A sharp bound for the area of minimal surfaces in the unit ball

0.09 MB·English

by  S. Brendle

A SHARP BOUND FOR THE AREA OF MINIMAL
SURFACES IN THE UNIT BALL
2
1 SIMON BRENDLE
0
2
Abstract. Let Σ be a k-dimensional minimal surface in the unit ball
n Bn which meets the boundary ∂Bn orthogonally. We show that the
a area of Σ is boundedfrom below bythevolume of the unit ball in Rk.
J

0
1

]
G 1. Introduction
D
One of the most important tools in minimal surface theory is the classical
.
h monotonicity formula, which asserts the following:
t
a Theorem 1 (cf. [1], [8]). Let Σ be a k-dimensional minimal submanifold of
m
Rn with boundary ∂Σ. Moreover, let y be a point in Rn and r be a positive
0
[
real number with the property that ∂Σ∩Br0(y) = ∅. Then the function
2
v |Σ∩Br(y)|
r 7→
4 rk
4
5 is monotone increasing for r ∈ (0,r0).
4
. We notethatTheorem1extends tothemoregeneral settingof stationary
8
0 varifolds; see [1], Theorem 8.5. The monotonicity formula plays a funda-
1 mental role in the analysis of singularities. Moreover, it has a number of
1
interesting geometric consequences (see e.g. [4]). In particular, it directly
:
v implies the following classical results:
i
X
Corollary 2. Let Σ be a k-dimensional minimal surface in the unit ball Bn
ar which passes through the origin and satisfies ∂Σ ⊂ ∂Bn. Then |Σ∩Bn| ≥
|Bk|.
Corollary 2 can be viewed as a sharp version of F. Almgren’s density
bound; see [2], p. 343, for details.
Corollary 3. Let Σˆ be a closed minimal submanifold of the unit sphere ∂Bn
of dimension k−1. Then |Σˆ|≥ |∂Bk|.
In order to deduce Corollary 3 from the monotonicity formula, one con-
sidersthek-dimensionalminimalconeΣ = {λx : x ∈ Σˆ, λ > 0} ⊂ Rn. Even
The author was supported in part by the National Science Foundation under grant
DMS-0905628.
1
2 SIMONBRENDLE
though the surface Σ is singular at the origin, the monontonicity formula
still holds, and we obtain
k|Σ∩B (y)| k|Σ∩B (y)|
|Σˆ| = lim r ≥ lim r = k|Bk|= |∂Bk|,
r→∞ rk r→0 rk
where y is arbitrary point on Σˆ.
In [5], A. Fraser and R. Schoen considered a free boundary value problem
for minimal surfaces in the unit ball. Specifically, they studied minimal
surfaces in the unit ball which meet the boundary orthogonally. In this
paper, we give an optimal lower bound for the area of such surfaces:
Theorem 4. Let Σ be a k-dimensional minimal surface in the unit ball Bn.
Moreover, suppose that the boundary of Σ lies in the unit sphere ∂Bn and
meets ∂Bn orthogonally. Then |Σ|≥ |Bk|. Moreover, if equality holds, then
Σ is contained in a k-dimensional subspace of Rn.
Applyingthe divergence theorem to the radial vector field V(x)= x gives
k|Σ|= div V = hV,xi = |∂Σ|.
Z Σ Z
Σ ∂Σ
Hence, Theorem 4 implies a sharp lower bound for the isoperimetric ratio
of Σ:
Corollary 5. Let Σ be a k-dimensional minimal surface in the unit ball Bn.
Moreover, suppose that the boundary of Σ lies in the unit sphere ∂Bn and
meets ∂Bn orthogonally. Then
|∂Σ|k |∂Bk|k
≥ .
|Σ|k−1 |Bk|k−1
Moreover, if equality holds, then Σ is contained in a k-dimensional subspace
of Rn.
Theorem 4 was conjectured by R. Schoen (see e.g. [7]), following a ques-
tion posed earlier by L. Guth. The k = 2 case of Theorem 4 was verified by
A. Fraser and R. Schoen (cf. [5], Theorem 5.4).
The proof of Theorem 4 is inspired by the classical monotonicity formula
for minimal submanifolds, and its analogue for the mean curvature flow (cf.
[3], [6]). In order to prove the classical monotonicity formula, one applies
the divergence theorem to the vector field x−y . This vector field can
|x−y|k
be interpreted as the gradient of the Newton potential in Rk. Similarly,
Huisken’s monotonicity formula for the mean curvature flow involves an
integral of the backward heat kernel in Rk. In order to prove Theorem 4,
we apply the divergence theorem to a suitably defined vector field W. This
vectorfieldagreeswiththegradientoftheGreen’sfunctionfortheNeumann
boundary value problem on Bk, up to a factor.
The author would like to thank Professor Frank Morgan and Professor
Brian White for comments on an earlier version of this paper.
MINIMAL SURFACES IN THE UNIT BALL 3
2. Proof of Theorem 4
Let us fix a point y ∈ ∂Bn. We define a vector field W in Bn\{y} by
1 x−y k−2 1 tx−y
W(x) = x− − dt.
2 |x−y|k 2 Z |tx−y|k
0
Lemma6. Foreverypointx ∈ Bn andeveryorthonormal k-frame{e ,...,e } ⊂
1 k
Rn, we have
k
k
hD W,e i≤ .
ei i 2
X
i=1
Proof. We have
k k
k k
hD W,e i = − |x−y|2− hx−y,e i2
X ei i 2 |x−y|k+2 (cid:16) X i (cid:17)
i=1 i=1
k−2 1 tk k
− |tx−y|2− htx−y,e i2 dt
2 Z0 |tx−y|k+2 (cid:16) Xi=1 i (cid:17)
k
≤ ,
2
as claimed.
Lemma 7. The vector field W is tangential along the boundary ∂Bn.
Proof. A straightforward computation gives
1 hx−y,xi k−2 1 htx−y,xi
hW,xi = |x|2− − dt
2 |x−y|k 2 Z |tx−y|k
0
1 hx−y,xi 1 1 d 1
= |x|2− + dt
2 |x−y|k 2Z0 dt(cid:16)|tx−y|k−2(cid:17)
1 hx−y,xi 1 1 1
= |x|2− + −
2 |x−y|k 2 |x−y|k−2 2
1 1
= (1−|x|2) −1 .
2 (cid:16)|x−y|k (cid:17)
In particular, hW,xi = 0 for all x ∈ ∂Bn\{y}.
Lemma 8. We have
x−y 1
W(x)= − +o
|x−y|k (cid:16)|x−y|k−1(cid:17)
as x → y.
Proof. Using the inequality
|tx−y|2 = t|x−y|2+(1−t)(1−t|x|2)≥ t|x−y|2+(1−t)2,
4 SIMONBRENDLE
we obtain
x−y 1 k−2 1 tx−y
W(x)+ = x− dt
(cid:12) |x−y|k(cid:12) (cid:12)2 2 Z |tx−y|k (cid:12)
(cid:12) (cid:12) (cid:12) 0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)1 k−2 1 1 (cid:12)
≤ + dt
2 2 Z |tx−y|k−1
0
1 k−2 1 1 k−1
2
≤ + dt.
2 2 Z0 (cid:16)t|x−y|2+(1−t)2(cid:17)
It follows from the dominated convergence theorem that
1 |x−y|2 k−1
2
dt → 0
Z0 (cid:16)t|x−y|2+(1−t)2(cid:17)
as x → y. Thus, we conclude that
x−y 1
W(x)+ = o
|x−y|k (cid:16)|x−y|k−1(cid:17)
as x → y. This completes the proof.
We now give the proof of Theorem 4. To that end, let us fix a point
y ∈ ∂Σ, and let W bethevector field definedabove. Note that W is smooth
on Bn\{y}. Using the divergence theorem, we obtain
k
−div W
ZΣ\Br(y)(cid:16)2 Σ (cid:17)
k
(1) = |Σ\B (y)|− hW,νi− hW,xi,
r
2 Z Z
Σ∩∂Br(y) ∂Σ\Br(y)
where ν denotes the inward pointing unit normal to the region Σ∩B (y)
r
within the submanifold Σ. In particular, the vector ν is tangential to Σ, but
normal to Σ∩∂B (y). It is easy to see that
r
x−y
ν = − +o(1)
|x−y|
for x ∈ Σ∩∂B (y). Using Lemma 8, we obtain
r
1 1
hW,νi = +o
rk−1 (cid:16)rk−1(cid:17)
for x ∈ Σ∩∂B (y). Since
r
1
|Σ∩∂B (y)| = |∂Bk|rk−1+o(rk−1),
r
2
we conclude that
1 k
(2) lim hW,νi = |∂Bk|= |Bk|.
r→0ZΣ∩∂Br(y) 2 2
MINIMAL SURFACES IN THE UNIT BALL 5
On the other hand, it follows from Lemma 7 that hW,xi = 0 for all x ∈
∂Bn\{y}. This implies
(3) hW,xi = 0.
Z
∂Σ\Br(y)
Combining (1), (2), and (3), we obtain
k k
lim −div W = (|Σ|−|Bk|).
r→0ZΣ\Br(y)(cid:16)2 Σ (cid:17) 2
On the other hand, we have
k
−div W ≥ 0
Σ
2
by Lemma 6. Putting these facts together, we conclude that |Σ|−|Bk| ≥ 0.
It remains to analyze the case of equality. Suppose that |Σ|−|Bk| = 0.
This implies
k
−div W = 0
Σ
2
at each point x ∈ Σ\{y}. Consequently, for each point x ∈ Σ\{y}, we have
k
|x−y|2− hx−y,e i2 = 0,
i
X
i=1
where {e ,...,e } is an orthonormal basis of T Σ. From this, we deduce
1 k x
that x−y ∈ T Σ for all points x ∈ Σ\{y}. Since y ∈ ∂Σ is arbitrary, we
x
conclude that ∂Σ is contained in a k-dimensional affine subspace of Rn. By
the maximum principle, Σ is contained in a k-dimensional affine subspace
of Rn.
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Department of Mathematics, Stanford University, 450 Serra Mall, Bldg
380, Stanford, CA 94305