Advances in Mathematics 321 (2017) 205–220

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Advances in Mathematics www.elsevier.com/locate/aim

Necessary conditions and nonexistence results for connected submanifolds in a Riemannian manifold Keomkyo Seo Department of Mathematics, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Yongsan-gu, Seoul, 04310, Republic of Korea

a r t i c l e

i n f o

Article history: Received 7 September 2016 Received in revised form 15 June 2017 Accepted 21 September 2017 Available online xxxx Communicated by the Managing Editors MSC: 49Q05 53A10 53C42

a b s t r a c t In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L2 norm of the mean curvature vector in a Riemannian manifold. © 2017 Elsevier Inc. All rights reserved.

Keywords: Minimal submanifold Bryant surface Mean curvature Density estimate Nonexistence

1. Introduction Douglas [8] and Radó [18] gave the first solution to the Plateau problem independently, which says that any simple closed curve in R3 bounds at least one minimal disk. In order E-mail address: [email protected]. URL: http://sites.google.com/site/keomkyo. https://doi.org/10.1016/j.aim.2017.09.038 0001-8708/© 2017 Elsevier Inc. All rights reserved.

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to generalize the original Plateau problem, one may ask whether any given two disjoint simple closed curves Γ1 and Γ2 in R3 bound a compact connected minimal surface or not. Douglas [9] showed that if D1 and D2 are the least area minimal disks bounded by Γ1 and Γ2 respectively, and if the following condition (so-called Douglas condition) is satisfied: inf{Area(S)} < Area(D1 ) + Area(D2 ), then there exists a minimal annulus bounded by Γ1 and Γ2 . Here the infimum is taken over all surfaces of annular type spanning Γ1 and Γ2 . However, the solution to this generalized Plateau problem does not exist in general. For instance, consider a pair of coaxial circles of fixed radii lying in parallel planes in R3 . It is well-known that if the distance of the planes is sufficiently small, then there exists a minimal annulus spanning two circles. Indeed, it is a part of a catenoid. It is obvious that if the planes are far apart, then there does no longer exist a catenoid bounded by the given pair of circles. Therefore it is interesting to give a quantitative description for the necessary condition on the boundary of compact connected minimal surfaces. In his series of papers [13–16], Nitsche obtained necessary conditions that two disjoint simple closed curves Γ1 and Γ2 lying in parallel planes in R3 bound a minimal annulus. In fact, he showed that if the diameter of Γi is di for i = 1, 2, and the distance of the two planes is r, then r≤

3 max(d1 , d2 ). 2

In other words, if the distance between two planes is bigger than 32 max(d1 , d2 ), then there is no minimal annulus bounded by Γ1 ∪ Γ2 . More generally, consider higher-dimensional minimal submanifolds in Euclidean space. Almgren [1] obtained the existence of a number r > 0 such that, for two disjoint (n − 1)-dimensional compact submanifolds Γ1 and Γ2 in Rn+k , if dist(Γ1 , Γ2 ) > r, then there does not exist n-dimensional compact connected minimal submanifold in Rn+k bounded by Γ1 ∪ Γ2 . Hildebrandt [11] was able to generalize the Nitsche’s result by using the maximum principle for subharmonic functions on the minimal surface in R3 . His idea is based on the following observations: (1) The coordinate functions x, y, z are harmonic on a minimal surface Σ ⊂ R3 , and hence the quadratic polynomial Q(x, y, z) := x2 + y 2 − z 2 is subharmonic on Σ. (2) The convex-hull property of a compact minimal surface shows that if two simple closed curves Γ1 and Γ2 are separated by the two disjoint components of the solid

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cone {(x, y, z) ∈ R3 : Q(x, y, z) < 0}, there is no compact connected minimal surface Σ ⊂ R3 bounded by Γ1 ∪ Γ2 . Moreover, Osserman–Schiffer [17] improved the Hildebrandt’s nonexistence theorem by proving that the same result holds in the larger optimal cone {(x, y, z) ∈ R3 : x2 + y 2 − z 2 sinh2 τ < 0}, where the constant τ  1.199 is given by the solution of the equation cosh τ − τ sinh τ = 0. Later, Dierkes [6] and Dierkes–Schwab [7] extended the results by Hildebrandt and Osserman–Schiffer to n-dimensional compact minimal submanifolds of arbitrary codimension in Euclidean space. These results on nonexistence of compact connected minimal submanifolds are given as follows. Theorem ([6,7,11]). (I) Let Bi ⊂ Rn+k be the closed ball centered at xi ∈ Rn+k with radius ri for i = 1, 2. If  dist(x1 , x2 ) >

n n−1

 12 (r1 + r2 ),

then there exists no n-dimensional compact connected minimal submanifold Σ ⊂ Rn+k with ∂Σ ⊂ B1 ∪ B2 and ∂Σ ∩ Bi = ∅ for i = 1, 2. (II) Let Ci be a compact subset of diameter di in Rn+k . If 1 dist(C1 , C2 ) > 2



2n(n + k) (n − 1)(n + k + 1)

 12 (d1 + d2 ),

then there exists no n-dimensional compact connected minimal submanifold Σ ⊂ Rn+k with ∂Σ ⊂ C1 ∪ C2 and ∂Σ ∩ Ci = ∅ for i = 1, 2. In this paper, we investigate necessary conditions and non-existence results for n-dimensional compact connected submanifolds in a Riemannian manifold M of sectional curvature bounded above by a constant K. Unlike the Euclidean space, one cannot construct a suitable cone arising from the quadratic form, since there is no global coordinates on a minimal submanifold Σ in a Riemannian manifold. However, there is a natural subharmonic function on Σ ⊂ M , which comes from the Green’s function on the space form of constant sectional curvature K. Motivated by this observation, we are able to obtain a density estimate (Proposition 3.1) for a submanifold with variable mean curvature in M by using the Laplacian of functions depending on the distance. In fact, the density of a surface played an important role in proving the embeddedness of minimal surfaces in the Euclidean space with total boundary curvature at most 4π by Ekholm–White–Wienholtz [10]. Thereafter, this embeddedness result was extended to minimal surfaces in a nonpositively curved manifold by Choe–Gulliver [5]. Our density estimate enables us to obtain an upper bound of the distance between the boundaries of a compact connected minimal submanifold (Theorem 3.2). In the

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2-dimensional case, we can obtain a distance estimate (Theorem 3.5) for surfaces with bounded mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. To our knowledge, this result is new even for constant mean curvature surface in the hyperbolic space. By making use of the distance estimate, we prove a nonexistence result for Bryant surfaces (i.e., surfaces with constant mean curvature one) in the 3-dimensional hyperbolic space (Corollary 3.7). Finally we obtain another density estimate using the second order of the length of the mean curvature vector, which provide necessary conditions for the existence of compact connected submanifolds in a Riemannian manifold (Theorem 4.2). As a corollary, we give a quantitative condition that the union of two disjoint closed curves in a Riemannian manifold bounds no compact connected surface (Corollary 4.3). 2. Laplacian estimates for functions of distance Let Σ be an n-dimensional submanifold of an (n + k)-dimensional Riemannian manifold M . For orthonormal vector fields {e1 , · · · , en } on a neighborhood of a point of Σ, one can extend these vector fields smoothly to orthonormal vector fields {e1 , · · · , en+k } on M . The mean curvature vector H of Σ in M is given by

H=

n 

¯ e ei − ∇ i

i=1

n 

∇ e i ei ,

i=1

¯ denote the connections of Σ and M , respectively. Then, for a smooth where ∇ and ∇ function f defined on M , we see that

Δf =

n 

(ei ei f − ∇ei ei f )

i=1

¯ + Hf − = Δf

k 

¯ 2 f (eα , eα ), ∇

α=n+1

¯ denote the Laplacians on Σ and M , respectively. From this observawhere Δ and Δ tion and the standard Laplacian comparison theorem, we have the following well-known Laplacian estimates. Lemma 2.1 ([3,4,21]). Let Σ be an n-dimensional submanifold in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Denote by H, ∇, and Δ the mean curvature vector, the connection, and the Laplacian ¯ the connection on M . Define the distance function on Σ, respectively, and denote by ∇ r(·) = dist(p, ·) for a fixed point p ∈ M . Then we have the following.

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(a) If K = 0, then ¯ Δr2 ≥ 2n + 2r H, ∇r , Δr ≥

n − |∇r|2 ¯ + H, ∇r . r

(b) If K = −κ2 < 0, then ¯ Δr ≥ κ(n − |∇r|2 ) coth κr + H, ∇r , Δ log sinh κr ≥ −nK cosh2 κr + (c) If K = κ2 > 0 and r <

π 2κ

K|∇r|2 ¯ (1 + cosh2 κr) + κ coth κr H, ∇r . sinh κr

on Σ, then

¯ Δr ≥ κ(n − |∇r|2 ) cot κr + H, ∇r , Δ log sin κr ≥ nK cot2 κr −

K|∇r|2 ¯ (1 + cos2 κr) + κ cot κr H, ∇r . sin2 κr

Denote by G(r) the Green’s function of the n-dimensional space form MK of constant sectional curvature K, where r(·) = dist(p, ·) is the distance function on MK for a fixed point p ∈ MK . In fact, the gradient of G(r) is given by ⎧ 1−n ⎪ ∇r ⎨r ∇G(r) = κn−1 sinh1−n κr∇r ⎪ ⎩ κn−1 sin1−n κr∇r

if if if

K = 0, K = −κ2 < 0, K = κ2 > 0,

where ∇ denotes the connection of the space form MK and κ > 0. It is well-known that the Green’s function of Rn is given by  G(r) =

log r r2−n

if if

n = 2, n ≥ 3.

Recall that in the 2-dimensional hyperbolic space H2 (−κ2 ) and the 2-dimensional sphere S2 (κ2 ) of constant sectional curvature −κ2 and κ2 , respectively,  G(r) =

2 2 log tanh( κr 2 ) on H (−κ ), κr 2 2 log tan( 2 ) on S (κ ).

Lemma 2.2. Let Σ be an n-dimensional submanifold with mean curvature vector H in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Denote by ∇ and Δ the connection and the Laplacian on Σ, ¯ the connection on M . For a fixed point p ∈ M , define the respectively, and denote by ∇ distance function r(·) = dist(p, ·) in M . Then we have the following.

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(a) If K = 0, then Δ log r ≥

2(1 − |∇r|2 ) 1 ¯ + H, ∇r r2 r

¯ Δr2−n ≤ n(n − 2)r−n (|∇r|2 − 1) + (2 − n)r1−n H, ∇r

if n = 2, if n ≥ 3.

(b) If K = −κ2 < 0, then ¯ ΔG(r) ≥ nκn (1 − |∇r|2 ) sinh−n κr cosh κr + κn−1 sinh1−n κr H, ∇r . (c) If K = κ2 > 0 and r <

π 2κ

on Σ, then

¯ ΔG(r) ≥ nκn (1 − |∇r|2 ) sin−n κr cos κr + κn−1 sin1−n κr H, ∇r . Proof. For n = 2, since Δr ≥ Δ log r = div

2−|∇r|2 r

¯ by Lemma 2.1(a), + H, ∇r

Δr |∇r|2 2(1 − |∇r|2 ) 1 ∇r ¯ = − ≥ + H, ∇r . r r r r2 r

For n ≥ 3, Δr2−n = div((2 − n)r1−n ∇r) 1 (2 − n)div(r−n ∇r2 ) 2 1 1 = n(n − 2)r−n−1 ∇r, 2r∇r + (2 − n)r−n Δr2 . 2 2

=

Since ¯ Δr2 ≥ 2n + 2r H, ∇r by Lemma 2.1(a), one can prove (a). Using Lemma 2.1(b), we see that ¯ Δr ≥ κ(n − |∇r|2 ) coth κr + H, ∇r , which implies that ΔG(r) = div(κn−1 sinh1−n κr∇r) = κn (1 − n) sinh−n κr cosh κr|∇r|2 + κn−1 sinh1−n κrΔr ¯ ≥ nκn (1 − |∇r|2 ) sinh−n κr cosh κr + κn−1 sinh1−n κr H, ∇r , which proves (b). A similar computation using Lemma 2.1(c) gives the proof of (c). 2

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3. Density and distance estimates via the first order of the length of mean curvature vector Let Σ be an n-dimensional submanifold in an (n + k)-dimensional Riemannian manifold M . The density of Σ at p is defined by ΘΣ (p) = lim

r→0

Vol(Σ ∩ Bp (r)) , ωn r n

where Bp (r) ⊂ M denotes the geodesic ball centered at p with radius r and ωn denotes the volume of the n-dimensional unit ball in Rn (see [22] for more details). If Σ is smooth, then the density ΘΣ (p) can be also computed as follows: ΘΣ (p) = lim

r→0

Vol(Σ ∩ ∂Bp (r)) . nωn rn−1

As mentioned in the introduction, density estimates for minimal surfaces in a Riemannian manifold have been used to prove the embeddedness of minimal surfaces under the curvature assumption on the boundary [5,10]. In this section, we give a general density estimates for submanifolds in a Riemannian manifold in terms of the first order of the length of mean curvature vector. Proposition 3.1 (Density estimate I). Let Σ be an n-dimensional compact connected submanifold with mean curvature vector H in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Then we have the following. (a) If K = 0, then nωn ΘΣ (p) ≤

|H|r1−n .

r1−n + ∂Σ

Σ

(b) If K = −κ2 < 0, then



nωn ΘΣ (p) ≤ κn−1

Σ

∂Σ

(c) If K = κ2 > 0 and r <

π 2κ

|H| sinh1−n κr.

sinh1−n κr + κn−1

on Σ, then

nωn ΘΣ (p) ≤ κn−1 ∂Σ

|H| sin1−n κr.

sin1−n κr + κn−1 Σ

Here ωn denotes the volume of the n-dimensional unit ball in Rn .

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Proof. Denote by Bp (ε) ⊂ M the geodesic ball centered at p ∈ M of radius ε > 0. Define Σε := Σ \ Bp (ε). For n = 2, applying Lemma 2.2(a) and the divergence theorem, we have

¯ H, ∇r ≤ r

Σε





1 ∂r = r ∂ν

Δ log r = Σε

∂Σε





1 ∂r − r ∂ν

∂Σ

1 ∂r , r ∂ν

Σ∩∂Bp (ε)

where ν denotes the outward unit normal vector tangent to Σε along ∂Σε . As ε → 0, we see that ∂r 1 ∂r ¯ = ∇r, ν → −1 and hence lim = 2πΘΣ (p). ε→0 ∂ν r ∂ν Σ∩∂Bp (ε)

¯ = 1, we get Using |∇r|

1 + r

2πΘΣ (p) ≤



|H| . r

Σ

∂Σ

For n ≥ 3, applying Lemma 2.2(a) and the divergence theorem again give ¯ ≥ Δr2−n (2 − n) r1−n H, ∇r Σε

Σε



= (2 − n)

r

1−n ∂r

∂ν

− (2 − n)

∂Σ

Σ∩∂Bp (ε)

Since lim

ε→0 Σ∩∂Bp (ε)

r1−n

∂r = nωn ΘΣ (p), ∂ν

we have nωn ΘΣ (p) ≤

|H|r1−n ,

r1−n + Σ

∂Σ

which proves (a). From Lemma 2.2(b), it follows that 1−n n−1 ¯ κ sinh κr H, ∇r ≤ ΔG(r) Σε

Σε



∇G, ν −

= ∂Σ

Σ∩∂Bp (ε)

∇G, ν

r1−n

∂r . ∂ν

K. Seo / Advances in Mathematics 321 (2017) 205–220

sinh1−n κr

= κn−1

∂r − κn−1 ∂ν

∂Σ

213

sinh1−n κr

∂r . ∂ν

Σ∩∂Bp (ε)

Using the fact that sinh1−n κr

lim κn−1

ε→0

∂r = nωn ΘΣ (p), ∂ν

Σ∩∂Bp (ε)

we obtain (b). Similarly, one can prove (c), applying Lemma 2.2(c) as above. 2 From the above density estimate, one can obtain the following distance estimate for the boundaries of a compact connected submanifold in a Riemannian manifold of sectional curvature bounded above by a constant. Theorem 3.2. Let C1 and C2 be two disjoint closed sets in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Suppose that there exists an n-dimensional compact connected minimal submanifold Σ ⊂ M with ∂Σ ⊂ C1 ∪ C2 and ∂Σ ∩ Ci = ∅ for i = 1, 2. Define d := dist(C1 , C2 ). Then we have the following. (a) If K = 0, then  d≤2

Vol(∂Σ) nωn

1  n−1

.

(b) If K = −κ2 < 0, then κd ≤κ sinh 2 (c) If K = κ2 > 0 and diam(Σ) <

π 2κ ,



Vol(∂Σ) nωn

1  n−1

.

then

κd ≤κ sin 2



Vol(∂Σ) nωn

1  n−1

,

where diam(Σ) denotes the diameter of Σ. Proof. Define the function f : Σ → R by f (x) = dist(x, C1 ) − dist(x, C2 ). Obviously, f < 0 on C1 and f > 0 on C2 . From the connectedness of Σ and the continuity of the function f , there exists a point p ∈ Σ satisfying that f (p) = 0. Since ΘΣ (p) ≥ 1 and

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K. Seo / Advances in Mathematics 321 (2017) 205–220

dist(p, q) ≥ dist(p, C1 ) = dist(p, C2 ) ≥

d 2

for any q ∈ ∂Σ, applying Proposition 3.1 gives the conclusion. 2 Remark 3.3. In the Euclidean space, the distance estimate for the boundaries of minimal submanifolds was already observed in [23]. As a consequence of Theorem 3.2, we are able to obtain the following quantitative condition for nonexistence of a compact connected minimal submanifold with two boundary components in a Riemannian manifold. Corollary 3.4. Let Γ1 and Γ2 be two (n − 1)-dimensional disjoint compact manifolds in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Define d := dist(Γ1 , Γ2 ). Assume that (a)  d>2

Vol(Γ1 ) + Vol(Γ2 ) nωn

1  n−1

if K = 0,

(b)

sinh

κd >κ 2



Vol(Γ1 ) + Vol(Γ2 ) nωn

1  n−1

if K = −κ2 < 0,

(c) κd >κ sin 2



Vol(Γ1 ) + Vol(Γ2 ) nωn

1  n−1

if K = κ2 > 0.

Then there is no n-dimensional compact connected minimal submanifold Σ ⊂ M bounded by Γ1 ∪ Γ2 for K ≤ 0. In case K > 0, the same conclusion holds under additional π assumption diam(Σ) < 2κ . In particular, if we restrict the dimension of a submanifold Σ to 2, then we have the following distance estimates for a surface with bounded mean curvature in terms of Length(∂Σ), Area(Σ), and the upper bound of |H|. Theorem 3.5. Let C1 and C2 be two disjoint closed sets in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Suppose that there exists a compact connected surface Σ ⊂ M with ∂Σ ⊂ C1 ∪ C2 and ∂Σ ∩ Ci = ∅ for i = 1, 2. Assume that the mean curvature vector H on Σ satisfies |H| ≤ α for some constant α ≥ 0. Define d := dist(C1 , C2 ). Then we have the following.

K. Seo / Advances in Mathematics 321 (2017) 205–220

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(a) If K = 0 and αLength(∂Σ) + α2 Area(Σ) < 2π, then d≤

2Length(∂Σ) . 2π − αLength(∂Σ) − α2 Area(Σ)

(b) If K = −κ2 < 0 and αLength(∂Σ) + (α2 + K)Area(Σ) < 2π, then tanh

κLength(∂Σ) κd ≤ . 2 2π − αLength(∂Σ) − (α2 + K)Area(Σ)

(c) If K = κ2 > 0, αLength(∂Σ) + (α2 + K)Area(Σ) < 2π and diam(Σ) <

π 2κ ,

then

κLength(∂Σ) κd ≤ . 2 2π − αLength(∂Σ) − (α2 + K)Area(Σ)

tan

Proof. As in the proof of Theorem 3.2, there exist a point p ∈ Σ satisfying that ΘΣ (p) ≥ 1 and dist(p, Ci ) ≥ d2 for i = 1, 2. By Proposition 3.1, we see that 2π ≤ 2πΘΣ (p) ≤

1 + r



2Length(∂Σ) |H| ≤ +α r d

Σ

∂Σ



1 . r

(1)

Σ

Lemma 2.1(a) gives Δr ≥

1 ¯ ≥ 1 − α. + H, ∇r r r

Integrating this inequality over Σ shows   1 ∂r − α ≤ Δr = ≤ Area(Σ). r ∂ν Σ

Σ

(2)

Σ

Combining (1) with (2), we obtain (a). From Lemma 2.1(b), it follows that Δ log sinh κr ≥ −K − ακ coth κr. Integrating this inequality over Σε = Σ \ Bp (ε), −K

1 − ακ

Σε

coth κr ≤

Σε

Δ log sinh κr Σε



=

κ coth κr ∂Σ

Letting ε → 0, we have

∂r − ∂ν



Σ∩∂Bp (ε)

κ coth κr

∂r . ∂ν

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2π ≤ 2πΘΣ (p) ≤ κ

coth κr + ακ

coth κr + K

Σ

∂Σ

κd Length(∂Σ) + ακ ≤ κ coth 2

1 Σ



coth κr + KArea(Σ). Σ

On the other hand, applying Lemma 2.1(b), we see that Δr ≥ κ coth κr − α. Thus



coth κr ≤

κ Σ

Δr + αArea(Σ) ≤ Length(∂Σ) + αArea(Σ). Σ

Therefore 2π ≤ κ coth

κd Length(∂Σ) + α(Length(∂Σ) + αArea(Σ)) + KArea(Σ), 2

which proves (b). By making use of Lemma 2.1(c), we have Δ log sin κr ≥ −K − ακ cot κr. Using this inequality, one can prove (c) in the same manner. 2 Remark 3.6. In case of compact minimal surfaces, Theorem 3.5(a) is the same as Theorem 3.2(a). Theorem 3.5 seems new even for surfaces with constant mean curvature in space forms. We recall that a surface with constant mean curvature one in the 3-dimensional hyperbolic space is called a Bryant surface. (Refer to [2,19,20] and the references therein.) It is well-known that minimal surfaces in the 3-dimensional Euclidean space are locally isometric to Bryant surfaces in the 3-dimensional hyperbolic space [12] and hence they share several common properties. As an interesting application of Theorem 3.5, we show a nonexistence result for Bryant surfaces in the 3-dimensional hyperbolic space in the following. Corollary 3.7. Let Γ1 and Γ2 be two disjoint simple closed curves in the hyperbolic space H3 (−1) of constant sectional curvature −1 satisfying that Length(Γ1 ∪Γ2 ) < 2π. Assume that   Length(Γ1 ∪ Γ2 ) dist(Γ1 , Γ2 ) > . tanh 2 2π − Length(Γ1 ∪ Γ2 ) Then there is no compact connected Bryant surface Σ ⊂ H3 (−1) bounded by Γ1 ∪ Γ2 .

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4. Density and distance estimates via the second order of the length of mean curvature vector In this section, we obtain density estimates in terms of the second order of |H|, which gives an application to surfaces with sufficiently small L2 norm of the mean curvature vector. Proposition 4.1 (Density estimate II). Let Σ be an n-dimensional compact connected submanifold with mean curvature vector H in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Then we have the following. (a) If K = 0, then nωn ΘΣ (p) ≤

r1−n +



1 4n

|H|2 r2−n . Σ

∂Σ

(b) If K = −κ2 < 0, then nωn ΘΣ (p) ≤ κn−1

κn−2 4n

sinh1−n κr +

π 2κ

|H|2 sinh2−n κr cosh−1 κr.

Σ

∂Σ

(c) If K = κ2 > 0 and r <



on Σ, then

nωn ΘΣ (p) ≤ κn−1

sin1−n κr +

κn−2 4n



|H|2 sin2−n κr cos−1 κr.

Σ

∂Σ

Proof. Denote by Bp (ε) ⊂ M the geodesic ball centered at p ∈ M of radius ε > 0. Define Σε := Σ \ Bp (ε). For n = 2, applying Lemma 2.2(a), we get  2(1 − |∇r|2 ) 1 ¯ H, ∇r Δ log r ≥ + r2 r Σε

Σε

   ¯  ∇r − ∇r H 2 |H|2 +  − = 2  r 4 8 Σε

≥−

1 8

|H|2 , Σε

¯ − ∇r|2 = |∇r| ¯ 2 − |∇r|2 = 1 − |∇r|2 . Thus where we used the identity |∇r 1 1 1 ∂r − . − |H|2 ≤ Δ log r ≤ 8 r r ∂ν Σε

Σε

∂Σ

Σ∩∂Bp (ε)

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218

Letting ε → 0, we obtain the proof of (a) when n = 2. For n ≥ 3, it follows from Lemma 2.2(a) that ¯ Δ(−r2−n ) ≥ n(n − 2)r−n (1 − |∇r|2 ) + (n − 2)r1−n H, ∇r    2 ¯ − ∇r  ∇r H  |H|2 2−n  + n − = (n − 2)r r 2n  4n ≥−

n − 2 2−n r |H|2 . 4n

Integrating this inequality over Σε shows −

n−2 4n







|H|2 r2−n ≤ (n − 2) Σε

r1−n − (n − 2)

∂Σ

r1−n

∂r . ∂ν

Σ∩∂Bp (ε)

Letting ε → 0, we see that nωn ΘΣ (p) ≤

r1−n +

1 4n

|H|2 r2−n , Σ

∂Σ

which proves (a). Applying Lemma 2.2(b), we get ¯ ΔG(r) ≥ nκn (1 − |∇r|2 ) sinh−n κr cosh κr + κn−1 sinh1−n κr H, ∇r   2  2 ¯   ∇r − ∇r H |H|  − + = κn−1 sinh2−n κr nκ cosh κr  sinh κr 2κ cosh κr  4nκ cosh κr ≥−

κn−2 |H|2 sinh2−n κr cosh−1 κr. 4n

Thus −

κn−2 4n

Σε

|H|2 sinh2−n κr cosh−1 κr ≤

ΔG(r) Σε



≤κ

n−1 ∂Σ

sinh

1−n

κr − κ

n−1

sinh1−n κr

∂r . ∂ν

Σ∩∂Bp (ε)

Letting ε → 0, we obtain (b). Using Lemma 2.2(c), one can prove (c) in the same manner. 2 When the dimension of the submanifolds is equal to 2, one can estimate the distance of the boundaries in terms of the boundary length and the L2 norm of |H| as follows.

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Theorem 4.2. Let C1 and C2 be two disjoint closed set in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Suppose that there exists a compact connected surface Σ ⊂ M with ∂Σ ⊂ C1 ∪ C2 and ∂Σ ∩ Ci = ∅ for i = 1, 2. Define d = dist(C1 , C2 ). Assume that the mean curvature vector H satisfies  |H|2 < 16π. Then we have the following. Σ (a) If K = 0, then d≤

2 Length(∂Σ)  . 2π − 18 Σ |H|2

(b) If K = −κ2 < 0, then sinh

κLength(∂Σ) κd  ≤ . 2 2π − 18 Σ |H|2

Proof. Choose a point p ∈ Σ as in the proof of Theorem 3.2 and Theorem 4.2. Define the distance function r(·) = dist(p, ·) in M . Note that ΘΣ (p) ≥ 1 and dist(p, Ci ) ≥ d2 for i = 1, 2. Proposition 4.1(a) shows that 1 2 1 1 2 + 2π ≤ |H| ≤ Length(∂Σ) + |H|2 , r 8 d 8 Σ

∂Σ

Σ

which proves (a). Similarly, using Proposition 4.1(b), one can prove (b). 2 As a consequence, we get the following nonexistence result for surfaces with small L2 norm of the length of the mean curvature vector in a Riemannian manifold with sectional curvature bounded above a nonpositive constant. Corollary 4.3. Let Γ1 and Γ2 be two disjoint simple closed curves in a complete simply connected Riemannian manifold M of sectional curvature bounded above by a constant K. Define d := dist(Γ1 , Γ2 ). Assume that (a) d>

2Length(Γ1 ∪ Γ2 )  if K = 0, 2π − 18 Σ |H|2

(b) sinh

κLength(Γ1 ∪ Γ2 ) κd  > if K = −κ2 < 0. 2 2π − 18 Σ |H|2

Then there is no compact connected surface Σ ⊂ M bounded by Γ1 ∪ Γ2 satisfying that  |H|2 < 16π. Σ

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