DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 29–34

A LOWER BOUND FOR THE RADIUS OF THE SMALLEST

BALL CONTAINING A MANIFOLD WITH CURVATURE BOUNDED FROM ABOVE

FERNANDO GIMÉNEZ

1. Introduction

Let F be a certain family of triples (φ, N, M) where N and M are riemannian manifods and φ : N → M is an isometric immersion. We look for bounds for the function ρ : F → ℝ^{+} which assigns to each (φ, N, M) the radius of the smallest M-ball containing φ(N) (extrinsic radius).

Spruck ([Sp]) proved that if F is the family of triples (φ, N, ℝ^{n}) such that N is a compact, connected riemannian manifold of dimension n − 1 with sectional curvature K ≥ β > 0, then the supremum of the function ρ is \frac{π}{2\sqrt{β}}.

Menninga in [Me] has generalized Spruck’s result by considering a family of triples (φ, N, M) where N is a compact, connected riemannian manifold of dimension s ≥ 2 with sectional curvature K ≥ β > 0 and M is a riemannian manifold of dimension n with injectivity radius i(M) ≥ \frac{π}{\sqrt{β}} and sectional curvature \overline{K} ≤ \frac{β}{4} .

For the infimum of the function ρ Jorge and Koutroufiotis have obtained in [JK] a result which implies that if F is the family of triples (φ, N, M) where N is a complete manifold of dimension s with scalar curvature bounded from below and sectional curvature K ≤ β, and M is a complete connected riemannian manifold of dimension n > 2s with sectional curvature λ ≥ \overline{K} ≥ α and the diameter of φ(N) is less than \frac{π}{2\sqrt{λ}} if λ > 0, then β > α and a lower bound for the function ρ is

[\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − α})

where

S_{λ}(t) = \{ \begin{matrix}\frac{sin(\sqrt{λ}t)}{\sqrt{λ}} & if λ > 0 \\ t & if λ = 0 \\ \frac{sinh(\sqrt{|λ|}t)}{\sqrt{|λ|}} & if λ < 0 \end{matrix}

1991 Mathematics Subject Classification. 53C40, 53C42, 53C21.

Key words and phrases. Isometric immersion, sectional curvature bounded from above, geodesic ball, extrinsic radius.

Work partially supported by a DGICYT N0. PB94-0972.

This paper is in final form and no version of it will be submitted for publication elsewhere.

and C_{λ}(t) = S′_{λ}(t).

This bound is a minimum if λ = α. In fact, it is attained at (φ, N, M) ≡ (i, S_{R}^{s}, 𝕂^{n}(λ)), where 𝕂^{n}(λ) is the simply connected space of constant sectional

curvature λ, S_{R}^{s} is the geodesic sphere of radius R = [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − λ}) in 𝕂^{s+1}(λ)

(where 𝕂^{s+1}(λ) is considered as a totally geodesic submanifold of 𝕂^{n}(λ)) and i : S_{R}^{s} → 𝕂^{n}(λ) is the canonical immersion. One has

ρ(i, S_{R}^{s}, 𝕂^{n}(λ)) = R = [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − λ})

The aim of this note is to obtain lower bounds for the function ρ defined on some families of triples (φ, N, ℂK^{n}(λ)) where ℂK^{n}(λ) is the simply connected Kähler manifold with constant holomorphic sectional curvature 4λ and real dimension 2n.

Now we state the main results of this paper. Let N be a complete riemannian manifold of dimension s, and let φ : N → ℂK^{n}(λ) be an isometric immersion, and suppose that the scalar curvature of N is bounded from below. Let us denote by K and \overline{K} the sectional curvatures of N and ℂK^{n}(λ), respectively. Let B_{r}(p) be the geodesic ball of radius r about p in ℂK^{n}(λ). From now on, λ will denote a nonzero real number and β ∈ ℝ. In the next theorems, d(φ(N)) will denote the extrinsic diameter of φ(N).

1.1 Theorem. Let λ > 0. Let us suppose that s > n + 1, K ≤ β and d(φ(N)) < \frac{π}{2\sqrt{λ}}. Then β > λ and

ρ(φ, N, M) ≥ [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − λ})

Let us recall that a holomorphic plane is that generated by two vectors of the form X, JX, and an antiholomorphic plane is that generated by two vectors X, Y such that Y is orthogonal to both X and JX.

1.2 Theorem. Let us suppose that s > n, and that, for each plane σ in T N such that φ_{∗}(σ) is a holomorphic plane we have K(σ) ≤ β. When λ > 0, let us assume, also, that d(φ(N)) < \frac{π}{2\sqrt{λ}} if s > n + 1, or d(φ(N)) < \frac{π}{4\sqrt{λ}} if s = n + 1. Then β > 4λ and

(a) for λ < 0, or λ > 0 and s > n + 1, then

ρ(φ, N, M) ≥ [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − 4λ})

and

(b) for λ > 0 and s = n + 1, then

ρ(φ, N, M) ≥ [\frac{C_{4λ}}{S_{4λ}} ]^{−1} (\sqrt{β − 4λ}) .

1.3 Theorem. Let us suppose that s > n + 1, and that for each plane σ in T N such that φ_{∗}(σ) is an antiholomorphic plane, we have K(σ) ≤ β. When λ > 0, let us assume, also, that d(φ(N)) < \frac{π}{2\sqrt{λ}} if s > n + 2, or d(φ(N)) < \frac{π}{4\sqrt{λ}} if s = n + 2. Then β > λ and

(a) for λ < 0, or λ > 0 and s > n + 2, then

ρ(φ, N, M) ≥ [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − λ})

and

(b) for λ > 0 and s = n + 2, then

ρ(φ, N, M) ≥ [\frac{C_{4λ}}{S_{4λ}} ]^{−1} (\sqrt{β − λ})

2. Proof of the main theorems

We shall use the following

Omori’s lemma([Om]). Let M be a complete Riemannian manifold whose sectional curvature is bounded from below. Let f : M ⟶ ℝ be a smooth function bounded from above. Then, for every m ∈ M and every ϵ > 0, there is an m′ ∈ M with the properties

f(m′) ≥ f(m), ∥grad f∥ (m′) < ϵ, ∇^{2}f(X, X) < ϵ ∥X∥^{2}

for every X ∈ T_{m′}M, where ∇ is the riemannian connection on M.

In the computations below, we shall denote by the same symbol a tangent vector X in N and its image φ_{∗}(X). For any fixed p ∈ ℂK^{n}(λ) let r : ℂK^{n}(λ) → ℝ be the distance to p. Let \overline{∇} and ∇ be the riemannian connections on ℂK^{n}(λ) and N respectively, and let α be the second fundamental form of the immersion φ. The following formula is well known ([JK])

(1)  (∇^{2}r ∘ φ) (X, Y) = (\overline{∇}^{2}r) (X, Y)+ < ∂r, α(X, Y) >

where ∂_{r} denotes the gradient of r in ℂK^{n}(λ). We shall also use the fact that −\overline{∇}^{2}r is the second fundamental form of the geodesic sphere of radius r in ℂK^{n}(λ) ([GW]) and ([Gr, p.138])

(\overline{∇}^{2}r) (X, X) = \frac{C_{λ}(r)}{S_{λ}(r)} (||X||^{2}− < X, J∂_{r} >^{2})

+\frac{C_{4λ}(r)}{S_{4λ}(r)} < X, J∂_{r} >^{2} (2)

2.1 Proof of theorem 1.1 We shall follow the ideas in [JK] and [Gi]. Given any p ∈ ℂK^{n}(λ), let R be any real number such that φ(N) ⊂ B_{R}(p).

Pick a point m ∈ N − φ^{−1}({p}). Observe that, if φ^{−1}({p}) = ∅, then r ∘ φ is smooth; if φ^{−1}(\{p\}) ≠ ∅ let δ := \frac{1}{2}r(m) and redefine now r such that r is smooth in φ(N) and coincide with the old r_{|φ(N)} outside φ(N) ∩ B_{\frac{δ}{2}} (p). Thus, we may apply Omori’s lemma to r ∘ φ, whence, for every ϵ′ > 0, there exists a point m_{ϵ′} ∈ N such that

0 < r ∘ φ(m) ≤ r ∘ φ(m_{ϵ′}) ≤ R, ∥grad(r ∘ φ)∥_{(m_{ϵ′})} < ϵ′,

∇^{2} (r ∘ φ) (X, X) < ϵ′ ∥X∥^{2} (3)

for every X ∈ T_{m_{ϵ′}} N. Using Cauchy-Schwarz inequality and the fact that ∥∂_{r}∥ = 1 outside B_{\frac{δ}{2}} (p) and m_{ϵ′} ∉ B_{\frac{δ}{2}} (p), we get

∇^{2} (r ∘ φ) (X, X) = \overline{∇}^{2} (r) (X, X)+ < ∂_{r}, α(X, X) >
(4)  ≥ \overline{∇}^{2} (r) (X, X) − ∥α(X, X)∥

for every X ∈ T_{m_{ϵ′}} N.

From (4) and the third inequality of (3) we get that

\frac{\overline{∇}^{2} (r) (X, X)}{∥X∥^{2}} − ϵ′ < \frac{∥α(X, X)∥}{∥X∥^{2}} (5)

for all X ∈ T_{m_{ϵ′}} N, X ≠ 0.

Using (2), the first inequality of (3), inequality (5) and the fact that t ⟼ \frac{C_{λ}(t)}{S_{λ}(t)}

and t ⟼ \frac{C_{4λ}(t)}{S_{4λ}(t)} are decreasing , we obtain

\frac{∥α(X, X)∥}{∥X∥^{2}} ≥ (\frac{C_{λ}(R)}{S_{λ}(R)}) (1 − \frac{< X, J∂_{r} >^{2}}{∥X∥^{2}} ) +

+ (\frac{C_{4λ}(R)}{S_{4λ}(R)}) \frac{< X, J∂_{r} >^{2}}{∥X∥^{2}} − ϵ′. (6)

We claim that there are X, Y ∈ φ_{∗}(T_{m_{ϵ′}} N) such that

α(X, X) = α(Y, Y) ≠ 0 and α(X, Y) = 0, < X, J∂_{r} >= 0. (7)

To prove (7) we identify W = φ_{∗}(T_{m_{ϵ′}} N) and φ_{∗}(T_{m_{ϵ′}} N)^{⊥} with ℝ^{s} and ℝ^{2n−s} respectively. We may extend α_{|W ×W} to a symmetric bilinear map ℂ^{s} × ℂ^{s} → ℂ^{2n−s}. Since s > 2n − s + 2 (because s > n + 1 by hypothesis), the system of 2n − s + 2 equations

α(Z, Z) = 0
< Re(Z), Im(Z) > = 0
< Re(Z), JIm(Z) > = 0

of s unknowns has a nontrivial solution Z = X + iY which satisfies (7).

We have from Gauss equation that

K(X, Y) − \overline{K}(X, Y) = \frac{< α(X, X), α(Y, Y) > − ∥α(Y, Y)∥^{2}}{∥X∥^{2} ∥Y ∥^{2} − < X, Y >^{2}}

≥ \frac{∥α(X, X)∥}{∥X∥^{2}} \frac{∥α(Y, Y)∥}{∥Y ∥^{2}}

≥ [\frac{C_{λ}(R)}{S_{λ}(R)} − ϵ′]^{2}

Now, let ϵ > 0, taking ϵ′ > 0 such that

[\frac{C_{λ}(R)}{S_{λ}(R)} − ϵ′]^{2} = (\frac{C_{λ}(R)}{S_{λ}(R)})^{2} − ϵ,

we conclude that

K(X, Y) − \overline{K}(X, Y) ≥ (\frac{C_{λ}(R)}{S_{λ}(R)})^{2} − ϵ,

Since K(X, Y) − \overline{K}(X, Y) ≤ β − α we have that

(\frac{C_{λ}(R)}{S_{λ}(R)})^{2} − ϵ ≤ β − α.

Since the inequality holds for every ϵ > 0 and the function (\frac{C_{λ}}{S_{λ}} )^{2} is decreasing then

R ≥ [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − α})

This finishes the proof of theorem 1.1.

2.2 Proof of theorems 1.2 and 1.3

It is similar to that of the theorem 1.1, but using the fact that \frac{C_{λ}}{S_{λ}} ≥ \frac{C_{4λ}}{S_{4λ}} if λ > 0 and \frac{C_{λ}}{S_{λ}} ≤ \frac{C_{4λ}}{S_{4λ}} if λ < 0.

3. Tightness of bounds

Here we shall prove that the bounds obtained in theorems 1.2 and 1.3 are the best possible when s = dim(N) is odd. Let β > 4λ. If s = 2s′ −1 we take N = S_{R}^{s},

the geodesic sphere of radius R = [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − 4λ}) on ℂK^{s′}(λ). Let φ : N →

ℂK^{s′}(λ) be the canonical immersion. It it obvious that ρ(φ, S_{R}^{s}, ℂK^{n}(λ)) = R. Moreover (φ, S_{R}^{s}, ℂK^{n}(λ)) satisfies the conditions of theorem 2.1. In fact, let σ =< {X, Y } >, ||X|| = ||Y || = 1, a plane in T N such that φ_{∗}(σ) is a holomorphic plane, then from Gauss equation,

K(σ) = \overline{K}_{H}(φ_{∗}(σ)) + [\frac{C_{λ}(R)}{S_{λ}(R)}]^{2} (1− < X, J∂_{r} >^{2})

+ [\frac{C_{4λ}(R)}{S_{4λ}(R)}]^{2} < X, J∂_{r} >^{2}

≤ 4λ + [\frac{C_{λ}(R)}{S_{λ}(R)}]^{2}
= 4λ + (β − 4λ) = β,

where ∂_{r} is the gradient of the distance (in ℂK^{s′}(λ)) to the center of the sphere S_{R}^{s}, considered as a vector in the tangent space to ℂK^{n}(λ). The second inequality follows from the fact that

[\frac{C_{λ}(R)}{S_{λ}(R)}]^{2} ≥ [\frac{C_{4λ}(R)}{S_{4λ}(R)}]^{2} .

It is obvius that

ρ(φ, S_{R}^{s}, ℂK^{n}(λ)) = R = [\frac{C_{λ}}{S_{λ}} ]^{−1} (\sqrt{β − 4λ})

thus, in this case, the bound obtained in the theorem 1.2 is a minimum. For the theorem 1.3 a similar argument shows that there is a riemannian manifold N of odd dimension s and an isometric immersion φ : N → ℂK^{n}(λ) satisfying the conditions of theorem 1.3 and such that the bound is just ρ(φ, N, ℂK^{n}(λ)).

References

[Gi] F. Giménez, ”Estimates for the curvature of a submanifold which lies inside a tube”. Preprint. [Gr] A. Gray, ”Tubes”. Addison-Wesley (1990)

[GW] R. E. Greene and H. Wu, ”Function theory on manifolds which possess a pole”. 699, Springer-Verlag (1979)

[JK] L. Jorge and D. Koutroufiotis, ”An estimate for the curvature of bounded submanifolds”, American J. of Math. 103, No. 4, (1981) pp. 711-725

[Me] N. Menninga, ”Immersions of positively curved manifolds into manifolds with curvature bounded above”. Trans. A. M. S. 318, No. 2, (1990), 809-821.

[Om] H. Omori, ”Isometric immersions of Riemannian manifold”, J. Math. Soc. of Japan, 19 (1967) pp. 205-214

[Sp] J. Spruck, ”On the radius of the smallst ball containing a compact manifold of positive curvature”. J. Differential Geom. 8, (1973), 257-258.

Departamento de Matemática Aplicada, E. T. S. I. Industriales Universidad Politécnica de Valencia, C/ Camino de Vera, 14 46022 Valencia (Spain)

E–mail address : fgimenez@mat.upv.es