Math. Nachr. 285, No. 10, 1264 – 1273 (2012) / DOI 10.1002/mana.201100078
Relative isoperimetric inequalities for minimal submanifolds outside a convex set Keomkyo Seo∗ Department of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku, Seoul, 140-742, Korea Received 22 March 2011, revised 4 July 2011, accepted 5 December 2011 Published online 6 March 2012 Key words Isoperimetric inequality, minimal submanifold, convex set MSC (2010) 58E35, 49Q20 Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C and ∂Σ ∼ ∂C is radially connected from a point p ∈ ∂Σ ∩ ∂C. We introduce a modified volume Mp (Σ) of Σ and obtain a sharp isoperimetric inequality 2πMp (Σ) ≤ Length(∂Σ ∼ ∂C)2 , where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume. c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction Let Σ be a domain in a complete simply connected surface with constant Gaussian curvature K. The classical isoperimetric inequality says that 4πArea(Σ) − KArea(Σ)2 ≤ Length(∂Σ)2 , where equality holds if and only if Σ is a geodesic disk. One natural way to extend this optimal inequality is to find the corresponding relative isoperimetric inequality. Let C be a closed convex set in a complete simply connected surface S with constant Gauussian curvature K ≤ 0. It has been known that if Σ is a relatively compact subset in S ∼ C, then 2πArea(Σ) − KArea(Σ)2 ≤ Length(∂Σ ∼ ∂C)2 ,
(1.1)
where equality holds if and only if Σ is a geodesic half disk [2]. Here ∼ denotes the set minus operator. The inequality (1.1) is called the relative isoperimetric inequality for Σ. Recently the author [8] extended the inequality (1.1) for a relatively compact subset Σ in S ∼ C of Gaussian curvature bounded above by a nonpositive constant. Surprisingly, the relative isoperimetric inequality (1.1) still holds for a compact minimal surface Σ with partially free boundary in a Riemannian manifold M with sectional curvature bounded above by a nonpositive constant, if the relative boundary ∂Σ ∼ ∂C is radially connected [7], [9]. Note that a curve Γ is said to be radially connected from a point p ∈ M if {dist(p, q) : q ∈ Γ} is a connected interval. Although we do not know whether the inequality (1.1) holds for Σ in a Riemannian manifold with sectional curvature bounded above by a positive constant or not, we introduce a concept of the modified volume Mp (Σ) to obtain the following isoperimetric ∗
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inequality (Theorem 2.6) without the additional curvature term: 2πMp (Σ) ≤ Length(∂Σ ∼ ∂C)2 . Notice that there is no curvature term of the ambient space in the above inequality. In Section 3, we study relative isoperimetric inequalities for an n-dimensional minimal submanifold Σ outside a closed convex set C in a Riemannian manifold with sectional curvature bounded above by a constant. Under the strong assumption that the relative boundary ∂Σ ∼ ∂C lies on a geodesic sphere centered at p ∈ ∂Σ ∩ ∂C, we prove the following relative isoperimetric inequality (Theorem 3.2) 1 n n ωn Mp (Σ)n −1 ≤ Vol(∂Σ ∼ ∂C)n , 2 where ωn is the volume of a unit ball in Rn . For the higher dimensional case, let Σ be a domain in hyperbolic space Hn or an n-dimensional minimal submanifold of Hm . Then the following linear isoperimetric inequality is well-known [5], [10]: (n − 1)Vol(Σ) ≤ Vol(∂Σ). However, it is not yet known whether this linear isoperimetric inequality holds for the ambient space with sectional curvature bounded above by a positive constant. Using the concept of the modified volume, we obtain the modified linear isoperimetric inequality (Theorem 3.3). Let Γ be a submanifold in a Riemannian manifold M . Given p ∈ M , we define a cone p× ×Γ from p over Γ to be the union of the geodesic segments from p to the points of Γ. Denote by St (p) the geodesic sphere of radius t with center at p. Then, in case that Γ is a curve, the angle of a curve Γ viewed from p is defined as follows and denoted by Angle(Γ, p) 1 ×Γ) ∩ St (p)). Angle(Γ, p) = lim Length((p× t→0 t
2
Two-dimensional relative isoperimetric inequalities
Let p be a point in the m-dimensional sphere Sm ⊂ Rm +1 and let r(x) be the distance from p to x in Sm . Choe and Gulliver [5] defined the modified volume Mp (Σ) of Σ with center at p as Mp (Σ) = cos r. Σ
Similarly for Σ in the m-dimensional hyperbolic space Hm , they defined the modified volume of Σ by cosh r. Mp (Σ) = Σ
Using the concept of the modified volume, they were able to prove the isoperimetric inequalities for minimal submanifolds in Sm or Hm . Motivated by this, we introduce the modified volume of a submanifold in a Riemannian manifold of variable curvature. Definition 2.1 Let M be a complete simply connected Riemannian manifold with sectional curvature bounded above by a positive constant K = k 2 for k > 0. Let p be a point in M and let r(x) be the distance from p to x in M . For a given n-dimensional submanifold Σ in M , the modified volume Mp (Σ) of Σ with center at p is defined as Mp (Σ) = cos kr. Σ
Similarly, when M is a complete simply connected Riemannian manifold with sectional curvature bounded above by a negative constant K = −k 2 for k > 0, we define the modified volume of Σ by cosh kr. Mp (Σ) = Σ
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It is easy to see that for a submanifold Σ in a manifold M with sectional curvature bounded above by a positive constant Mp (Σ) ≤ Vol(Σ) and in a manifold M with sectional curvature bounded above by a negative constant Mp (Σ) ≥ Vol(Σ). Remark 2.2 When M is the m-dimensional sphere Sm or the m-dimensional hyperbolic space Hm , the definition of the modified volume of Σ is exactly same as that of Choe-Gulliver [5]. Thus it can be thought of as an extension of their definition. Geometrically, if M is Sm ⊂ Rn +1 with p the north pole (0, . . . , 0, 1), the modified volume of a domain Ω ⊂ M is the usual Euclidean volume of the orthogonal projection of Ω into the horizontal hyperplane xn +1 = 0. (See [5, Lemma 1] for details.) In order to study a Riemannian manifold with sectional curvature bounded above by a constant, we need the following useful lemmas on the Laplacians of some functions of distance. Since the proof is a simple application of the Hessian comparison theorem, we shall omit the proof of the lemmas. (See [4] and [9].) Lemma 2.3 Let Σ be an n-dimensional compact minimal submanifold in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a nonzero constant K. Let Δ be the Laplacian on Σ. For fixed p ∈ M , define the extrinsic distance function r(·) = dist(p, ·) in M . Then we have (a) If K = k 2 for k > 0 and r <
π 2k
on Σ, then
Δ cos kr ≤ −nk 2 cos kr. (b) If K = −k 2 for k > 0, then Δ cosh kr ≥ nk 2 cosh kr. Lemma 2.4 Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonzero constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. Define the extrinsic distance function r(·) = dist(p, ·) for any p ∈ ∂Σ ∩ ∂C. (a) If K = k 2 for k > 0 and r < Δ log
π 2k
on Σ, then
sin kr ≥ πδp . 1 + cos kr
(b) If K = −k 2 for k > 0, then Δ log
sinh kr ≥ πδp . 1 + cosh kr
For a minimal submanifold Σ with boundary in Sm or Hm and any point p ∈ Sm or Hm , Choe and Gulliver [5] proved the following: Mp (Σ) ≤ Mp (p× ×∂Σ). Outside a convex set C in Sm or Hm , one can extend this result as follows: Proposition 2.5 Let C be a closed convex set in Sm or Hm . Assume that Σ is an n-dimensional compact minimal submanifold outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C in Sm or Hm . For any point p ∈ ∂Σ ∩ ∂C, Mp (Σ) ≤ Mp (p× ×(∂Σ ∼ ∂C)). c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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P r o o f. First suppose that the ambient space is Sm . Define r(x) = dist(p, x) for x ∈ Σ and the given point p ∈ ∂Σ ∩ ∂C. Let ν and η be the unit conormals to ∂Σ on Σ and p× ×∂Σ, respectively. Note that for any ¯ is ¯ q ∈ ∂Σ ∼ ∂C, η(q) makes the smallest angle with ∇r(q) among all vectors perpendicular to Tq ∂Σ, where ∇ m the connection on the ambient space S . As in [3], one can see that ∂r ∂r ≤ . ∂ν ∂η Moreover, since C is convex and p ∈ ∂Σ ∩ ∂C, one can see that ∇r points outward of C for p ∈ ∂Σ ∩ ∂C. Thus it follows from the orthogonality condition that ∂r (x) = ∇r(x), ν(x) ≤ 0 ∂ν for all x ∈ ∂Σ ∩ ∂C. Using the fact that Δ cos r = −n cos r on Σ ⊂ Sm , we get 1 1 1 ∂r ∂r Mp (Σ) = − + Δ cos r = sin r sin r n Σ n ∂ Σ∩∂ C ∂ν n ∂ Σ∼∂ C ∂ν ∂r 1 sin r ≤ n ∂ Σ∼∂ C ∂ν ∂r 1 sin r ≤ n ∂ Σ∼∂ C ∂η 1 =− Δ cos r = Mp (p× ×(∂Σ ∼ ∂C)), n p× ×(∂ Σ∼∂ C ) ×(∂Σ ∼ ∂C)) ∼ (∂Σ ∼ ∂C) which consists of geodesics where we used the fact that ∂∂ ηr = ∇r, η = 0 on ∂(p× emanating from p. A similar proof holds for Σ ⊂ Hm . For a minimal surface Σ outside a closed convex set C in a complete connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant K, the relative isoperimetric inequality 2πArea(Σ) − KArea(Σ)2 ≤ Length(∂Σ ∼ ∂C)2 was proved whenever ∂Σ is radially connected from p ∈ ∂Σ ∼ ∂C in [9]. We shall prove an analogous result for the ambient space with sectional curvature bounded above by a positive constant in terms of the modified volume. Theorem 2.6 Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K = k 2 for k > 0. Assume that Σ is a compact π . If ∂Σ ∼ ∂C is minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C and diam(Σ) < 2k radially connected from a point p ∈ ∂Σ ∩ ∂C, then we have 2πMp (Σ) ≤ Length(∂Σ ∼ ∂C)2 ,
(2.1)
where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. Remark 2.7 As mentioned in the introduction, the modified isoperimetric inequality (2.1) does not contain the additional curvature term of the ambient space. P r o o f. Define r(x) = dist(p, x) in M . From Lemma 2.3 (a), it follows that 1 Mp (Σ) = cos kr ≤ − 2 Δ cos kr 2k Σ Σ 1 1 ∂r ∂r + = sin kr sin kr , 2k ∂ Σ∩∂ C ∂ν 2k ∂ Σ∼∂ C ∂ν www.mn-journal.com
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where ν is the outward unit conormal to ∂Σ. Since Σ is orthogonal to ∂C along ∂Σ ∩ ∂C by assumption, one can see, as in the proof of Proposition 2.5, that ∂∂ νr (x) ≤ 0 for x ∈ ∂Σ ∩ ∂C. Thus 1 ∂r sin kr . Mp (Σ) ≤ 2k ∂ Σ∼∂ C ∂ν Let η be the unit conormal to p× ×∂Σ. Then as in the proof of Proposition 2.5, we have ∂r ∂r ≤ = 1 − ∇r, τ 2 , ∂ν ∂η where τ is a unit tangent to ∂Σ ∼ ∂C. Hence we get 1 Mp (Σ) ≤ sin kr 1 − ∇r, τ 2 . 2k ∂ Σ∼∂ C
(2.2)
Now we carry the last integral above over to the simply connected space form M of sectional curvature K. To do this, we construct a certain 2-dimensional cone in M . Let C1 , . . . , Cm be the components of ∂Σ ∼ ∂C such p, y), y ∈ M , and that each Ci is parametrized by ci (s) with arclength parameter s. Fix p¯ ∈ M , define r¯ = dist(¯ choose qi ∈ Ci for each i = 1, . . . , m. Then choose q¯1 , . . . , q¯m ∈ M in such a way that r(qi ) = r¯(¯ qi ). Suppose that each curve C¯i is parametrized by c¯i (s) with arclength parameter s. Then through developing under a local isometry as in [4], we construct a unique curve C¯i ⊂ M satisfying ci (s)), r(ci (s)) = r¯(¯ ∇r, ci (s) = ∇¯ r, c¯i (s) ,
(2.3)
where ci (s) and c¯i (s) are the unit tangent vectors of ci (s) and c¯i (s), respectively. Also it follows from the definition of C¯i that Length(Ci ) = Length(C¯i ). From the inequalities (2.2) and (2.3), we have 1 Mp (Σ) ≤ sin kr 1 − ∇r, τ 2 2k ∂ Σ∼∂ C m 2 1 = sin kr 1 − ∇r, ci (s) 2k i=1 C i m 2 1 = sin k¯ r 1 − ∇¯ r, c¯i (s) . 2k i=1 C¯ i Let η¯ be the outward unit conormal to C¯i on p¯× ×C¯i . It is easy to see that 2 ∂ r¯ . 1 − ∇¯ r, c¯i (s) = ∂ η¯ ¯ , the boundary of p¯× Moreover, if C¯i is not a closed curve in M ×C¯i consists of C¯i and the two geodesic segments ∂ r¯ = 0 on the geodesic segments. Thus it follows that from p¯ to c¯i (0) and c¯i (Li ). Obviously ∂ η¯ m 1 ∂ r¯ sin k¯ r 2k i=1 C¯ i ∂ η¯ m m 1 ×C¯i = Mp¯ p¯× ×C¯ , Δ cos k¯ r= Mp¯ p¯× =− 2 2k i=1 p¯× ¯ ×C i i=1
Mp (Σ) ≤
where C¯ =
(2.4)
m ¯ i=1 Ci .
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On the other hand, integrating Lemma 2.4 (a) gives k ∂r k ∂r sin kr π≤ ≤ ≤ Δ log 1 + cos kr ∂ Σ∼∂ C sin kr ∂ν ∂ Σ∼∂ C sin kr ∂η Σ k ∂ r¯ sin k¯ r ¯ p¯). = = Angle(C, Δ log = r ∂ η¯ 1 + cos k¯ r C¯ sin k¯ p¯× ×C¯ ¯ the inequality (2.4) and the following Lemma Since C¯ is also radially connected from p¯ by our construction of C, 2.8 give ¯ ≤ Length(C) ¯ 2 = Length(∂Σ ∼ ∂C)2 , 2πMp (Σ) ≤ 2πMp¯ (¯ p× ×C) where equality holds if and only if equality holds in Lemma 2.8. Therefore equality holds if and only if Σ is a totally geodesic half disk with constant Gaussian curvature K. Lemma 2.8 Let M be a complete simply connected Riemannian manifold with sectional curvature bounded above by a positive constant K = k 2 for k > 0. Let Γ be a piecewise smooth compact 1-dimensional manifold with (possibly) boundary in M . Assume that Γ is radially connected from a point p and Angle(Γ, p) ≥ π. If π , then we have diam(Γ) < 2k 2πMp (p× ×Γ) ≤ Length(Γ)2 , where equality holds if and only if p× ×Γ is a totally geodesic half disk with constant Gaussian curvature K. Remark 2.9 Unlike [7, Lemma 3], the above inequality does not contain the additional curvature term of the ambient space. P r o o f. We shall make use of doubling argument as in [7]. Consider an n-dimensional sphere Sn k12 ⊂ 2 Rn +1 of radius k12 as a model space of constant sectional ×Γ is a 2-dimensional cone, one curvature k . Since p× 2 1 n 1 can develop it onto a 2-dimensional great sphere S k 2 ⊂ S k 2 in such a way to preserve the area, the length of the boundary and the angle as in the proof of Theorem 2.6. (See also [4].) Therefore, if we let Γ1 , Γ2 , . . . , Γm 2 1 and a local isometry from p× ×Γi into be the connected components of Γ, then we can find a curve γ in S 2 i k p¯× ×γi where p¯ is the north pole of S2 k12 . When ∂Γi = ∅, γi is a curve in S2 k12 given by the geodesic polar coordinates r = γi (θ) satisfying 0 ≤ θ ≤ Θi
and
γi (0) = γi (Θi ) = dist(p, Γi ) = dist(¯ p, γi ), where Θi := Angle(Γi , p) and θ is the angle parameter of the cone. Define the doubling γ i of γi by γi (θ), 0 ≤ θ ≤ Θi r=γ i (θ) = (2.5) γi (θ − Θi ), Θi ≤ θ ≤ 2Θi . ×Γi can be identified with p¯× ×γi in S2 k12 as above. Here γi is a curve in S2 k12 When ∂Γi = ∅, the cone p× given by the geodesic polar coordinates r = γi (θ) for 0 ≤ θ ≤ Θi . Choose θi∗ ∈ [0, Θi ] such that γi (θi∗ ) = dist(p, Γi ) = dist(¯ p, γi ). Now define the doubling γ i of γi by ⎧ ∗ ⎪ ⎨γi (θ + θi ), r=γ i (θ) = γi (2Θi − θ − θi∗ ), ⎪ ⎩ γi (θ + θi∗ − 2Θi ), www.mn-journal.com
0 ≤ θ ≤ Θi − θi∗ , Θi − θi∗ ≤ θ ≤ 2Θi − θi∗ , 2Θi − θi∗ ≤ θ ≤ 2Θi .
(2.6)
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From our construction of γ i , it follows that Mp¯ (¯ p× × γi ) = cos kr = 2 cos kr = 2Mp (p× ×Γi ), p¯× ×γi p× ×Γ i Length( γi ) = 2Length(Γi ), Angle( γi , p¯) = 2Angle(Γi , p), γ i (0) = γ i (Θi ) = dist(¯ p, γ i ) = dist(p, Γi ). Hence if we let γ =
m i=1
γ i , then we obtain
Mp¯ (¯ p× × γ ) = 2Mp (p× ×Γ), Length( γ ) = 2Length(Γ),
(2.7)
Angle( γ , p¯) = 2Angle(Γ, p) ≥ 2π. Using the remark below Definition 2.1, one can see that Mp¯ (¯ p× × γ ) is equal to the Euclidean area of the standard projection of p¯× × γ onto the plane containing the equator of S2 k12 . Thus if we denote the projection by π, we get Mp¯ (¯ p× × γ ) = Area(0× ×π( γ )), Angle( γ , p¯) = Angle(π( γ ), 0), Length( γ ) ≥ Length(π( γ )), where 0 is the origin of the plane and we used the fact that the standard projection π is a length decreasing map in the last inequality. Applying Theorem 1 of [3], we see that 4πArea(0× ×π( γ )) ≤ Length(π( γ ))2 , where equality holds if and only if π( γ ) is a circle. Therefore we finally obtains that p× × γ ) = 4πArea(0× ×π( γ )) ≤ Length(π( γ ))2 ≤ Length( γ )2 . 4πMp¯ (¯ From (2.7), it immediately follows that 2πMp (p× ×Γ) ≤ Length(Γ)2 , where equality holds if and only if Length(π( γ )) = 2Length(Γ) and Angle(Γ, p) = π. This implies that Σ is a totally geodesic half disk with constant Gaussian curvature K. If ∂Σ ∼ ∂C is connected, it is radially connected. Further, when ∂Σ ∼ ∂C has two connected components, one can immediately obtain the following from the same arguments as in the proof of Corollary 1 of [7]. Corollary 2.10 Let Σ be a compact minimal surface satisfying the same assumptions as in Theorem 2.6 except the radial connectedness. Then (2.1) holds if ∂Σ ∼ ∂C is connected or has two components that are connected by a component Γ of ∂Σ ∩ ∂C.
3 Higher dimensional relative isoperimetric inequalities One of the most important properties of an n-dimensional minimal submanifold Σ in Rm is the monotonicity property which says that for a ball B(p, r) of radius r centered at p, the volume of Σ ∩ B(p, r) divided by the volume of n-dimensional ball of radius r is a nondecreasing function of r. This property also holds in the mdimensional hyperbolic space Hm [1], but does not hold in Sm [6]. The author [9] proved that this monotonicity property holds for minimal submanifolds outside a closed convex set in Rm . The following result shows that the monotonicity property still holds for modified volume in a complete simply connected Riemannian manifold with sectional curvature bounded above by a nonzero constant. c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Proposition 3.1 Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonzero constant K. Assume that Σ is an n-dimensional compact minimal submanifold outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. Suppose that r(·) = dist(p, ·) in M for any p ∈ ∂Σ ∩ ∂C. Denote by B(p, r) the geodesic ball of radius r centered at p. π , dist(p, ∂Σ ∼ ∂C) (a) If K = k 2 for k > 0, then for 0 < r < min 2k Mp (Σ ∩ B(p, r)) sinn kr is a monotonically nondecreasing function of r. (b) If K = −k 2 for k > 0, then for 0 < r < dist(p, ∂Σ ∼ ∂C) Mp (Σ ∩ B(p, r)) sinhn kr is a monotonically nondecreasing function of r. P r o o f. For (a), define Σr = Σ ∩ B(p, r). Then 1 Mp (Σr ) = cos kr ≤ − 2 Δ cos kr nk Σ r Σr 1 1 ∂r ∂r + = sin kr sin kr . nk ∂ Σ r ∼∂ C ∂ν nk ∂ Σ r ∩∂ C ∂ν Since
∂r ∂ν
= ∇r, ν ≤ 0 on ∂Σr ∩ ∂C by the orthogonality condition, one sees that sin kr 1 ∂r = Mp (Σr ) ≤ sin kr |∇r|. nk ∂ Σ r ∼∂ C ∂ν nk ∂ Σ r ∼∂ C
Denote the volume forms on Σ and ∂Σr by dv and dΣr , respectively. Then dv = Thus d dr
1 dΣr dr. |∇r|
cos kr|∇r|2 dv = Σr
Using the fact that r ≤
= ≤ =
d log dr
r
|∇r|.
cos kr|∇r|dΣr dr = cos kr 0
∂ Σr
∂ Σr
and |∇r| ≤ 1 on Σ, we get 1 sin kr cos kr |∇r| nk cos kr ∂ Σr 1 sin kr d cos kr|∇r|2 nk cos kr dr Σ r 1 sin kr d cos kr nk cos kr dr Σ r 1 sin kr d Mp (Σr ). nk cos kr dr
π 2k
Mp (Σr ) ≤
Therefore
d dr
Mp (Σr ) sinn kr
≥ 0,
Mp (Σr ) is monotonically nondecreasing. which implies that the function sinn kr A similar proof holds for (b). www.mn-journal.com
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From the above monotonicity property, one can prove the following isoperimetric inequality when the relative boundary ∂Σ ∼ ∂C lies on a geodesic sphere. Although the assumption is quite strong, the inequality is true for any dimension. Theorem 3.2 Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonzero constant K. Assume that Σ is an n-dimensional compact minimal submanifold outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C and ∂Σ ∼ ∂C lies on a geodesic sphere π . Then centered at a fixed point p ∈ ∂Σ ∩ ∂C. Furthermore if K = k 2 for k > 0, assume that diam(Σ) < 2k 1 n n ωn Mp (Σ)n −1 ≤ Vol(∂Σ ∼ ∂C)n , 2 where ωn is the volume of a unit ball in Rn . Equality holds if and only if Σ is a totally geodesic half ball with constant sectional curvature K. P r o o f. Suppose that K = k 2 for k > 0. Let r(·) = dist(p, ·) in M . Let R be the radius of the geodesic sphere on which ∂Σ ∼ ∂C lies. It follows that 1 Δ cos kr Mp (Σ) ≤ − 2 nk Σ 1 1 ∂r ∂r + = sin kr sin kr nk ∂ Σ∼∂ C ∂ν nk ∂ Σ∩∂ C ∂ν ∂r 1 sin kr ≤ nk ∂ Σ∼∂ C ∂ν sin kR ∂r sin kR ≤ Vol(∂Σ ∼ ∂C). = nk nk ∂ Σ∼∂ C ∂ν Since lim
r →0
ωn Mp (Σ ∩ B(p, r)) = n, n sin kr 2k
we see from Proposition 3.1 that ωn Mp (Σ) . ≤ n 2k sinn kR Thus we have Mp (Σ) ≤
2k n ωn
n1
1
(Mp (Σ)) n
1 Vol(∂Σ ∼ ∂C), nk
which gives the desired inequality. Moreover, equality holds if and only if Σ is a cone with density at p equal to 12 with constant sectional curvature K and ∂Σ ∩ ∂C is totally geodesic, or equivalently Σ is a totally geodesic half ball with constant sectional curvature K. Similarly one can prove the above theorem in case of K = −k 2 . Let Σ be a domain in n-dimensional hyperbolic space Hn or an n-dimensional minimal submanifold of Hm . Then the following linear isoperimetric inequality was proved by Yau [10] and Choe-Gulliver [5] respectively: (n − 1)Vol(Σ) ≤ Vol(∂Σ). Recently the author [9] obtained the relative version of this linear isoperimetric inequality. That is, for an n-dimensional minimal submanifold Σ outside a convex set C in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a negative constant K = −k 2 for k > 0, we have k(n − 1)Vol(Σ) ≤ Vol(∂Σ ∼ ∂C). When the sectional curvature of the ambient space is bounded above by a positive constant, we have the following linear isoperimetric inequality using the modified volume. c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Theorem 3.3 Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K = k 2 for k > 0. Assume that Σ is an n-dimensional compact minimal submanifold outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. Suppose that π . Then we have diam(Σ) < 2k nkMp (Σ) ≤ Vol(∂Σ ∼ ∂C). P r o o f. By Lemma 2.3 (a), 1 Mp (Σ) = cos kr ≤ − 2 Δ cos kr nk Σ Σ 1 ∂r ≤ sin kr nk ∂ Σ∼∂ C ∂ν 1 ≤ 1 nk ∂ Σ∼∂ C 1 = Vol(∂Σ ∼ ∂C). nk The above linear isoperimetric inequality still holds even when we do not consider a relative isoperimetric problem. More precisely, we have Theorem 3.4 Let Σ be an n-dimensional compact minimal submanifold of a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K = k 2 for k > 0. If π , then we have diam(Σ) < 2k nkMp (Σ) ≤ Vol(∂Σ). The proof is analogous to the proof of Theorem 3.3. Acknowledgements The author would like to thank the referees for the valuable comments and suggestions. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20110005520).
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