g − ( || ∆ b g φ || + ρ S ,φ ) ! ν for φ ∈ Imm ∞ W ( S V ( r ) , B S ), where g is the W -invariant metric on S V ( r ) induced from h , i by φ , dv b g is the volume element of b g , ∆ b g is the Laplace operator with respectto b g and || · || is the norm of ( · ) with respect to h , i . We consider the followingevolution equation(E S ) ∂φ t ∂t = D S ( φ t )for a C ∞ -family φ t in Imm ∞ W ( S V ( r ) , B S ). Since || ∆ b g t φ t || is the mean curvatureof φ t , this evolution equation is interpreted as a modified volume-preserving meancurvature flow equation in V , where b g t is the W -invariant metric on S V ( r ) inducedfrom h , i by φ t . Denote by ι r the inclusion map of S V ( r ) into V . It is clear that ι r is W -equivariant.First we prove the following result for the evolution equation ( E S ). Theorem A.
Let S = ( V, ( △ , { m α | α ∈ △} , ε )) be a weighted root system and ι r theinclusion map of S V ( r ) into V . Then there exists a positive constant R smaller than r S such that, if r < R , then the solution φ t of the evolution equation (E S ) satisfyingthe initial condition φ = ι r uniquely exists in infinite time and φ t i converges to a W -equivariant C ∞ -embedding φ ∞ of S V ( r ) into B S (in the C ∞ -topology) as i → ∞ forsome sequence { t i } ∞ i =1 in [0 , ∞ ) with lim i →∞ t i = ∞ . Furthermore, φ t ( ≤ t < ∞ )remain to be strictly convex and hence so is also φ ∞ . G/K be an irreducible simply connected rank l ( ≥
2) symmetric space ofcompact type or non-compact type. A weighted root system of rank l is defined for G/K in a natural manner (see Section 3). We call this system the weighted rootsystem associated with
G/K and denote it by S G/K . Let S G/K = ( V, ( △ , { m α | α ∈△} , ε )), where we note that ε = (cid:26) G/K is of compact type) − G/K is of non-compact type).The vector space V is identified with the tangent space T eK T of a maximal flattotally geodesic submanifold T in G/K through eK , where e is the identity elementof G . In the case where G/K is of compact type, T is identified with the quotientspace V /
Π of V by a lattice Π in V , and in the case where G/K is of non-compacttype, it is identified with V oneself. For convenience, let Π = { } in the case where G/K is of non-compact type. Let W be the reflection group associated with S G/K .Then it is shown that Π is W -invariant. Denote by π the quotient map of V onto V /
Π = T and r ( G/K ) the injective radius of
G/K . Note that r ( G/K ) = ∞ in thecase where G/K is of non-compact type. It is easy to show that r ( G/K ) = r S G/K .The main theorem in this paper is as follows.
Theorem B.
Let S = ( V, ( △ , { m α | α ∈ △} , ε )) and ι r as in Theorem A. Assumethat r < R and S is isomorphic to one associated to an irreducible simply connectedrank l ( ≥ symmetric space G/K of compact type or non-compact type, where R is as in Theorem A. Then the following statements (i) − (iii) hold. (i) The hypersurface M := K · π ( φ ∞ ( S V ( r ))) in G/K is a K -invariant strictlyconvex spherical soap bubble in G/K , where φ ∞ is as in Theorem A. (ii) Let C ( ⊂ V ) be a Weyl domain (i.e., a fundamental domain of the reflectiongroup W ) and θ the element of (0 , π ) defined by θ := max P max Z ∈ P max max Z ∈ P min ,Z ∠ Z Z (cid:18) = max P max Z ∈ P min max Z ∈ P max ,Z ∠ Z Z (cid:19) , where ∠ Z Z denotes the angle between −−→ Z and −−→ Z , P moves over the Grass-mannian of all two-planes in V , P max (resp. P min ) denotes the set of all the maximal(resp. minimal) points of the function ρ S ,ι r over S V ( r ) ∩ C ∩ P ( C : the closureof C ) and P max ,Z ( ⊂ P max ) (resp. P min ,Z ( ⊂ P min ) ) denotes the set (which is atmost two-points set) of all the maximal (resp. minimal) points neighboring Z in S V ( r ) ∩ C ∩ P of the function. Then we have M ⊂ B (cid:18) r cos θ (cid:19) \ B ( r cos θ ) , here B ( r cos θ ) (resp. B ( r cos θ ) ) is the closed geodesic ball of radius r cos θ (resp. r cos θ ) in G/K centered at eK . (iii) Let η max and η min be the functions defined by η max ( s ) := max Z ∈ S V (1) (cid:18) ρ S ,ι s ( Z ) + l − s (cid:19) and η min ( s ) := min Z ∈ S V (1) (cid:18) ρ S ,ι s ( Z ) + l − s (cid:19) . Then the constant mean curvature H M of M satisfies η min ( r ) ≤ H M ≤ η max ( r ) . Remark 1.1.
For convenience, denote by M ( r ) and H ( r ) the soap bubble M andthe mean curvature H M as in the statement of Theorem A, respectively. Since thevolume of the domain surrounded by M ( r ) is strictly increasing (and continuous)with respect to r , it is shown that H ( r ) is strictly decreasing (and continuous)with respect to r . Easily we can show lim r → η max ( r ) = lim r → η min ( r ) = ∞ and hencelim r → H ( r ) = ∞ . On the other hand, in the case where G/K is of non-compact type,we have lim r →∞ η max ( r ) = max Z ∈ S V (1) X α ∈△ + m α | α ( Z ) | (=: b max ( G/K ))and lim r →∞ η min ( r ) = min Z ∈ S V (1) X α ∈△ + m α | α ( Z ) | (=: b min ( G/K )) . Hence we obtain b min ( G/K ) ≤ lim r →∞ H ( r ) ≤ b max ( G/K ) . By using (ii) of Theorem B, we can derive the following result.
Corollary C.
Under the hypothesis of Theorem B, set θ G/K := max ( Z ,Z ) ∈ S V ( r ) ∩ C ∠ Z Z . Then we have M ⊂ B (cid:18) r cos θ G/K (cid:19) \ B ( r cos θ G/K ) . In particular, we obtain the following result in the case where
G/K is of ranktwo. 5 orollary D.
Under the hypothesis of Theorem B, assume that the rank of
G/K is equal to two. Then we have M ⊂ B (cid:18) r √ (cid:19) \ B √ r ! ( △ : ( a ) − type or ( g ) − type) B ( √ r ) \ B (cid:18) r √ (cid:19) ( △ : ( b ) − type) . According to Corollary D, when the root system of
G/K is of type a , φ ∞ ( S V ( r ))is as in Figure 1 for example. VS V ( r √ ) φ ∞ ( S V ( r )) CS V ( √ r ) The six corners of φ ∞ ( S V ( r )) are smooth.The six edges of φ ∞ ( S V ( r )) are curved. Figure 1.
In this section, we shall recall the definition of the volume-preserving mean curvatureflow and the result of N.D. Alikakos and A. Freire ([AF]) for this flow. Let M be an n -dimensional compact oriented manifold, ( f M , e g ) be an ( n + 1)-dimensional orientedRiemannian manifold and f an immersion of M into f M . Also, let { f t } t ∈ [0 ,T ) be a C ∞ -family of immersions of M into f M , where T is a positive constant or T = ∞ .Define a map F : M × [0 , T ) → f M by F ( x, t ) = f t ( x ) (( x, t ) ∈ M × [0 , T )). Denoteby π M the natural projection of M × [0 , T ) onto M . For a vector bundle E over M , denote by π ∗ M E the induced bundle of E by π M . Denote by g t and N t theinduced metric and the outward unit normal vector of f t , respectively. Also, denoteby H t the mean curvature of f t for − N t . Define the function H over M × [0 , T ) by H ( x,t ) := ( H t ) x (( x, t ) ∈ M × [0 , T )), the section g of π ∗ M ( T (0 , M ) by g ( x,t ) := ( g t ) x (( x, t ) ∈ M × [0 , T )) and the section N of π ∗ M ( T f M ) by N ( x,t ) := ( N t ) x (( x, t ) ∈ × [0 , T )), where T (0 , M is the tensor bundle of (0 , M and T f M is thetangent bundle of f M . The average mean curvature H (: [0 , T ) → R ) is defined by H t := R M H t dv g t R M dv g t , where dv g t is the volume element of g t . G. Huisken ([Hu3]) called the flow { f t } t ∈ [0 ,T ) a volume-preserving mean curvature flow if it satisfies F ∗ (cid:18) ∂∂t (cid:19) = ( H − H ) N. He studied this flow in [Hu3]. Along this flow, the volume of (
M, g t ) decreases butthe volume of the domain D t sorrounded by M t := f t ( M ) is preserved invariantly.In particular, if f t ’s are embeddings, then we call { M t } t ∈ [0 ,T ) rather than { f t } t ∈ [0 ,T ) a volume-preserving mean curvature flow.Assume that f M is compact. Let S be a geodesic sphere in f M such that, forany point p of the domain surrounded by S , S lies in a geodesically convex domainof p (in f M ), and ι S the inclusion map of S into f M . Then, according to MainTheorem of [AF], it is shown that, if S is a small geodesic sphere of radius smallerthan some positive constant among the above geodesic spheres, then for any strictlyconvex C ∞ -embedding f of S into f M which is sufficiently close to ι S , there existsthe volume-preserving mean curvature flow f t starting from f in infinite time, each f t is strictly convex and f t i converges to a strictly convex C ∞ -embedding f ∞ ofconstant mean curvature (in C ∞ -topology) as i → ∞ for some sequence { t i } ∞ i =1 withlim i →∞ t i = ∞ . Furthermore, if all critical points of the scalar curvature functionsof f t ’s are non-degenerate, then f t converges to the embedding f ∞ (in C ∞ -topology)as t → ∞ . Note that the positive constants δ i and ε i ( i = 0 , , ,
3) in the statementof Main Theorem of [AF] are taken as δ < δ < δ / < δ / ε < ε < ε / < ε / δ and ε are taken as in P257 of [AF]. Bythe compactness of f M , the existenceness of the positive constant δ in Page 257 of[AF] is assured. Hence the existencenesses of δ i ( i = 1 , ,
3) also are assured. In thecase where f M is homogeneous, it is clear that their existencenesses are assured evenif it is not compact. Hence the statement of Main Theorem of [AF] is valid in thecase where f M is a (not necessarily compact) homogeneous space. Also, we note thatthe statement of Main Theorem of [AF] does not hold without the assumption that f is sufficiently close to ι S . For example, in the case where f is a strictly convexembedding of a small geodesic sphere S into f M = S n +1 (1) (which is not close to7 S ) as in Figure 2, the volume-preserving mean curvature flow f t starting from f (0 ≤ t < ∞ ) does not remain to be strictly convex. This fact is stated in Remarksof Page 38 of [Hu3]. f ( S ) (strictly convex) f t ( S ) (not strictly convex) f M = S n +1 (1) f is not close to ι S . Figure 2.
Let
G/K be an irreducible simply connected symmetric space of compact type ornon-compact type. In this section, we shall first define the weighted root systemassociated with
G/K . Set n := dim( G/K ) − l := rank( G/K ). Let g (resp. k ) be the Lie algebra of G (resp. K ) and g = k + p the canonical decompositionassociated with the symmetric pair ( G, K ). The space p is identified with the tangentspace T eK ( G/K ) of
G/K at eK , where e is the identity element of G . Take amaximal abelian subspace a of p . Let △ ( ⊂ a ∗ ) be the (restricted) root system ofthe symmetric pair ( G, K ) with respect to a and p α the root space for α ∈ △ . See[He] about the definitions of these notions. Then we have the following root spacedecomposition: p = a ⊕ (cid:18) ⊕ α ∈△ + p α (cid:19) , where △ + is the positive root system under some lexicographic ordering of a ∗ . Set m α := dim p α ( α ∈ △ ). Let W be the Weyl group of G/K with respect to a (i.e., thegroup generated by the reflections with respect to the hyperplanes α − (0)’s ( α ∈ △ )in a ), which acts on a , C ( ⊂ a ) be a Weyl domain (i.e., a fundamental domain of theaction W y a ) and S a ( r ) be the sphere of radius r centered at the origin in a . Theisotropy group K acts on G/K naturally. This action is called the isotropy actionof
G/K . Denote by exp the exponential map of G and Exp the exponential map of G/K at eK . Set T := Exp( a ), which is a maximal flat totally geodesic submanifoldof G/K . Note that T is identified with the quotient a / Π by a W -invariant latticeΠ in a in the case where G/K is of compact type, and it is identified with a oneself8n the case where G/K is of non-compact type. Set(3 . ε := (cid:26) G/K is of compact type) − G/K is of non-compact type).Then the system S := ( a , ( △ , { m α | α ∈ △} , ε )) is a weighted root system. We callthis system the weighted root system associated with G/K and denote it by S G/K .Next we shall describe explicitly the mean curvatures of K -invariant hypersur-faces in G/K in terms of the roots. As a special case, those of geodesic spheresin
G/K are described explicitly. Let M a be a W -invariant star-shaped hypersur-face (at the origin ) in a . Assume that, in the case where G/K is of compact type,max Z ∈ M a d ( , Z ) is smaller than the injective radius r ( G/K ) of
G/K , where d is theEuclidean distance of a . Set M T := Exp( M a ) and M := K · M T (= K · (Exp( M a ∩ C )).Note that M T and M are hypersurfaces in T and G/K , respectively. Denote by N the outward unit normal vector field of M ( ⊂ G/K ), and A and H the shape oper-ator and the mean curvature of M ( ⊂ G/K ) for the inward unit normal vector field − N , respectively. Also, denote by A T and H T those of M T ( ⊂ T ) for the inwardunit normal vector − N | M T , respectively, and b A and b H those of M a ( ⊂ a ) for theinward unit normal vector, respectively. Take any Z ∈ M a ∩ C . Denote by L Z the K -orbit K · Exp Z , which is a principal orbit of the K -action because Z ∈ C . Wehave T Exp Z L Z = (exp Z ) ∗ (cid:18) ⊕ α ∈△ + p α (cid:19) . Denote by A Z the shape tensor of L Z ( ⊂ G/K ). Take X α ∈ p α ( α ∈ △ + ). Then wehave(3 . A ZN Exp Z ((exp Z ) ∗ X α ) = − √ εα ((exp Z ) − ∗ ( N Exp Z ))tan( √ εα ( Z )) (exp Z ) ∗ X α by suitably rescaling of the metric of G/K (see [GT],[K1],[K2]), where √ εα ( Z )) =0 in case of √ εα ( Z ) = π . In the sequel, we give G/K this rescaled metric. Thevector field N | L Z is a K -equivariant normal vector field along L Z . Hence, sincethe K -action is a hyperpolar action, it is a parallel normal vector field of L Z (seeTheorem 5.5.12 of [PT]). Hence, we have(3 . A ZN Exp( Z ) = − A Exp Z | T Exp Z L Z , where we note that A Exp Z denotes the value of A at Exp Z . In the sequel, we express A Exp Z as A for simplicity. From (3 .
2) and (3 . . A ((exp Z ) ∗ X α ) = √ εα ((exp Z ) − ∗ ( N Exp Z ))tan( √ εα ( Z )) (exp Z ) ∗ X α . X ∈ a ⊖ Span { N Exp Z } . Since T is totally geodesic in G/K , we have(3 . A ((exp Z ) ∗ X ) = A T ((exp Z ) ∗ X ) = (exp Z ) ∗ ( b AX ) . From (3 .
4) and (3 . . H Exp Z = X α ∈△ + m α √ εα ((exp Z ) − ∗ ( N Exp Z ))tan( √ εα ( Z )) + b H Z ( Z ∈ M a ∩ C ) . Take any Z ′ ∈ M a ∩ ∂C , where ∂C is the boundary of C . Set △ Z ′ + := { α ∈△ + | α ( Z ′ ) = 0 } . Then it follows from (3 .
6) and the continuity of H that(3 . H Exp Z ′ = X α ∈△ + \△ Z ′ + m α √ εα ((exp Z ′ ) − ∗ ( N Exp Z ′ ))tan( √ εα ( Z ′ )) + 1 || Z ′ || X α ∈△ Z ′ + m α + b H Z ′ , where we note that (exp Z ′ ) − ∗ ( N Exp Z ′ ) = || Z ′ || Z ′ . The hypersuface M is of constantmean curvature κ if and only if H Exp Z = κ ( Z ∈ M a ) because M is K -invariant.We consider the case where G/K is of rank two. Let c : [0 , b ) → a be the curveparametrized by the arclength whose image is equal to M a , where b is the length of M a . Then it follows from (3 .
6) that M is of constant mean curvature κ if and onlyif the following relation holds: − X α ∈△ + m α √ εα ( c ′′ ( s ) / || c ′′ ( s ) || )tan( √ εα ( c ( s ))) + c ′′ ( s ) = κ ( s ∈ [0 , b )) . This relation is equivalent to the relation (26) of [Hs, Page 164]. W.Y. Hsiang ([Hs])derived some facts by using the relation (26).In particular, we consider the case where M is a geodesic sphere. Let S ( r )be the geodesic sphere of radius r ( >
0) centered at eK and S a ( r ) the sphere ofradius r centered at the origin in a . Assume that r is smaller than the firstconjugate radius of G/K in the case where
G/K is of compact type. Then we have S ( r ) = K · (Exp( S a ( r ) ∩ C )). Denote by N r the outward unit normal vector of S ( r ) and H r the mean curvature of S ( r ) for − N r . Take any Z ∈ S a ( r ) ∩ C . Since( N r ) Exp Z = r (exp Z ) ∗ ( Z ), it follows from (3 .
6) and (3 .
7) that(3 .
8) ( H r ) Exp Z = X α ∈△ + m α √ εα ( Z ) r tan( √ εα ( Z )) + l − r ( Z ∈ S a ( r ) ∩ C ) X α ∈△ + \△ Z + m α √ εα ( Z ) r tan( √ εα ( Z ))+ 1 r X α ∈△ Z + m α + l − ( Z ∈ S a ( r ) ∩ ∂C ) . Proof of Theorem A
In this section, we shall prove Theorem A stated in Introduction. We use thenotations in Sections 1-3. Let S = ( V, ( △ , { m α | α ∈ △} , ε )) , φ t and ι r be as inthe statement of Theorem A. Assume that there exists a solution φ t ( t ∈ [0 , T ))of the ( E S ) satisfying φ = ι r . Denote by b g t and ν t the induced metric and theoutward unit normal vector of φ t , respectively. Also, denote by b h t and b A t thesecond fundamental form and the shape operator of φ t for − ν t , respectively. Definethe sections b g (resp. b h ) of π ∗ S V ( r ) ( T (0 , S V ( r )) by b g ( x,t ) := ( b g t ) x (resp. b h ( x,t ) := ( b h t ) x )(( x, t ) ∈ S V ( r ) × [0 , T )), where π S V ( r ) denotes the natural projection of S V ( r ) × [0 , T )onto S V ( r ). Also, denote by b ∇ t the Riemannian connection of b g t and ∆ b g t the Laplaceoperator with respect to b g t . Define a function b r t ( t ∈ [0 , T )) over S V ( r ) by b r t ( Z ) := || φ t ( Z ) || ( Z ∈ S V ( r ))and a diffeomorphism b c t of S V ( r ) by b c t ( Z ) := rφ t ( Z ) || φ t ( Z ) || ( Z ∈ S V ( r )) . Then we can derive the following fact for the evolution of b r t . Lemma 4.1.
The functions { b r t } t ∈ [0 ,T ) satisfies the following evolution equation: (4 . ∂ b r∂t = R S V ( r ) (cid:0) || ∆ b g t φ t || + ρ S ,φ t (cid:1) dv b g t R S V ( r ) dv b g t − ( || ∆ b g t φ t || + ρ S ,φ t ) ! × b r t · || c t ∗ (grad t b r t ) || p || grad t b r t || + b r t || c t ∗ (grad t b r t ) || . Proof.
By a simple calculation, we have ∂ b r∂t = h D S ( φ t ) , b r t φ t i = R S V ( r ) (cid:0) || ∆ b g t φ t || + ρ S ,φ t (cid:1) dv b g t R S V ( r ) dv b g t − ( || ∆ b g t φ t || + ρ S ,φ t ) ! × h ν t , φ t i b r t ν t = − || grad t b r t || || c t ∗ (grad t b r t ) || p || grad t b r t || + b r t || c t ∗ (grad t b r t ) || · c t ∗ (grad t b r t )+ || c t ∗ (grad t b r t ) || p || grad t b r t || + b r t || c t ∗ (grad t b r t ) || · φ t . From these relations, we obtain the desired evolution equation. q.e.d.The following evolution equation holds for b h t . Lemma 4.2.
The families { b h t } t ∈ [0 , ∞ ) satisfies ∂ b h∂t − ∆ b g t b h t = b ∇ t d ( ρ S ,φ t ) + Tr( b A t ) b h t + R S V ( r ) (cid:0) || ∆ b g t φ t || + ρ S ,φ t (cid:1) dv b g t R S V ( r ) dv b g t − || ∆ b g t φ t || − ρ S ,φ t ! b h t ( b A t • , • ) . Proof.
For simplicity, set b H S t := || ∆ b g t φ t || + ρ S ,φ t and H S t := R S V ( r ) (cid:0) || ∆ b g t φ t || + ρ S ,φ t (cid:1) dv b g t R S V ( r ) dv b g t . By a simple calculation, we have ∂ b h∂t = b ∇ t d b H S t + ( H S t − b H S t ) b h t ( b A t • , • ) . Also, by using the Simon’s identity, we have∆ b g t b h t = b ∇ t d b H t + b H t b h t ( b A t • , • ) − Tr( b A t ) b h t . From these relations, we obtain the desired evolution equation. q.e.d.Also, we prepare the following lemma, which will be used in the proof of thestatement (iii) of Theorem B.
Lemma 4.3.
Let H S t and b H S t be as in the proof of Lemma 4.2. The family b H S t } t ∈ [0 , ∞ ) satisfies the following evolution equation: (4 . ∂ b H S ∂t − ∆ b g t b H S t = X α ∈△ + m α √ εα ( φ t ∗ (grad b g t b H S t ))tan( √ ε ( α ◦ φ t ))+( H S t − b H S t ) X α ∈△ + m α √ ε ( α ◦ ν t ) (3 cos ( √ ε ( α ◦ φ t )) − ( √ ε ( α ◦ φ t )) . Proof.
The family { b g t } t ∈ [0 , ∞ ) satisfies ∂ b g∂t = 2( H S t − b H S t ) b h t . From the evolution equation in Lemma 4.2 and this evolution equation, we have ∂ b H S ∂t − ∆ b g t b H S t = ∂ρ S ,φ t ∂t + 3( H S t − b H S t )Tr( b A t ) . On the other hand, we have ∂ρ S ,φ t ∂t = X α ∈△ + m α − ( H S t − b H S t ) √ ε ( α ◦ ν t ) sin ( √ ε ( α ◦ φ t )) + √ εα ( φ t ∗ (grad b g t b H S t ))tan( √ ε ( α ◦ φ t )) ! . From these relations, we can derive the desired evolution equation. q.e.d.By using Lemmas 4.1 and 4.2, we shall prove Theorem A.
Proof of Theorem A.
Since L t := φ t ( S V ( r )) ( t ∈ [0 , T )) are W -invariant, theirbarycenter are equal to the origin of V . Hence the barycenter ξ ( t ) of L t is equalto the origin of V and the diffeomorphisms e ( t ) in the barysentric system (3 . S V (1) under theidentification of T V and V . Hence the left-hand sides of the first and the secondrelations in (3 .
1) are equal to zero. On the other hand, it is clear that the right-handsides in the first and the second relations are equal to zero in our setting. Thus thefirst and the second relations in (3 .
1) (of [AF]) are trivial. Also, it is easy to showthat (4 .
1) corresponds to the third relation in (3 .
1) (of [AF]), where we regard b r t as a function over S V (1) under the natural identification of S V ( r ) and S V (1). Herewe note that the term E in the right-hand side of (3 .
1) (of [AF]) vainishes in oursetting beacuse E is defined by E = h w, ν − e i in Page 283 of [AF] and, in oursetting, w is equal to by the W -invariantness of φ t . According to Lemma 3.6 of13AF], there exists a positive constant R such that, if r < R , then the solution b r t of the evolution equation (4 .
1) satisfying the initial condition b r = r uniquely existsin infinite time. According to the discussion in Page 299(Line 3 from the bottom)-300(Line 8) of [AF], b r t i converges to a W -equivariant C ∞ -function b r ∞ over S V (1) (inthe C ∞ -topology) as t → ∞ for some sequence { t i } ∞ i =1 in [0 , ∞ ) with lim i →∞ t i = ∞ .This fact implies that the solution φ t of ( E S ) satisfying φ = ι r exists uniquely ininfinite time and that φ t i converges to a W -equivariant C ∞ -embedding φ ∞ of S V ( r )into V (in the C ∞ -topology) as i → ∞ . The positive constant δ (which correspondsto the above R ) in Lemma 3.6 of [AF] is smaller than the positive constant δ inLemma 3.2 of [AF] , δ is smaller than the half of the positive constant δ (= δ M ) inLemma 1.1 of [AF] and δ is smaller than the half of the positive constant δ definedin Page 257 of [AF]. Also, according to the definition of δ (see P257 of [AF]), we seethat the positive constant corresponding to δ is equal to r S in our setting. Thus R is smaller than r S . Denote by P ( b h t ) the right-hand side of the evolution equationin Lemma 4.2. Assume that v ∈ Ker ( b h t ) Z . We may assume that Z ∈ S V ( r ) ∩ C without loss of generality. Then we have(4 . P ( b h t ) Z ( v, v ) = ( b ∇ t d ( ρ S ,φ t )) Z ( v, v ) . Let γ be the ∇ t -geodesic in S V ( r ) with γ ′ (0) = v . Then we have( b ∇ t d ( ρ S ,φ t )) Z ( v, v ) = d ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ρ S ,φ t ( γ ( s ))= X α ∈△ + m α √ εα ( e ∇ ν t ◦ γ ∂∂s | s =0 ν t ∗ ( γ ′ ( s )))tan( √ εα ( φ t ( Z ))) − εα ( φ t ( Z )) α ( ν t ∗ ( v ))sin ( √ εα ( φ t ( Z ))) − √ ε α ( ν t ( Z )) α ( e ∇ φ t ◦ γ ∂∂s | s =0 φ t ∗ ( γ ′ ( s )))sin ( √ εα ( φ t ( Z )))+ 2 √ ε α ( φ t ∗ ( v )) α ( ν t ( Z ))sin ( √ εα ( φ t ( Z ))) tan( √ εα ( φ t ( Z ))) ! = X α ∈△ + m α √ εα ( φ t ∗ (( b ∇ tv b A t )( v ))) − b h t ( b A t ( v ) , v ) α ( ν t ( Z ))tan( √ εα ( φ t ( Z ))) − εα ( φ t ( Z )) α ( φ t ∗ ( b A t ( v )))sin ( √ εα ( φ t ( Z ))) − εα ( ν t ( Z )) b h t ( v, v )sin ( √ εα ( φ t ( Z )))+ 2 √ ε α ( φ t ∗ ( v )) α ( ν t ( Z ))sin ( √ εα ( φ t ( Z ))) tan( √ εα ( φ t ( Z ))) ! , where e ∇ φ t ◦ γ (resp. e ∇ ν t ◦ γ ) is the pullback connection of the connection e ∇ of V by14 t ◦ γ (resp. ν t ◦ γ ). Hence, since v ∈ Ker b h t , we obtain(4 .
4) ( ∇ t d ( ρ S ,φ t )) Z ( v, v ) = X α ∈△ + m α √ ε tan( √ εα ( φ t ( Z ))) × (cid:18) α ( φ t ∗ (( b ∇ tv b A t )( v ))) + 2 εα ( φ t ∗ ( v )) α ( ν t ( Z ))sin ( √ εα ( φ t ( Z ))) (cid:19) . According to the proof of Lemma 3.6 of [AF], we may assume thatsup t ∈ [0 , ∞ ) ||| φ t ( Z ) || < R , sup t ∈ [0 , ∞ ) || φ t ∗ ( b A t ( v )) || < ε and sup t ∈ [0 , ∞ ) || φ t ∗ ( v ) || < || v || + ε for sufficiently small positive constants ε and ε , where we note that the statementof Lemma 3.6 is done in the space W T •• . See Page 251-252 of [AF] about the definitionof W T •• , where we note that T = ∞ according to the statement of this lemma. Byimitating the discussion in the proof of Lemma 8.3 of [Hu1], it follows from theabove second inequality thatsup t ∈ [0 , ∞ ) || φ t ∗ (( b ∇ tv b A t )( v )) || < ε ′ , where ε ′ also is a sufficiently small positive constant because ε is sufficiently small.Also, we see that α ( ν t ( Z )) ≥ t ∈ [0 , ∞ )) because the statement of Lemma 3.6 isdone in the space W ∞•• . Hence, by taking R as a sufficiently small positive constant,we can show that the right-hand side of (4 .
4) is greater than or equal to zero in alltimes. Hence we have P ( b h t ) Z ( v, v ) ≥ t ∈ [0 , ∞ )). Therefore, since φ = ι r isstrictly convex (i.e., b h > b h t > φ t is strictly convex for all t ∈ [0 , ∞ ) and hence so is also φ ∞ . This completesthe proof. q.e.d. In this section, we shall prove Theorem B stated in Introduction. We use the no-tations in Sections 1-4. Let S = ( V, ( △ , { m α | α ∈ △} , ε )) , φ t and ι r be in thestatement of Theorem B.First we shall prove the statement (i) of Theorem B. Proof of (i) of Theorem B.
Let φ t ( t ∈ [0 , ∞ )) be the solution of ( E S ) satisfying φ = ι r . The existence of this flow is assured by Theorem A. Define a map f t of the15eodesic sphere S ( r ) := K · π ( S V ( r )) in G/K into
G/K by(5 . f t ( kπ ( Z )) := kπ ( φ t ( Z )) ( k ∈ K, Z ∈ S V ( r )) . Denote by N t the outward unit normal vector field of f t and A t , H t , H t the shapeoperator, the mean curvature and the average mean curvature of f t for the inwardunit normal vector field − N t , respectively. According to (3 . .
2) ( H t − H t ) N t ◦ π | S V ( r ) = D S ( φ t ) , where we use the fact that || ∆ b g t φ t || is the mean curvature of φ t . Here we note that(( H t − H t ) N t ) x ∈ T f t ( x ) T (= a = V ) ( x ∈ π ( S V ( r ))) and hence ( H t − H t ) N t ◦ π | S V ( r ) is regarded as a map from S V ( r ) to V . Since φ t ( t ∈ [0 , ∞ )) is the solution of( E S ) starting from ι r , it follows from (5 .
2) that { f t } t ∈ [0 , ∞ ) is the volume-preservingmean curvature flow starting from the inclusion map ι S ( r ) : S ( r ) ֒ → G/K . Hence,since r < R by the assumption, it follows from Theorem A that φ t i converges to astrictly convex embedding φ ∞ (in C ∞ -topology) as i → ∞ for some sequence { t i } ∞ i =1 in [0 , ∞ ) with lim i →∞ t i = ∞ and that φ t (0 ≤ t < ∞ ) remain to be strictly convexand hence so is also φ ∞ . Let f ∞ be the map the geodesic sphere S ( r ) into G/K defined as in (5 .
1) for φ ∞ instead of φ t . Since φ ∞ is strictly convex, it follows from(3 .
4) and (3 .
5) that so is also f ∞ . Also, since { f t } t ∈ [0 , ∞ ) is the volume-preservingmean curvature flow, it follows from Main theorem of [AF] that f ∞ is of constantmean curvature. This completes the proof. q.e.d.To prove the statements (ii) and (iii) of Theorem B, we prepare the followinglemma. Lemma 5.1.
Let Z max (resp. Z min ) be one of maximum (resp. minimum) pointsof ρ S ,ι r (hence H ◦ π | S V ( r ) ). Then the curves t φ t ( Z max ) ( t ∈ [0 , ∞ )) and t φ t ( Z min ) ( t ∈ [0 , ∞ )) are described as (5 . φ t ( Z max ) = (cid:18) r Z t ( H t − H t ( π ( Z max ))) dt (cid:19) Z max ( t ∈ [0 , ∞ )) ,φ t ( Z min ) = (cid:18) r Z t ( H t − H t ( π ( Z min ))) dt (cid:19) Z min ( t ∈ [0 , ∞ )) . Proof.
Denote by b g t and ν t the induced metric and the outward unit normal vectorfield of φ t , respectively. First we shall calculate ∂ν∂t . Since h ν t , ν t i = 1, we have h ∂ν∂t , ν t i = 0. Hence ( ∂ν∂t ) Z is tangent to φ t ( S V ( r )) at φ t ( Z ) for each Z ∈ S V ( r ).16ix ( Z , t ) ∈ S V ( r ) × [0 , ∞ ). Let { e i } l − i =1 be an orthonormal base of T Z S V ( r ) withrespect to ( b g t ) Z and e i the tangent vector field of S V ( r ) × [0 , ∞ ) along { Z }× [0 , ∞ )defined by ( e i ) ( Z ,t ) := ( e i ) L ( Z ,t ) (( Z , t ) ∈ { Z } × [0 , ∞ )), where ( e i ) L ( Z ,t ) is thehorizontal lift of e i to ( Z , t ) (with respect to the natural projection of S V ( r ) × [0 , ∞ )onto S V ( r )). Then we have(5 . (cid:18) ∂ν∂t (cid:19) ( Z ,t ) = l − X i =1 *(cid:18) ∂ν∂t (cid:19) ( Z ,t ) , φ t ∗ ( e i ) + φ t ∗ ( e i )= − l − X i =1 * ( ν t ) Z , (cid:18) ∂φ t ∗ ( e i ) ∂t (cid:19) ( Z ,t ) + φ t ∗ ( e i )= − l − X i =1 * ( ν t ) Z , ∂∂t ( e i φ ) (cid:12)(cid:12)(cid:12)(cid:12) t = t + φ t ∗ ( e i )= − l − X i =1 * ( ν t ) Z , e i ∂φ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = t !+ φ t ∗ ( e i )= − l − X i =1 h ( ν t ) Z , e i ( H t − ( H t ◦ π | S V ( r ) ))( ν t ) Z i φ t ∗ ( e i )= l − X i =1 e i ( H t ◦ π | S V ( r ) ) φ t ∗ ( e i )= l − X i =1 b g t ((grad b g t ( H t ◦ π | S V ( r ) )) Z , e i ) φ t ∗ ( e i )= ( φ t ) ∗ ((grad b g t ( H t ◦ π | S V ( r ) )) Z ) , where we use [ ∂∂t , e i ] = 0.Now we shall derive (5 .
3) in terms of (5 . C ∞ -curve t Z max t ( t ∈ [0 , ε )) such that Z max0 = Z max and that Z max t is a maximumpoint of H t ◦ π | S V ( r ) for each t ∈ [0 , ε ), where ε is a positive constant. Similarly,there exists a C ∞ -curve t Z min t ( t ∈ [0 , b ε )) such that Z min0 = Z min and that Z min t is a minimum point of H t ◦ π | S V ( r ) for each t ∈ [0 , b ε ), where b ε is a positiveconstant. According to (5 . t ∈ [0 , ε ), ∂ || φ t || ∂t | ( Z max t ,t ) < Z max t isa minimum point of Z d || φ t || ∂t | ( Z,t ) . On the other hand, according to (5 . d ( ν t ) dt | Z max t = 0, that is, ( ν t ) Z max t = ( ν ) Z max ( t ∈ [0 , ε )). From these facts, wecan derive that φ t ( Z max t ) = λ ( t ) Z max ( t ∈ [0 , ε )) for some positive funcion λ over[0 , ε ) and that Z max t = Z max ( t ∈ [0 , ε )) holds. Similarly, we we can show that φ t ( Z min t ) = λ ( t ) Z min ( t ∈ [0 , b ε )) for some positive funcion λ over [0 , b ε ) and that17 min t = Z min ( t ∈ [0 , b ε )) holds. Hence it follows from (5 .
3) that ∂λ ∂t = 1 r (cid:0) H t − H t ( π ( Z max )) (cid:1) . and ∂λ ∂t = 1 r (cid:0) H t − H t ( π ( Z min )) (cid:1) . Therefore we can derive φ t ( Z max ) = (cid:18) r Z t ( H t − H t ( π ( Z max ))) dt (cid:19) Z max ( t ∈ [0 , ε )) φ t ( Z min ) = (cid:18) r Z t ( H t − H t ( π ( Z min ))) dt (cid:19) Z min ( t ∈ [0 , b ε )) . It is easy to show that these relations hold over [0 , ∞ ). This completes the proof.q.e.d.According to this lemma, in the case where △ is of type ( a ), the flow φ t ( S V ( r ))is as in Figure 3 or 4 for example. By using this lemma, we prove the statements(ii) and (iii) of Theorem B. Proof of (ii) and (iii)
Theorem B.
Let P be a two-plane in V with P max = ∅ and P min = ∅ . Take Z max ∈ P max and Z min ∈ P min ,Z max . According to Lemma 5.1, φ t ( Z max ) and φ t ( Z min ) are as in (5 . θ be the angle between −−−−→ Z max and −−−−→ Z min .From the convexity of φ t ( S V ( r )), we can derive that the part between φ t ( Z max ) and φ t ( Z min ) of the curve φ ∞ ( S V ( r )) ∩ P ∩ C is included by ( B V ( r cos θ ) \ B V ( r cos θ )) ∩ P ∩ C (see Figure 5). From this fact and the definition of θ , we can derive M ⊂ B ( r cos θ ) \ B ( r cos θ ). Thus the statement (ii) of Theorem B follows. Since r is asufficiently small positive constant smaller than R , max S V ( r ) b r t is sufficiently smallfor all t ∈ [0 , ∞ ). Hence we have 3 cos ( √ ε ( α ◦ φ t )) ≥ t ∈ [0 , ∞ ). Therefore,according to the maximum principle, it follows from the evolution equation (4 . b H S t that min S V ( r ) b H S ≤ b H S t ≤ max S V ( r ) b H S , which implies that min S ( r ) H r ≤ H t ≤ max S ( r ) H r and hence min S ( r ) H r ≤ H M ≤ max S ( r ) H r . On the other hand,according to (3 . η min ( r ) ≤ H r ≤ η max ( r ). Therefore we obtain η min ( r ) ≤ H M ≤ η max ( r ). q.e.d.18 min Z max zoom inzoom in Z min Z max VS V ( r ) φ t ( S V ( r )) C φ t ( Z min ) φ t ( Z max )The case where △ is of type ( a ). Figure 3. Z max Z min zoom inzoom in Z min VS V ( r ) φ t ( S V ( r )) C Z max φ t ( Z min ) φ t ( Z max )The case where △ is of type ( a ). Figure 4. φ t ( Z min ) can moveThe range which φ t ( Z max ) can move C ∩ PPS V ( r ) ∩ P Z max Z min S V ( r cos θ ) ∩ PS V ( r cos θ ) ∩ Pθ Figure 5.
Proof of Corollary C.
Let θ be as in the statement (ii) of Theorem B. Clearly wehave θ ≤ θ G/K . Hence we obtain the desired inclusion relation from (ii) of TheoremB. q.e.d.
Proof of Corollary D.
Since
G/K is of rank two, we have θ G/K = π G/K is of type a ) π G/K is of type b ) π G/K is of type g ).In the case where △ is of type ( a ), it is symmetric for two simple roots withconsidering the multiplicities. Hence, in this case, it follows that θ in Theorem Bare smaller than or equal to the half of θ G/K (= the half of the length of S V (1) ∩ C ),that is, they are smaller than or equal to π (see Figure 6). Therefore, from (ii) ofTheorem B, we obtain M ⊂ B (cid:18) r √ (cid:19) \ B √ r ! ( △ : ( a ) − type or ( g ) − type) B ( √ r ) \ B (cid:18) r √ (cid:19) ( △ : ( b ) − type) . q.e.d.20oom inThe range which φ t ( Z max ) can moveThe range which φ t ( Z min ) can move π π CS V ( r ) V Z max Z min θ S V ( r cos θ ) S V ( r cos θ )The case where △ is of ( a )-type Figure 6.References [AF] N.D. Alikakos and A. Freire, The normalized mean curvature flow for a small bubblein a Riemannian manifold, J. Differential Geom. (2003) 247-303.[CM] E. Cabezas-Rivas and V. Miquel, Volume preserving mean curvature flow in thehyperbolic space, Indiana Univ. Math. J. (2007) 2061-2086.[GT] O. Goertsches and G. Thorbergsson, On the Geometry of the orbits of Hermann actions,Geom. Dedicata (2007) 101-118.[He] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press,New York, 1978.[Hs] W.Y. Hsiang, On soap bubbles and isoperimetric regions in non-compact symmetricspaces, I, Tohoku Math. J. (1992) 151-175.[HH] W.T. Hsiang and W.Y. Hsiang, On the uniqueness of isoperimetric solutions andimbedded soap bubbles in non-compact symmetric spaces, I, Invent. Math. (1989)39-58.[Hu1] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. DifferentialGeom. (1984) 237-266.[Hu2] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their meancurvature, Invent. math. (1986) 463-480.[Hu3] G. Huisken, The volume preserving mean curvature flow, J. reine angew. Math. (1987) 35-48.[K1] N. Koike, Actions of Hermann type and proper complex equifocal submanifolds,Osaka J. Math. (2005) 599-611.[K2] N. Koike, Collapse of the mean curvature flow for equifocal submanifolds, Asian J.Math. (2011) 101-128. PT] R.S. Palais and C.L. Terng, Critical point theory and submanifold geometry, LectureNotes in Math. , Springer, Berlin, 1988.Department of Mathematics, Faculty of ScienceTokyo University of Science, 1-3 KagurazakaShinjuku-ku, Tokyo 162-8601 Japan([email protected]), Springer, Berlin, 1988.Department of Mathematics, Faculty of ScienceTokyo University of Science, 1-3 KagurazakaShinjuku-ku, Tokyo 162-8601 Japan([email protected])