CUBO A Mathematical Journal Vol.20, No¯ 3, (13–29). October 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000300013 Mean curvature flow of certain kind of isoparametric foliations on non-compact symmetric spaces Naoyuki Koike Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka Shinjuku-ku, Tokyo 162-8601, Japan koike@rs.kagu.tus.ac.jp ABSTRACT In this paper, we investigate the mean curvature flows starting from all leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal leaf of the foliation in C∞-topology. These results give the geometric information between the leaves. RESUMEN En este art́ıculo, investigamos el flujo por curvatura media comenzando desde cualquier hoja de una foliación isoparamétrica dada por la acción de un cierto grupo soluble en un espacio simétrico de tipo no-compacto. Demostramos que el flujo por curvatura media comenzando desde cualquier hoja no mı́nima de la foliación existe para tiempo infinito, si la foliación no admite hojas mı́nimas, entonces el flujo es asintótico al flujo autosemejante comenzando desde otra hoja; en cambio si el flujo admite una hoja mı́nima (en este caso, se muestra que la hoja mı́nima es única), entonces el flujo converge a dicha hoja mı́nima de la foliación en la topoloǵıa C∞. Estos resultados entregan información geométrica entre las hojas. Keywords and Phrases: error function based activation function, multivariate quasi-interpolation neural network approximation, Kantorovich-Shilkret type operator. http://dx.doi.org/10.4067/S0719-06462018000300013 14 Naoyuki Koike CUBO 20, 3 (2018) 2010 AMS Mathematics Subject Classification: 41A17, 41A25, 41A30, 41A35. CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 15 1 Introduction In [6], we proved that the mean curvature flow starting from any non-minimal compact isopara- metric (equivalently, equifocal) submanifold in a symmetric space of compact type collapses to one of its focal submanifolds in finite time. Here we note that parallel submanifolds and focal ones of the isoparametric submanifold give an isoparametric foliation consisting of compact leaves on the symmetric space, where an isoparametric foliation means a singular Riemannian foliation satisfying the following conditions: (i) The mean curvature form is basic, (ii) The regular leaves are submanifolds with section. A singular Riemannian foliation satisfying only the first condition is called a generalized isoparamet- ric foliation. Recently, M. M. Alexandrino and M. Radeschi [1] investigated the mean curvature flow starting from a regular leaf of a generalized isoparametric foliation consisting of compact leaves on a compact Riemannian manifold. In particular, they [1] generalized our result to the mean curvature flow starting from a regular leaf of the foliation in the case where the foliation is isoparametric and the ambient space curves non-negatively. On the other hand, we [7] proved that the mean curvature flow starting from a certain kind of non-minimal (not necessarily compact) isoparametric submanifold in a symmetric space of non-compact type (which curves non-positively) collapses to one of its focal submanifolds in finite time. Here we note that the isoparametric folia- tion associated with this isoparametric submanifold consists of curvature-adapted leaves. See the next paragraph about the definition of the curvature-adaptedness. In this paper, we study the mean curvature flow starting from leaves of the isoparametiric foliation given by the action of a certain kind of solvable subgroup (see Examples 1 and 2) of the (full) isometry group of a symmetric space of non-compact type. Here we note that this isoparametric foliation consists of (not necessarily curvature-adapted) non-compact regular leaves. We shall explain the solvable group action which we treat in this paper. Let G/K be a symmetric space of non-compact type, g = k + p (k := Lie K) be the Cartan decomposition associated with the symmetric pair (G, K), a be the maximal abelian subspace of p, ã be the Cartan subalgebra of g containing a and g = k + a + n be the Iwasawa’s decomposition. Let A, Ã and N be the connected Lie subgroups of G having a, ã and n as their Lie algebras, respectively. Let π : G → G/K be the natural projection. Given metric. In this paper, we give G/K the G-invariant metric induced from the restriction B|p×p of the Killing form B of g to p × p. The symmetric space G/K is identified with the solvable group AN with a left-invariant metric through π|AN. Fix a lexicographic ordering of a. Let g = g0 + ∑ λ∈△ gλ, p = a + ∑ λ∈△+ pλ and 16 Naoyuki Koike CUBO 20, 3 (2018) k = k0 + ∑ λ∈△+ kλ be the root space decompositions of g, p and k with respect to a, where we note that gλ = {X ∈ g | ad(a)X = λ(a)X for all a ∈ a} (λ ∈ △), pλ = {X ∈ p | ad(a)2X = λ(a)2X for all a ∈ a} (λ ∈ △+), kλ = {X ∈ k | ad(a)2X = λ(a)2X for all a ∈ a} (λ ∈ △+ ∪ {0}). Note that n = ∑ λ∈△+ gλ. Let G = KAN be the Iwasawa decomposition of G. Now we shall give examples of a solvable group contained in AN whose action on G/K(= AN) is (complex) hyperpolar. Since G/K is of non-compact type, π gives a diffeomorphism of AN onto G/K. Denote by 〈 , 〉 the left-invariant metric of AN induced from the metric of G/K by π|AN. Also, denote by 〈 , 〉G the bi-invariant metric of G induced from the Killing form B. Note that 〈 , 〉 6= ι∗〈 , 〉G, where ι is the inclusion map of AN into G. Denote by Exp the exponential map of the Riemannian manifold AN(= G/K) at e and by expG the exponential map of the Lie group G. Let l be a r-dimensional subspace of a + n and set s := (a + n) ⊖ l, where (a + n) ⊖ l denotes the orthogonal complement of l in a + n with respect to 〈 , 〉e (e : is the identity element of G). According to the result in [5], if s is a subalgebra of a + n and lp := prp(l) (prp : the orthogonal projection of g onto p) is abelian, then the S-action (S := expG(s)) gives an isoparametric foliation without singular leaf. We [5] gave examples of such a subalgebra s of a + n. Example 1. Let b be a r(≥ 1)-dimensional subspace of a and sb := (a + n) ⊖ b. It is clear that bp(= b) is abelian and that sb is a subalgebra of a + n. Example 2. Let {λ1, · · · , λk} be a subset of a simple root system Π of △ such that Hλ1, · · · , Hλk are mutually orthogonal, b be a subspace of a ⊖ Span{Hλ1, · · · , Hλk} (where b may be {0}) and li (i = 1, · · · , k) be a one-dimensional subspace of RHλi + gλi with li 6= RHλi, where Hλi is the element of a defined by 〈Hλi, ·〉 = λi(·) and RHλi is the subspace of a spanned by Hλi. Set l := b + k∑ i=1 li. Then, it is shown that lp is abelian and that sb,l1,··· ,lk := (a + n) ⊖ l is a subalgebra of a + n. In Example 2, a unit vector of li is described as 1 cosh(||λi||ti) ξi − 1 ||λi|| tanh(||λi||ti)Hλi for a unit vector ξi of gλi and some ti ∈ R, where ||λi|| := ||Hλi||. Then we denote li by lξi,ti if neces- sary and set ξiti := 1 cosh(||λi||ti) ξi − 1 ||λi|| tanh(||λi||ti)Hλi. Set Sb := expG(sb) and Sb,,l1,··· ,lk := expG(sb,l1,··· ,lk). Denote by Fb and Fb,l1,··· ,lk the isoparametric foliations given by the Sb-action and the Sb,l1,··· ,lk-one, respectively. A submanifold in a Riemannian manifold is said to be curvature-adapted if, for each normal vector v of the submanifold, the normal Jacobi operator R(v) := R(·, v)v preserves the tangent space of the submanifold invariantly and the restriction of R(v) to the tangent space commutes with the shape operator Av, where R is the curvature tensor of the ambient Riemannian manifold. According to the results in[5], the following facts hold for CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 17 isoparametric foliations Fb and Fb,l1,··· ,lk: (i) All leaves of Fb are curvature-adapted. (ii) Let λ1, · · · , λk (∈ △+) be as in Example 2. If the root system △ of G/K is non-reduced and 2λi0 ∈ △+ for some i0 ∈ {1, · · · , k}, then all leaves of Fb,l1,··· ,lk are not curvature-adapted. (iii) If b 6= {0}, then Fb,l1,··· ,lk admits no minimal leaf. On the other hand, if b = {0}, then this action admits the only minimal leaf. (iv) Let l1, · · · , lk be as in Example 2 and li (i = 1, · · · , k) be the orthogonal projection of li onto gλi. Then Fb,l1,··· ,lk is congruent to Fb,l1,··· ,lk. In more detail, we have Lb·γ ξ1 (t1)· ··· ·γξk(tk) (Sb,l1,··· ,lk · e) = Sb,l1,··· ,lk · (b · γξ1(t1) · · · · · γξk(tk)), where γξi (i = 1, · · · , k) is the geodesic in AN(= G/K) with γ′ξi(0) = ξ i, b is an element of exp(b) and Lb·γ ξ1 (t1)· ··· ·γξk(tk) is the left translation by b · γξ1(t1) · · · · · γξk(tk). For example, in case of k = 1 and b = e, the positional relation among the leaves of these foliations is as in Figure 1. Sb,l1 · e S b,l1 · e Sb,l1 · γξ1(t1) = Lγξ1(t1)(Sb,l1 · e) Exp(b + l1) Exp(b + l1) γξ1 γξ1(t1) γξ1 t1 e Figure 1. According to the above facts (i) and (ii), the leaves of Fb,l1,··· ,lk give examples of interesting isoparametric submanifolds in G/K. In this paper, we shall prove the following facts for the mean curvature flows starting from the non-minimal leaves of F b,l1,··· ,lk . Theorem A. Assume that b 6= {0}. Let M be any leaf of F b,l1,··· ,lk . and Mt (0 ≤ t < T) be the mean curvature flow starting from M. Then the following statements (i) − (iii) hold. (i) T = ∞ holds. (ii) If M passes through exp(b), then the mean curvature flow Mt is self-similar. (iii) If M does not pass through exp(b), then the mean curvature flow Mt asymptotes the mean curvature flow starting from the leaf of F b,l1,··· ,lk passing through a point of exp(b). Remark 1.1. The mean curvature flow starting from any leaf of Fb is self-similar. 18 Naoyuki Koike CUBO 20, 3 (2018) Exp(l)Exp(b) M1 M3M2 The mean curvature flows starting from leaves M1 and M3 of Fb,̄l1,··· ,̄lk (b 6= {0}) asymptotes the mean curvature flow (which is self-similar) starting from a leaf M2 of Fb,̄l1,··· ,̄lk. Figure 2. Also, in case of b = {0}, we obtain the following fact. Theorem B. Let M be a leaf of F {0},l1,··· ,lk -action other than S {0},l1,··· ,lk · e and Mt (0 ≤ t < T) be the mean curvature flow starting from M. Then the following statements (i) − (ii) hold. (i) T = ∞ holds. (ii) Mt convergres to the only minimal leaf S{0},l1,··· ,lk · e (in C ∞-topology) as t → ∞. Exp(l) M1 M3 M2 The mean curvature flows starting from leaves M1, M2 and M3 of e F{0},̄l1,··· ,̄lk converge to the only minimal leaf M 0 of F{0},̄l1,··· ,̄lk. M0 Figure 3. The following question arises naturally. Question. Let F be an isoparametric foliation consisting of non-compact regular leaves on a non- positively curved Riemannian manifold. Assume that the leaves of F are cohomogeneity compact (i.e., each leaf L is invariant under some subgroup action HL of the isometry group of the ambient space and the quotient space L/HL is compact). In what case, does the result similar to Theorem A or B hold for F? CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 19 2 Mean curvature flow. In this section, we shall recall the notion of the mean curvature flow. Let ft’s (t ∈ [0, T)) be a one- parameter C∞-family of immersions of a manifold M into a Riemannian manifold M̃, where T is a positive constant or T = ∞. Define a map F : M×[0, T) → M̃ by F(x, t) = ft(x) ((x, t) ∈ M×[0, T)). Denote by π the natural projection of M × [0, T) onto M. For a vector bundle E over M, denote by π∗E the induced bundle of E by π. Also, denote by Ht and gt the mean curvature vector field and the induced metric of ft, respectively. Define a section g of π ∗(T(0,2)M) by g(x,t) := (gt)x ((x, t) ∈ M×[0, T)) and sections H of F∗TM̃ by H(x,t) := (Ht)x ((x, t) ∈ M×[0, T)), where T(0,2)M is the tensor bundle of degree (0, 2) of M and TM̃ is the tangent bundle of M̃. The family ft’s (0 ≤ t < T) is called a mean curvature flow if it satisfies (1.1) F∗ ( ∂ ∂t ) = H. In particular, if ft’s are embeddings, then we call Mt := ft(M)’s (0 ∈ [0, T)) rather than ft’s (0 ≤ t < T) a mean curvature flow. See [3], [4] and [2] and so on about the study of the mean curvature flow (treated as the evolution of an immersion). 3 The non-curvature-adaptedness of the leaves. In [5], we proved the following statement: (∗) If the root system △ of G/K is non-reduced and 2λi0 ∈ △+ for some i0 ∈ {1, · · · , k}, then all leaves of F b,l1,··· ,lk are not curvature-adapted. (see the statement (ii) of Proposition 3.5 in [5]). However, there is a gap in the second-half part of the proof. In this section, we shall close the gap by recalculating the normal Jacobi operators of the leaves (see Proposition 3.5). We shall use the notations in Introduction. According to the fact (iv) stated in Introduction, we have Lb·γ ξ1 (t1)· ··· ·γξk(tk) (Sb,l1,··· ,lk · e) = Sb,l1,··· ,lk · (b · γξ1(t1) · · · · · γξk(tk)). Hence we suffice to show that the leaves Sb,l1,··· ,lk · e’s are not curvature-adapted. As stated in Example 2, we set ξiti := 1 cosh(||λi||ti) ξi − 1 ||λi|| tanh(||λi||ti)Hλi. For the shape operator of Sb,l1,··· ,lk · e, we showed the following facts (see Lemma 3.2 of [5]). Lemma 3.1[5]. Let A be the shape tensor of Sb,l1,··· ,lk · e (⊂ AN). Then, for Aξ0 (ξ0 ∈ b) and Aξiti (i = 1, · · · , k), the following statements (i) ∼ (vii) hold: (i) For X ∈ a ⊖ (b + k∑ i=1 RHλi), we have Aξ0X = Aξiti X = 0 (i = 1, · · · , k). 20 Naoyuki Koike CUBO 20, 3 (2018) (ii) For X ∈ Ker(ad(ξi)|gλi ) ⊖ Rξ i, we have Aξ0X = 0 and Aξiti X = −||λi|| tanh(||λi||ti)X. (iii) Assume that 2λi ∈ △+. For X ∈ g2λi, we have Aξ0([θξi, X]) = 0 and Aξiti X = −2||λi|| tanh(||λi||ti)X − 1 2 cosh(||λi||ti) [θξi, X], Aξi ti ([θξi, X]) = − ||λi|| 2 cosh(||λi||ti) X − ||λi|| tanh(||λi||ti)[θξ i, X], where θ is the Cartan involution of g with Fix θ = k. (iv) For X ∈ (Rξi + RHλi) ⊖ li, we have Aξ0X = 0 and Aξiti X = −||λi|| tanh(||λi||ti)X. (v) For X ∈ (gλj ⊖ Rξj) + ((Rξj + RHλj) ⊖ lj) + g2λj (j 6= i), we have Aξ0X = Aξiti X = 0. (vi) For X ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have Aξ0X = µ(ξ0)X. (vii) Let ki := exp ( π√ 2||λi|| (ξi + θξi) ) , where exp is the exponential map of G. Then Ad(ki)◦ Aξiti = −Aξiti ◦ Ad(ki) holds over n⊖ k∑ i=1 (gλi +g2λi), where Ad is the adjoint representation of G. Remark 3.1. If λi ∈ △+, then we have ||λi|| = √ 2 from how to choose the metric of G/K (see Introduction). According to (5.3) in Page 310 of [8], we have the following fact. Lemma 3.2[8]. Let X and Y be left-invariant vector fields on AN and ∇ be the Levi-Civita connection of the left-invariant metric 〈 , 〉 of AN. Then we have (3.2) ∇XY = 1 2 ( [X, Y] − ad(X)∗(Y) − ad(Y)∗(X) ) , where ad(X)∗ (resp. ad(Y)∗) is the adjoint operator of ad(X) (resp. ad(Y)) with respect to 〈 , 〉e and (•)a+n is the the (a + n)-component of (•). Let pr1a+n (resp. pr 2 a+n) be the projection of g onto a + n with respect to the decomposition g = k + (a + n) (resp. g = (k0 + ∑ λ∈△+ pλ) + (a + n)). We [5] showed the following facts (see the proof of Lemma 3.2 in [5]). Lemma 3.3[5]. (i) For any H ∈ a, we have (3.3) ad(H)∗ = ad(H). CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 21 (ii) For any X ∈ gλ, we have (3.4) ad(X)∗ = −pra+n ◦ ad(θX) =    0 on a −〈X, ·〉e ⊗ Hλ − prn ◦ pr1a+n ◦ ad(Xk) +prn ◦ pr2a+n ◦ ad(Xp) on n, where (•)k (resp. (·)p) denotes the k-component (resp. p-component) of (•). According to (3.4), we have (3.5) ad(X)∗(Y) =    0 (λ − µ ∈ △+) −〈X, Y〉Hλ (λ = µ) −[θX, Y] (µ − λ ∈ △+) 0 (λ − µ /∈ △ ∪ {0}) for any X ∈ gλ (λ ∈ △+) and any Y ∈ gµ (µ ∈ △+). For each X ∈ a + n, we denote by X̃ the left-invariant vector field on AN with (X̃)e = X. By using Lemma 3.2, (3.3), (3.4) and (3.5), we can derive the facts directly. Lemma 3.4. For any unit vector Xλ, Yλ of gλ (λ ∈ △+) and Hλ (λ ∈ △+), we have ∇ H̃λ H̃µ = ∇H̃λX̃µ = 0, ∇X̃λH̃µ = −λ(Hµ)X̃λ (λ, µ ∈ △+) and ∇ X̃λ Ỹµ =    1 2 ( [X̃λ, Ỹµ] + θ̃[Yµ, θXλ] ) (λ − µ ∈ △+) 1 2 [X̃λ, Ỹµ] + 〈X̃λ, Ỹµ〉H̃λ (λ = µ) 1 2 ( [X̃λ, Ỹµ] + θ̃[Xλ, θYµ] ) (µ − λ ∈ △+) 1 2 [X̃λ, Ỹµ] (λ − µ /∈ △ ∪ {0}) From Lemma 3.4 and (3.5), we can derive the following facts for the normal Jacobi operators by somewhat long calculations. Proposition 3.5. Let R be the curvature tensor of AN(= G/K). Then, for R(ξ0) (ξ 0 ∈ b) and R(ξiti) (i = 1, · · · , k), the following statements (i) ∼ (vi) hold: (i) For X ∈ a ⊖ (b + k∑ i=1 RHλi), we have R(ξ0)(X) = R(ξ i ti )(X) = 0 (i = 1, · · · , k). 22 Naoyuki Koike CUBO 20, 3 (2018) (ii) For X ∈ Ker(ad(ξi)|gλi ) ⊖ Rξ i, we have R(ξ0)(X) = 0 and R(ξ i ti )(X) = ||λi|| 2 2 (1 − 3 tanh2(||λi||ti))X. (iii) Assume that 2λi ∈ △+ (hence ||λi|| = √ 2). For X ∈ g2λi, we have R(ξ0)(X) = R(ξ0)([θξ i, X]) = 0 and R(ξiti)(X) = −||λi|| 2(1 + 3 tanh2(||λi||ti))X − 3||λi|| tanh(||λi||ti) 2 cosh(||λi||ti) [θξi, X] R(ξiti)([θξ i, X]) = − 6||λi|| tanh(||λi||ti) cosh(||λi||ti) X + √ 2||λi|| 4 (1 − 3 tanh2(||λi||ti))[θξ i, X]. (iv) For X ∈ (Rξi + RHλi) ⊖ li, we have R(ξ0)(X) = 0 and R(ξiti)(X) = −||λi|| 2X. (v) For X ∈ (gλj ⊖ Rξj)+((Rξj +RHλj)⊖ lj)+g2λj (j 6= i), we have R(ξ0)(X) = R(ξiti)(X) = 0. (vi) For X ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have R(ξ0)(X) = −µ(ξ0)2X. From Lemma 3.1 and Proposition 3.5, we can derive the following facts directly. Proposition 3.6. For [Aξ0, R(ξ0)] (ξ0 ∈ b) and [Aξiti , R(ξ i ti )] (i = 1, · · · , k), the following state- ments (i) ∼ (vi) hold: (i) For X ∈ a ⊖ (b + k∑ i=1 RHλi), we have [A, R(ξ0)](X) = [Aξiti , R(ξiti)](X) = 0 (i = 1, · · · , k). (ii) For X ∈ Ker(ad(ξi)|gλi ) ⊖ Rξ i, we have [Aξ0, R(ξ0)](X) = [Aξiti , R(ξiti)](X) = 0. (iii) Assume that 2λi ∈ △+ (hence ||λi|| = √ 2). For X ∈ g2λi, we have [Aξ0, R(ξ0)](X) = [Aξ0, R(ξ0)]([θξ i, X]) = 0 and [Aξi ti , R(ξiti)](X) = − 3 2 cosh3( √ 2ti) [θξi, X] [Aξiti , R(ξiti)]([θξ i, X]) = − 6 cosh3( √ 2ti) X. (iv) For X ∈ (Rξi + RHλi) ⊖ li, we have [Aξ0, R(ξ0)](X) = [Aξiti , R(ξ i ti )](X) = 0. (v) For X ∈ (gλj ⊖ Rξj) + ((Rξj + RHλj) ⊖ lj) + g2λj (j 6= i), we have [Aξ0, R(ξ0)](X) = [Aξiti , R(ξiti)](X) = 0. (vi) For X ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have [Aξ0, R(ξ0)](X) = [Aξiti , R(ξ i ti )](X) = 0. From (iv) of Proposition 3.6, we can derive the statement (∗). Also, we [5] showed the following fact in terms of Lemma 3.1. CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 23 Proposition 3.7[5]. If b = {0}, then Fb,l ξ1,t1 ,··· ,l ξk,tk admits the only minimal leaf. 4 Proof of Theorem A In this section, we shall prove Theorem A. We use the notations in Sections 1 and 3. Note that Exp|a = exp |a and Exp|n 6= exp |n. Set Σ := Exp(T⊥e Sb,l1,··· ,lk · e)(= Exp(b + R{ξ 1, · · · , ξk})), which is the flat section of the S b,l1,··· ,lk -action through e. Each leaf of F b,l1,··· ,lk meets Σ at the only one point. That is, Σ is regarded as the leaf space of this foliation. For ξ0 ∈ b and ti ∈ R (i = 1, · · · , k), we set xξ0,t1,··· ,tk := Expξ0 · γξ1(t1) · · · · · γξk(tk). Also, denote by D ds (•) the covariant derivative of vector fields (•) along curves in AN (with respect to the left-invariant metric). The following fact is well-known about the geodesics in rank one symmetric spaces of non-compact type but we shall give the proof. Lemma 4.1. The velocity vector γ′ ξi (s) (i = 1, · · · , k) is described as (4.1) γ′ ξi (s) = 1 cosh(||λi||s) (ξ̃i)γ ξi (s) − tanh(||λi||s) ||λi|| (H̃λi)γξi (s) and γ′ξ0(s) is described as (4.2). γ′ξ0(s) = (ξ̃0)γξ0(s) Proof. Set Y(s) := 1 cosh(||λi||s) (ξ̃i)γ ξi (s) − tanh(||λi||s) ||λi|| (H̃λi)γξi(s) . It is clear that Y(0) = ξi. By using Lemma 3.4, we can show D ds Y = 0. Hence we obtain Y(s) = γ′ ξi (s). Also, it is clear that (ξ̃0)γξ0 (0) = ξ0. By using Lemma 3.4, we can show D ds (ξ̃0)γξ0 (s) = 0. Hence we obtain (ξ̃0)γξ0 (s) = γ′ξ0(s). q.e.d. Next we shall show the following fact. Lemma 4.2. The point xξ0,t1,··· ,tk belongs to Σ. Proof. It is clear that Exp(ξ0) belongs to Σ. First we shall show that Exp(ξ0) · γξ1(t1) belongs to Σ. Let γξ0 be the geodesic in AN with γ ′ ξ0 (0) = ξ0. Since γξ1 is a geodesic in AN and LExp(ξ0) is an isometry of AN, LExp(ξ0) ◦ γξ1 is a geodesic in AN. Hence we suffice to show that (LExp(ξ0) ◦γξ1)′(0) = (ξ̃1)Exp(ξ0) is tangent to Σ. Denote by ξ̂1 the parallel vector field along γξ0. Take orthonormal bases {eλ1, · · · , eλmλ} of gλ (λ ∈ △+). Also, take an orthonormal base {e 0 1, · · · , e0r} 24 Naoyuki Koike CUBO 20, 3 (2018) of a. We describe ξ̂1 as ξ̂1(s) = r∑ i=1 a0i (s)(ẽ 0 i )γξ0 (s) + ∑ λ∈△+ mλ∑ i=1 aλi (s)(ẽ λ i )γξ0 (s) (s ∈ R), where a0i and a λ i are functions over R. Fix s0 ∈ R. By using Lemma 3.4, we can show D ds ∣∣∣∣ s=s0 ξ̂1 = r∑ i=1 ( (a0i ) ′(s0)(ẽ 0 i )γξ0(s0) + (a0i )(s0) D ds ∣∣∣∣ s=s0 ((ẽ0 i )γξ0 (s) ) ) + ∑ λ∈△+ mλ∑ i=1 ( (aλi ) ′(s0)(ẽ λ i )γξ0(s0) + aλi (s0) D ds ∣∣∣∣ s=s0 ((ẽλ i )γξ0 (s) ) ) = r∑ i=1 ( (a0i ) ′(s0)(ẽ 0 i )γξ0 (s0) + (a0i )(s0)∇γ′ξ0 (s0)((ẽ 0 i )γξ0(s0) ) ) + ∑ λ∈△+ mλ∑ i=1 ( (aλi ) ′(s0)(ẽ λ i )γξ0 (s0) + aλi (s0)∇γ′ξ0 (s0)((ẽ λ i )γξ0(s0) ) ) = r∑ i=1 ( (a0i ) ′(s0)(ẽ 0 i )γξ0 (s0) + (a0i )(s0)(∇ξ̃0ẽ 0 i )γξ0 (s0) ) + ∑ λ∈△+ mλ∑ i=1 ( (aλi ) ′(s0)(ẽ λ i )γξ0 (s0) + aλi (s0)(∇ξ̃0ẽ λ i )γξ0(s0) ) ) = r∑ i=1 (a0i ) ′(s0)(ẽ 0 i )γξ0 (s0) + ∑ λ∈△+ mλ∑ i=1 (aλi ) ′(s0)(ẽ λ i )γξ0 (s0) = 0, that is, (a0i ) ′(s0) = (a λ i ) ′(s0) = 0, where we use γ ′ ξ0 (s0) = ξ̃0γξ0(s0) . From the arbitrariness of s0, we see that a 0 i and a λ i are constant. Hence we obtain ξ̂ 1(s) = (ξ̃1)γξ0(s) . On the other hand, since ξ1 is tangent to Σ and Σ is totally geodesic, ξ̂1(1) also is tangent to Σ. Hence we see that (ξ̃1)Exp(ξ0) is tangent to Σ. Therefore Exp(ξ0) · γξ1(t1) belongs to Σ. Next we shall show that Exp(ξ0) · γξ1(t1) · γξ2(t2) belongs to Σ. Since γξ2 is a geodesic in AN and LExp(ξ0)·γξ1(t1) is an isometry of AN, LExp(ξ0)·γξ1(t1) ◦ γξ2 is a geodesic in AN. Hence we suffice to show that (LExp(ξ0)·γξ1(t1) ◦ γξ2) ′(0) = (ξ̃2)Exp(ξ0)·γξ1(t1) is tangent to Σ. Denote by ξ̂2 the parallel vector field along γξ1 := LExp(ξ0) ◦ γξ1 with ξ̂ 2(0) = (ξ̃2)Exp(ξ0). We describe ξ̂2 as ξ̂2(s) = r∑ i=1 b0i (s)(ẽ 0 i )γ ξ1 (s) + ∑ λ∈△+ mλ∑ i=1 bλi (s)(ẽ λ i )γ ξ1 (s) (s ∈ R), CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 25 where b0i and b λ i are functions over R. Fix s0 ∈ R. By using Lemma 3.4, we can show (4.3) D ds ∣∣∣∣ s=s0 ξ̂2 = r∑ i=1 ( (b0i ) ′(s0)(ẽ 0 i )γ ξ1 (s0) + (b 0 i )(s0) D ds ∣∣∣∣ s=s0 ((ẽ0 i )γ ξ1 (s)) ) + ∑ λ∈△+ mλ∑ i=1 ( (bλi ) ′(s0)(ẽ λ i )γ ξ1 (s0) + b λ i (s0) D ds ∣∣∣∣ s=s0 ((ẽλ i )γ ξ1 (s)) ) = r∑ i=1 ( (b0i ) ′(s0)(ẽ 0 i )γ ξ1 (s0) + (b 0 i )(s0)∇γ′ ξ1 (s0)((ẽ 0 i )γ ξ1 (s)) ) + ∑ λ∈△+ mλ∑ i=1 ( (bλi ) ′(s0)(ẽ λ i )γ ξ1 (s0) + b λ i (s0)∇γ ′ ξ1 (s0)((ẽ λ i )γ ξ1 (s)) ) = 0. Since γ′ ξ1 (s0) = 1 cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (H̃λ1)γξ1 (s0) by Lemma 4.1, γ′ξ1(s0) is de- scribed as γ′ ξ1 (s0) = (LExp(ξ0))∗(γ ′ ξ1 (s0)) = 1 cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (H̃λ1)γξ1 (s0) . Hence, by using Lemma 3.4, we have (4.4) ∇γ′ ξ1 (s0)((ẽ 0 i )γ ξ1 = 1 cosh(||λ1||s0) (∇ ξ̃1 ẽ0 i )γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (∇ H̃λ1 ẽ0 i )γ ξ1 (s0) = − λ1(e 0 i ) cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) and (4.5) ∇γ′ ξ1 (s0)((ẽ λ i )γ ξ1 = 1 cosh(||λ1||s0) (∇ ξ̃1 ẽλ i )γ ξ1 (s0) − tanh(||λ1||s0) ||λ1|| (∇ H̃λ1 ẽλ i )γ ξ1 (s0) =    1 2 cosh(||λ1||s0) ( [ξ̃1, ẽλi ] + θ̃[e λ i , θξ 1] ) (λ1 − λ ∈ △+) 1 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + 2〈ξ̃1, ẽλ i 〉H̃λ1 ) (λ1 = λ) 1 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + θ̃[ξ1, θeλi ] ) (λ − λ1 ∈ △+) 1 2 cosh(||λ1||s0) [ξ̃1, ẽλ i ] (λ1 − λ /∈ △ ∪ {0}). 26 Naoyuki Koike CUBO 20, 3 (2018) By substituting (4.4) and (4.5) into (4.3), we obtain (4.6) D ds ∣∣∣∣ s=s0 ξ̂2 = r∑ i=1 ( (b0i ) ′(s0)(ẽ 0 i )γ ξ1 (s0) − λ1(e 0 i )(b 0 i )(s0) cosh(||λ1||s0) (ξ̃1)γ ξ1 (s0) ) + ∑ λ∈△+ mλ∑ i=1 (bλi ) ′(s0)(ẽ λ i )γ ξ1 (s0) + ∑ λ1−λ∈△+ mλ∑ i=1 bλi (s0) 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + θ̃[eλ i , θξ1] ) + ∑ λ−λ1∈△+ mλ∑ i=1 bλi (s0) 2 cosh(||λ1||s0) ( [ξ̃1, ẽλ i ] + θ̃[ξ1, θeλi ] ) + ∑ λ−λ1 /∈△∪{0} mλ∑ i=1 bλi (s0) 2 cosh(||λ1||s0) [ξ̃1, ẽλ i ] + mλ1∑ i=1 b λ1 i (s0) 2 cosh(||λ1||s0) ( [ξ̃1, ẽ λ1 i ] + 2〈ξ̃1, ẽ λ1 i 〉H̃λ1 ) = 0. Without loss of generality, we may assume that eλ2 1 = ξ2. Hence we have bλ2 1 (0) = 1 and bλi (0) = 0 for any (λ, i) other than (λ2, 1). From (4.6) and these relations, we obtain b λ2 1 ≡ 1 and bλi ≡ 0 for any (λ, i) other than (λ2, 1), where we note that λ1 − λ2 /∈ △ ∪ {0}. Therefore we obtain ξ̂2 = (ξ̃2)γ ξ1 (s). On the other hand, since (ξ̂ 2)(0) is tangent to Σ and Σ is totally geodesic, ξ̂2(t1) also is tangent to Σ. Hence we see that (ξ̃2)Exp(ξ0)·γξ1(t1) is tangent to Σ. Therefore Exp(ξ0) · γξ1(t1) · γξ2(t2) belongs to Σ. In the sequel, by repeating the same discussion, we can derive that xξ0,t1,··· ,tk = Exp(ξ0)·γξ1(t1)· · · · ·γξk(tk) belongs to Σ. It is clear that any point of Σ is described as xξ0,t1,··· ,tk for some ξ0 ∈ b and some t1, · · · , tk ∈ R. Fix an orthonormal base {e01, · · · , e0m0} of b, where m0 := dim b. Define vector fields E 0 i (i = 1, · · · , m0) and Ej (j = 1, · · · , k) along Σ by (E0i )xξ0,t1,··· ,tk := (Lxξ0,t1,··· ,tk )∗(e 0 i )(= (ẽ 0 i )xξ0,t1,··· ,tk ) and (Ej)xξ0,t1,··· ,tk := (Lxξ0,t1,··· ,tk )∗(ξ j tj )(= (ξ̃ j tj )xξ0,t1,··· ,tk ). By imitating the discussions in the proofs of Lemmas 4.1 and 4.2, we can show the following fact for these vector fields. Lemma 4.3. The vector fields E0i (i = 1, · · · , m0) and Ej (j = 1, · · · , k) are tangent to Σ and they give a parallel orthonormal tangent frame field on Σ. Proof. Let (ξ̂i)j (resp. (ξ̂i)0) be the parallel vector field along γξj (i 6= j) (resp. γξ0) with (ξ̂i) j 0 = ξi (resp. (ξ̂i)00 = ξ i) and (ξ̂0) j be the parallel vector field along γξj with (ξ̂0) j 0 = ξ0. According to Lemma 4.1, we have (γξi) ′(t) = (Lγ ξi (t))∗(ξ i t) and (γξ0) ′(t) = (Lγξ0 (t) )∗(ξ0). Also, we can CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 27 show (ξ̂i) j γ ξj (t) = (Lγ ξj (t))∗(ξ i) (j 6= i), (ξ̂i)0 γξ0(t) = (Lγξ0 (t) )∗(ξ i) and (ξ̂0) j γ ξj (t) = (Lγ ξj (t))∗(ξ0) by imitating the discussion in the proof of Lemma 4.2. On the basis of these facts, we can derive the statement of this lemma, where we note that Σ is flat. ξ̃j ξ̃ j tj e e e Σ Σ Σ Hλj Ej γξj(tj) γξj(tj) γξj(tj) Hλj ξj ξj ξj Hλj Figure 4. By using these lemmas, we prove Theorem A. Proof of Theorem A. In this proof, we use the notations as in Example 2. Set Mxξ0,t1,··· ,tk := S b,l1,··· ,lk · xξ0,t1,··· ,tk. Denote by Hxξ0,t1,··· ,tk the mean curvature vector field of Mxξ0,t1,··· ,tk . Let {e01, · · · , e0m0} be an orthonormal base of b and (Hλ)b = ∑m0 i=1 Hiλe 0 i be the b-component of Hλ. According to the fact (iv) stated in Introduction, we have Mxξ0,t1,··· ,tk = Lxξ0,t1,··· ,tk (Sb,l ξ1,t1 ,··· ,l ξk,tk · e). Denote by Ĥξ0,t1,··· ,tk the mean curvature vector field of Sb,l ξ1,t1 ,··· ,l ξk,tk ·e. According to Lemma 3.1, we have (Ĥξ0,t1,··· ,tk)e = ∑ λ∈△+ mλ(Hλ)b − k∑ i=1 ||λi|| tanh(||λi||ti)(mλi + 2m2λi)ξ i ti and hence (4.7) (Hxξ0,t1,··· ,tk )xξ0,t1,··· ,tk = ∑ λ∈△+ m0∑ i=1 mλH i λ(E 0 i )xξ0,t1,··· ,tk − k∑ i=1 ||λi|| tanh(||λi||ti)(mλi + 2m2λi)(E i)xξ0,t1,··· ,tk . 28 Naoyuki Koike CUBO 20, 3 (2018) Define a tangent vector field Z over Σ by Zx := (H x)x (x ∈ Σ). According to (4.7), we have (4.8) Zxξ0,t1,··· ,tk = ∑ λ∈△+ m0∑ i=1 mλH i λ(E 0 i )xξ0,t1,··· ,tk − k∑ i=1 ||λi|| tanh(||λi||ti)(mλi + 2m2λi)(E i)xξ0,t1,··· ,tk . Define a coordinate φ = (u1, · · · , um0+k) : Σ → Rm0+k of Σ by φ(x∑m0 i=1 sie 0 i ,t1,··· ,tk ) := (s1, · · · , sm0, t1, · · · , tk) (s1, · · · , sm0, t1, · · · , tk ∈ R). We can show ∂∂ui = E 0 i (i = 1, · · · , m0) and ∂∂um0+j = E j (j = 1, · · · , k). Hence φ is a Euclidean coordinate of Σ. Under the identification of Σ and Rm0+k by φ, we regard Z as a tangent vector field on Rm0+k. Then Z is described as (4.9) Z(u1,··· ,um0+k) = ( ∑ λ∈△+ mλH 1 λ, · · · , ∑ λ∈△+ mλH m0 λ , −||λ1|| tanh(||λ1||um0+1)(mλ1 + 2m2λ1), · · · , −||λk|| tanh(||λk||um0+k)(mλk + 2m2λk)). Fix (a1, · · · , am0, t1, · · · , tk) ∈ Rm0+k. Let c be the integral curve of Z starting from (a1, · · · , am0, t1, · · · , tk) and let c = (c1, · · · , cm0+k). We suffice to investigate c to investigate the mean curvature flow starting from Mx∑m0 i=1 aie 0 i ,t1,··· ,tk From c′(t) = Zc(t), we have c ′ i(t) = ∑ λ∈△+ mλH i λ (i = 1, · · · , m0) and c′m0+j(t) = −(mλj + 2m2λj )||λj|| tanh (||λj||cm0+j(t)) (j = 1, · · · , k). By solving c′i(t) = ∑ λ∈△+ mλH i λ under the initial condition ci(0) = ai, we have (4.10) ci(t) = ai + t ∑ λ∈△+ mλH i λ. Also, by solving c′m0+j(t) = −(mλj + 2m2λj )||λj|| tanh(||λj||cm0+j(t)) under the initial condition cm0+j(0) = tj, we have (4.11) cm0+j(t) = 1 ||λj|| arcsinh ( e −||λj|| 2 (mλj +2m2λj)t sinh(||λj||tj) ) . From (4.10) and (4.11), we can derive T = ∞, lim t→∞ ∑m0 i=1 ci(t) 2 = ∞ (i = 1, · · · , m0) and lim t→∞ cm0+j(t) = 0 (j = 1, · · · , k). If t1 = · · · = tk = 0, then we have cm0+j ≡ 0 (j = 1, · · · , m0). Hence the mean curvature flow starting from Mxξ0,0,··· ,0 (xξ0,0,··· ,0 ∈ Exp(b)) consists of the leaves of F b,l1,··· ,lk through points of Exp(b). Also, according to the fact (iv) stated in Introduc- tion, the leaves of F b,l1,··· ,lk through points of Exp(b) are congruent to S b,l1,··· ,lk · e. Therefore, the mean curvature flow starting from Mxξ0,0,··· ,0 is self-similar. From limt→∞ ∑m0 i=1 ci(t) 2 = ∞ (i = 1, · · · , m0) and lim t→∞ cm0+j(t) = 0 (j = 1, · · · , k), we see that the mean curvature flow starting CUBO 20, 3 (2018) Mean curvature flow of certain kind of isoparametric . . . 29 from any leaf of F b,l1,··· ,lk asymptotes the mean curvature flow starting from the leaf of F b,l1,··· ,lk passing through a point of Exp(b). q.e.d. According to this proof, we obtain the following fact. Corollary 4.1. (i) The mean curvature flow starting from Mxξ0,0,··· ,0 is self-similar. (ii) The mean curvature flow starting from Mxξ0,t1,··· ,tk ((t1, · · · , tk) 6= (0, · · · , 0)) asymp- totes the flow starting from Mxξ0,0,··· ,0. In more detail, the distance between Mxξ0,t1,··· ,tk and Mxξ0,0,··· ,0 is equal to √√√√ k∑ j=1 1 ||λj|| 2 arcsinh2 ( e −||λj|| 2(mλj+2m2λj )t sinh(||λj||tj) ) , which converges to zero as t → ∞. Next we prove Theorem B. Proof of Theorem B. In case of b = {0}, the relation (4.9) is as follows: (4.12) Z(u1,··· ,uk) = (−||λ1|| tanh(||λ1||um0+1)(mλ1 + 2m2λ1 ), · · · , −||λk|| tanh(||λk||um0+k)(mλk + 2m2λk)). Hence, according to the dicussion in the proof of Theorem A, the mean curvature flow starting from any leaf of F b,l1,··· ,lk converges to the only minimal leaf S b,l,··· ,lk · e. Furthermore, the flow converges to the minimal leaf in C∞-topology because the flow consists of S b,l1,··· ,lk -orbits and the limit submanifold also is a S b,l1,··· ,lk -orbit. q.e.d. 30 Naoyuki Koike CUBO 20, 3 (2018) References [1] M. M. Alexandrino and M. Radeschi, Mean curvature flow of singular Riemannian foliations, J. Geom. Anal. 26 2204–2220 (2015). [2] B. Andrews and C. Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differ- ential Geom. 85 (2010) 357-396. [3] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984) 237-266. [4] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curva- ture, Invent. math. 84 (1986) 463-480. [5] N. Koike, Examples of a complex hyperpolar action without singular orbit, Cubo A Math. J. 12 (2010) 131-147. [6] N. Koike, Collapse of the mean curvature flow for equifocal submanifolds, Asian J. Math. 15 (2011) 101-128. [7] N. Koike, Collapse of the mean curvature flow for isoparametric submanifolds in a symmetric space of non-compact type, Kodai Math. J. 37 (2014) 355-382. [8] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329. Introduction Mean curvature flow. The non-curvature-adaptedness of the leaves. Proof of Theorem A