real space form

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実空間型内の一定な平均曲率をもつ部分多様体について On submanifolds with constant mean curvature in a

real space form

数学専攻菊地信吾 Shingo Kikuchi Let ˜M^n+p(c) be an (n+p)-dimensional complete, connected and simply connected Riemannian manifold with constant sectional curvature c. We call it a space form. A space form ˜M^n+p(c) is one of the following:

(i) If c > 0, then ˜M^n+p(c) is a Euclidean sphere S^n+p(c);

(ii) If c= 0, then ˜M^n+p(c) is a Euclidean space R^n+p; (iii) If c <0, then ˜M^n+p(c) is a hyperbolic spaceH^n+p(c).

LetMⁿbe ann-dimensional, connected and orientable submanifold iso- metrically immersed in ˜M^n+p(c). Denote byh^α_ij the local component of the second fundamental form for eachi, j, α(1≤i, j ≤n, n+ 1≤α≤n+p).

We set

S:=

n+p∑

α=n+1

∑n

i,j=1

(h^α_ij)² and H := 1 n

vu ut ^n+p∑

α=n+1

(∑ⁿ

i=1

h^α_ii⁾²

be the squared norm of the second fundamental form and the mean curvature of Mⁿ in ˜M^n+p(c), respectively. Mⁿ is called minimal if the mean curvatureH of Mⁿ is equal to zero.

Now, we denote by A_α the n×n matrix of h^α_ij with respect to indices i, j. Define linear maps ϕ_α :T_xM →T_xM by

⟨ϕ_αX, Y⟩:= 1

n traceA_α⟨X, Y⟩ − ⟨A_αX, Y⟩ for n+ 1≤α≤n+p, where ⟨ , ⟩ is the Riemannian metric of Mⁿ. Moreover, we define the bilinear mapϕ:T_xM ×T_xM →T_xM^⊥ by

ϕ(X, Y) =

n+p∑

α=n+1

⟨ϕ_αX, Y⟩e_α,

where {e_n+1, . . . , e_n+p} denotes an orthonormal basis. It is easy to check that traceϕ= 0 and that

|ϕ|² :=

n+p∑

α=n+1

traceϕ²_α =S−nH². Let

P_H(x) =x²+ n(n−2)

√

n(n−1)Hx−n(H²+c) 1

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and

Q_H(x) = 3

2x²+ n(n−2)

√

n(n−1)

Hx−n(H²+c)

be the polynomials for each real numberH ∈R. We putA_H the square of the positive root ofP_H(x) = 0 and B_H one of Q_H(x) = 0.

Besides, in the case of p = 1, we denote by hij the local component of the second fundamental form for each i, j (1 ≤ i, j ≤ n) and by A the n ×n matrix of h_ij with respect to indices i, j. We choose a local orthonormal frame field {e1, . . . , en} such that hij = λiδij. Then we have H = 1

n

∑n

i=1

λ_i and S =

∑n

i=1

λ²_i. In the hypersurface we may put ϕ =ϕ_n+1. Thenϕ :TxM →TxM satisfies

⟨ϕX, Y⟩:= 1

n traceA⟨X, Y⟩ − ⟨AX, Y⟩. It easily check that traceϕ= 0 and that

|ϕ|² := traceϕ² = 1 2n

∑n

i,j=1

(λ_i−λ_j)².

Hence we get that |ϕ|² = 0 if and only if Mⁿ is totally umbilic.

We study generalizations of the results of the following theorems. More- over, we also study in the case ofc=−1.

Theorem 0.1(see Alencar and do Carmo [1]). Let Mⁿ be a compact and orientable hypersurface with constant mean curvature H in Sⁿ⁺¹(1). As- sume that|ϕ|² ≤A_H for allx∈M. Then

(i) either |ϕ|² ≡0 andMⁿ is totally umbilic or |ϕ|² ≡A_H. (ii) |ϕ|² ≡AH if and only if

(A) H = 0 and Mⁿ is a Cliﬀord torus in Sⁿ⁺¹(1), i.e., Mⁿ is a product of spheresSⁿ¹(r₁)×Sⁿ²(r₂), n₁ +n₂ =n, of appropriate radii.

(B) H ̸= 0, n ≥ 3, and Mⁿ=Sⁿ⁻¹(1)×S¹(√

1−r²)⊂ Sⁿ⁺¹(1), where r² < ⁿ⁻_n¹.

(C) H ̸= 0, n = 2, and M² = S¹(1) × S¹(√

1−r²) ⊂ S³(1), where r² ̸= ¹₂.

Theorem 0.2(see Uchida and Matsuyama [10]). Let Mⁿ be a complete, connected and orientable submanifold with nonzero constant mean curva- ture H in Sⁿ⁺²(c). If |ϕ| satisfies |ϕ|² ≤ A_H for all x∈ Mⁿ, then Mⁿ lies in a totally geodesic hypersurfaceSⁿ⁺¹(c) of Sⁿ⁺²(c) and

(i) either |ϕ|² ≡0 andMⁿ is totally umbilic or |ϕ|² ≡A_H. 2

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(ii) |ϕ|² ≡A_H if and only if

(B)n ≥ 3and Mⁿ = Sⁿ⁻¹(r₁)×S¹(r₂) ⊂ Sⁿ⁺¹(c), where r²₁+r₂² = ¹_c andr₁² < ⁿ_nc⁻¹.

(C) n = 2 and M² = S¹(r1)×S¹(r2) ⊂ S³(c), where r²₁ +r₂² = ¹_c and r₁² ̸= _2c¹.

The purpose of this paper is to prove the following theorems:

Theorem 1. LetMⁿ be a complete, connected and orientable submanifold with nonzero constant mean curvatureH inS^n+p(c) (p≥3). If|ϕ|satisfies

|ϕ|² ≤ B_H for all x ∈ Mⁿ, then Mⁿ lies in a totally geodesic submanifold Sⁿ⁺¹(c) of S^n+p(c), and |ϕ|² ≡0 andMⁿ is totally umbilic.

Theorem 2.1. Let Mⁿ be a complete, connected and orientable hyper- surface with constant mean curvature H > 1 in Hⁿ⁺¹(−1). Assume that

|ϕ|² ≤AH for all x∈Mⁿ. Then

(i) either |ϕ|² ≡0 andMⁿ is totally umbilic or |ϕ|² ≡AH.

(ii) |ϕ|² ≡ A_H if and only if Mⁿ is isometric to Sⁿ⁻¹(r) ×H¹(−_r²¹₊₁) for somer >0.

Theorem 2.2. Let Mⁿ be a complete, connected and orientable subman- ifold with constant mean curvature H > 1 in Hⁿ⁺²(−1). If |ϕ| satisfies

|ϕ|² ≤ A_H for all x ∈Mⁿ, then Mⁿ lies in a totally geodesic hypersurface Hⁿ⁺¹(−1)of Hⁿ⁺²(−1)and

(i) either |ϕ|² ≡0 andMⁿ is totally umbilic or |ϕ|² ≡A_H.

(ii) |ϕ|² ≡ A_H if and only if Mⁿ is isometric to Sⁿ⁻¹(r) ×H¹(−_r²¹₊₁) for somer >0.

Theorem 2.3. Let Mⁿ be a complete, connected and orientable subman- ifold with constant mean curvature H > 1 in H^n+p(−1) (p ≥ 3). If |ϕ| satisfies|ϕ|² ≤ BH for all x ∈Mⁿ, then Mⁿ lies in a totally geodesic sub- manifold Hⁿ⁺¹(−1) of H^n+p(−1), and|ϕ|² ≡0and Mⁿ is totally umbilic.

The following generalized maximum principle due to Omori [8] and Yau [11] will be used in order to prove our theorems:

Generalized Maximum Principle(see Omori [8] and Yau [11]). LetMⁿ be a complete Riemannian manifold whose Ricci curvature is bounded from the below andf ∈C²(M)a function bounded from the above onMⁿ. Then, for any ϵ >0, there exists a pointp∈Mⁿ such that

f(p)≥supf −ϵ, ∥gradf∥(p)< ϵ and ∆f(p)< ϵ.

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