A sharp bound for the area of minimal surfaces in the unit ball
aa r X i v : . [ m a t h . DG ] J a n A SHARP BOUND FOR THE AREA OF MINIMALSURFACES IN THE UNIT BALL
SIMON BRENDLE
Abstract.
Let Σ be a k -dimensional minimal surface in the unit ball B n which meets the boundary ∂B n orthogonally. We show that thearea of Σ is bounded from below by the volume of the unit ball in R k . Introduction
One of the most important tools in minimal surface theory is the classicalmonotonicity formula, which asserts the following:
Theorem 1 (cf. [1], [8]) . Let Σ be a k -dimensional minimal submanifold of R n with boundary ∂ Σ . Moreover, let y be a point in R n and r be a positivereal number with the property that ∂ Σ ∩ B r ( y ) = ∅ . Then the function r
7→ | Σ ∩ B r ( y ) | r k is monotone increasing for r ∈ (0 , r ) . We note that Theorem 1 extends to the more general setting of stationaryvarifolds; see [1], Theorem 8.5. The monotonicity formula plays a funda-mental role in the analysis of singularities. Moreover, it has a number ofinteresting geometric consequences (see e.g. [4]). In particular, it directlyimplies the following classical results:
Corollary 2.
Let Σ be a k -dimensional minimal surface in the unit ball B n which passes through the origin and satisfies ∂ Σ ⊂ ∂B n . Then | Σ ∩ B n | ≥| B k | . Corollary 2 can be viewed as a sharp version of F. Almgren’s densitybound; see [2], p. 343, for details.
Corollary 3.
Let ˆΣ be a closed minimal submanifold of the unit sphere ∂B n of dimension k − . Then | ˆΣ | ≥ | ∂B k | . In order to deduce Corollary 3 from the monotonicity formula, one con-siders the k -dimensional minimal cone Σ = { λ x : x ∈ ˆΣ , λ > } ⊂ R n . Even The author was supported in part by the National Science Foundation under grantDMS-0905628. though the surface Σ is singular at the origin, the monontonicity formulastill holds, and we obtain | ˆΣ | = lim r →∞ k | Σ ∩ B r ( y ) | r k ≥ lim r → k | Σ ∩ B r ( y ) | r k = k | B k | = | ∂B k | , where y is arbitrary point on ˆΣ.In [5], A. Fraser and R. Schoen considered a free boundary value problemfor minimal surfaces in the unit ball. Specifically, they studied minimalsurfaces in the unit ball which meet the boundary orthogonally. In thispaper, 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 B n .Moreover, suppose that the boundary of Σ lies in the unit sphere ∂B n andmeets ∂B n orthogonally. Then | Σ | ≥ | B k | . Moreover, if equality holds, then Σ is contained in a k -dimensional subspace of R n . Applying the divergence theorem to the radial vector field V ( x ) = x gives k | Σ | = Z Σ div Σ V = Z ∂ Σ h V, x i = | ∂ Σ | . Hence, Theorem 4 implies a sharp lower bound for the isoperimetric ratioof Σ:
Corollary 5.
Let Σ be a k -dimensional minimal surface in the unit ball B n .Moreover, suppose that the boundary of Σ lies in the unit sphere ∂B n andmeets ∂B n orthogonally. Then | ∂ Σ | k | Σ | k − ≥ | ∂B k | k | B k | k − . Moreover, if equality holds, then Σ is contained in a k -dimensional subspaceof R n . 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 byA. Fraser and R. Schoen (cf. [5], Theorem 5.4).The proof of Theorem 4 is inspired by the classical monotonicity formulafor minimal submanifolds, and its analogue for the mean curvature flow (cf.[3], [6]). In order to prove the classical monotonicity formula, one appliesthe divergence theorem to the vector field x − y | x − y | k . This vector field canbe interpreted as the gradient of the Newton potential in R k . Similarly,Huisken’s monotonicity formula for the mean curvature flow involves anintegral of the backward heat kernel in R k . In order to prove Theorem 4,we apply the divergence theorem to a suitably defined vector field W . Thisvector field agrees with the gradient of the Green’s function for the Neumannboundary value problem on B k , up to a factor.The author would like to thank Professor Frank Morgan and ProfessorBrian White for comments on an earlier version of this paper. INIMAL SURFACES IN THE UNIT BALL 3 Proof of Theorem 4
Let us fix a point y ∈ ∂B n . We define a vector field W in B n \ { y } by W ( x ) = 12 x − x − y | x − y | k − k − Z tx − y | tx − y | k dt. Lemma 6.
For every point x ∈ B n and every orthonormal k -frame { e , . . . , e k } ⊂ R n , we have k X i =1 h D e i W, e i i ≤ k . Proof.
We have k X i =1 h D e i W, e i i = k − k | x − y | k +2 (cid:16) | x − y | − k X i =1 h x − y, e i i (cid:17) − k − Z tk | tx − y | k +2 (cid:16) | tx − y | − k X i =1 h tx − y, e i i (cid:17) dt ≤ k , as claimed. Lemma 7.
The vector field W is tangential along the boundary ∂B n . Proof.
A straightforward computation gives h W, x i = 12 | x | − h x − y, x i| x − y | k − k − Z h tx − y, x i| tx − y | k dt = 12 | x | − h x − y, x i| x − y | k + 12 Z ddt (cid:16) | tx − y | k − (cid:17) dt = 12 | x | − h x − y, x i| x − y | k + 12 1 | x − y | k − −
12= 12 (1 − | x | ) (cid:16) | x − y | k − (cid:17) . In particular, h W, x i = 0 for all x ∈ ∂B n \ { y } . Lemma 8.
We have W ( x ) = − x − y | x − y | k + o (cid:16) | x − y | k − (cid:17) as x → y . Proof.
Using the inequality | tx − y | = t | x − y | + (1 − t )(1 − t | x | ) ≥ t | x − y | + (1 − t ) , SIMON BRENDLE we obtain (cid:12)(cid:12)(cid:12)(cid:12) W ( x ) + x − y | x − y | k (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) x − k − Z tx − y | tx − y | k dt (cid:12)(cid:12)(cid:12)(cid:12) ≤
12 + k − Z | tx − y | k − dt ≤
12 + k − Z (cid:16) t | x − y | + (1 − t ) (cid:17) k − dt. It follows from the dominated convergence theorem that Z (cid:16) | x − y | t | x − y | + (1 − t ) (cid:17) k − dt → x → y . Thus, we conclude that W ( x ) + x − y | x − y | k = o (cid:16) | x − y | k − (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 be the vector field defined above. Note that W is smoothon B n \ { y } . Using the divergence theorem, we obtain Z Σ \ B r ( y ) (cid:16) k − div Σ W (cid:17) = k | Σ \ B r ( y ) | − Z Σ ∩ ∂B r ( y ) h W, ν i − Z ∂ Σ \ B r ( y ) h W, x i , (1)where ν denotes the inward pointing unit normal to the region Σ ∩ B r ( y )within the submanifold Σ. In particular, the vector ν is tangential to Σ, butnormal to Σ ∩ ∂B r ( y ). It is easy to see that ν = − x − y | x − y | + o (1)for x ∈ Σ ∩ ∂B r ( y ). Using Lemma 8, we obtain h W, ν i = 1 r k − + o (cid:16) r k − (cid:17) for x ∈ Σ ∩ ∂B r ( y ). Since | Σ ∩ ∂B r ( y ) | = 12 | ∂B k | r k − + o ( r k − ) , we conclude that(2) lim r → Z Σ ∩ ∂B r ( y ) h W, ν i = 12 | ∂B k | = k | B k | . INIMAL SURFACES IN THE UNIT BALL 5
On the other hand, it follows from Lemma 7 that h W, x i = 0 for all x ∈ ∂B n \ { y } . This implies(3) Z ∂ Σ \ B r ( y ) h W, x i = 0 . Combining (1), (2), and (3), we obtainlim r → Z Σ \ B r ( y ) (cid:16) k − div Σ W (cid:17) = k | Σ | − | B k | ) . On the other hand, we have k − div Σ W ≥ | Σ | − | B k | ≥ | Σ | − | B k | = 0.This implies k − div Σ W = 0at each point x ∈ Σ \ { y } . Consequently, for each point x ∈ Σ \ { y } , we have | x − y | − k X i =1 h x − y, e i i = 0 , where { e , . . . , e k } is an orthonormal basis of T x Σ. From this, we deducethat x − y ∈ T x Σ for all points x ∈ Σ \ { y } . Since y ∈ ∂ Σ is arbitrary, weconclude that ∂ Σ is contained in a k -dimensional affine subspace of R n . Bythe maximum principle, Σ is contained in a k -dimensional affine subspaceof R n . References [1] W. Allard,
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