Journal of Geometry and Physics 81 (2014) 87–91

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Journal of Geometry and Physics

journal homepage: www.elsevier.com/locate/jgp

Bernstein type theorem for entire weighted minimal graphsin Gn

× RDoan The Hieu a,∗, Tran Le Nam b

a Department of Mathematics, College of Education, Hue University, Hue, Viet Namb Department of Mathematics, Dong Thap University, Dong Thap, Viet Nam

a r t i c l e i n f o

Article history:Received 4 January 2014Received in revised form 17 March 2014Accepted 20 March 2014Available online 26 March 2014

MSC:primary 53C25secondary 53A1049Q05

Keywords:Gauss spaceWeighted minimal graphsBernstein theorem

a b s t r a c t

Based on a calibration argument, we prove a Bernstein type theorem for entire minimalgraphs over Gauss space Gn by a simple proof.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The classical Bernstein theorem asserts that an entire minimal graph over R2 is a plane (see [1–3]). The theorem hasbeen generalized to dimensions n ≤ 7 (see [4–6]) and has been proved to be false in dimensions n ≥ 8 (see [7]). Entireminimal graphs over Rn, n ≥ 8, that are not hyperplanes, were found by Bombieri, De Giorgi and Giusti in [7]. For arbitrarydimensions, the theorem is settled under some hypotheses on the growth of theminimal graph. The theorem is also studiedin product spaces Mn

× R (see [8]), where M is a Riemannian manifold, as well as for self-similar shrinkers. Ecker andHuisken [9] proved a Bernstein type theorem for self-similar shrinkers that are entire graphs and have at most polynomialvolume growth. Later the condition on volume growth is removed by Wang [10].

Self-similar shrinkers, simplest solutions to mean curvature flow that satisfy the following equation:

H =12⟨x⃗,n⟩,

are just minimal hypersurfaces in Rn+1 under the conformally changed metric gij = exp(−|−→x |

2/2n)δij (see [11]) or special

examples of weighted minimal hypersurfaces in Rn+1 with density e−|x|24 , a modified version of Gauss space. Gauss space

Gn+1, Euclidean space Rn+1 with Gaussian probability density e−f= (2π)−

n+12 e−

|x|22 , are typical examples of a manifold

with density and very interesting to probabilists.

∗ Corresponding author. Tel.: +84 543829200.E-mail addresses: dthehieu@yahoo.com (D.T. Hieu), lenamdongthapmuoi@gmail.com (T.L. Nam).

http://dx.doi.org/10.1016/j.geomphys.2014.03.0110393-0440/© 2014 Elsevier B.V. All rights reserved.

88 D.T. Hieu, T.L. Nam / Journal of Geometry and Physics 81 (2014) 87–91

It should be mentioned that the Bernstein type theorem for self-similar shrinkers can be applied for f -minimal graphs inGn+1

= Gn× G1.

Motivated by these works, we study a Bernstein type theorem for entire weighted minimal graphs in the product spaceGn

× R, that is Rn+1 with mixed Gaussian–Euclidean density.Based on a calibration argument, we prove a volume growth estimate of a weighted minimal graph in the product space

Gn× R and the theorem follows easily. Our proof is quite simple, elementary and holds for arbitrary dimensions without

any additional conditions. The proof is also adapted for the case in which Gauss space is replaced by Rn with a radial densityand finite weighted volume.

2. Weighted minimal surfaces in Gn × R

A manifold with density, also called a weighted manifold, is a Riemannian manifold with a positive function e−f used toweight both volume and perimeter area. The weighted mean curvature of hypersurfaces on such a manifold is defined asfollows:

Hf = H + ⟨∇f ,n⟩, (1)

where H is the Euclidean mean curvature and n is the unit normal vector field of the hypersurface. A hypersurface withHf = 0 is called a weighted minimal hypersurface or an f -minimal hypersurface. For more details about manifolds withdensity, we refer the reader to [12–14].

Let S be a regular hypersurface in Gn× R, M be a point on S and ρ denote the projection onto xn+1-axis. The following

lemma shows a geometric meaning of ⟨∇f ,n⟩.

Lemma 1.

|⟨∇f ,n⟩| = d(ρ(M), TMS),

where ⟨∇f ,n⟩ is evaluated at M.

Proof. Let n(a1, a2, . . . , an+1) be the unit normal vector of S at M . Then an equation of TMS is

n+1i=1

aixi + d = 0.

Therefore,

d(ρ(M), TMS) = |an+1xn+1 + d| = |⟨(x1, x2, . . . , xn, 0), (a1, a2, . . . , an, an+1)⟩| = |⟨∇f ,n⟩|,

where (x1, x2, . . . , xn, xn+1) are coordinates ofM and ⟨∇f ,n⟩ is evaluated atM. �

Below are simple examples of constant weighted mean curvatures and weighted minimal surfaces in G2× R.

1. Planes parallel to the z-axis have constant weighted mean curvature and planes containing the z-axis are weightedminimal.

2. Planes z = a are weighted minimal.3. Right circular cylinders about z-axis have constantweightedmean curvature and the one of radius 1 isweightedminimal.4. Consider the deformation determined by a family of parametric minimal surfaces given by

Xθ (u, v) = (x(u, v), y(u, v), z(u, v));

x(u, v) = cos θ sinh v sin u + sin θ cosh v cos u,y(u, v) = − cos θ sinh v cos u + sin θ cosh v sin u,z(u, v) = u cos θ + v sin θ;

where −π < u ≤ π, − ∞ < v < ∞ and the deformation parameter −π < θ ≤ π. A direct computation shows thatXθ is minimal with the normal vector field

N =

cos ucosh v

,sin ucosh v

, −sinh ucosh v

.

Since ∇f = (x, y, 0), ⟨∇f ,N⟩ = sin θ , Xπ/2 is the catenoid while X0 is the helicoid, it follows that(a) In G2

× R, Xθ has constant weighted curvature.(b) The helicoid X0 is a ruled weighted minimal surface in G2

× R.

(c) The catenoid Xπ/2 has constant weighted curvature 1 in G2× R.

D.T. Hieu, T.L. Nam / Journal of Geometry and Physics 81 (2014) 87–91 89

3. A Bernstein type theorem for weighted minimal graphs in Gn × R

3.1. Minimality of hyperplanes in Gn× (R, e−h)

Consider the product spaceGn×(R, e−h)with product density e−(f+h).Apoint inGn

×(R, e−h) can bewritten as (x, xn+1),where x = (x1, x2, . . . , xn) ∈ Rn. An equation of a non-vertical hyperplane in Gn

× (R, e−h) = Rn+1 is of the form

Σni=1aixi + xn+1 + c = 0, c, a1, a2, . . . , an ∈ R.

A direct computation shows that ⟨∇(f + h),n⟩ = 0 if and only if

Σni=1aixi + h′(xn+1) = 0.

Thus,

1. the plane is weighted minimal and horizontal if and only if h′(−c) = 0,2. the plane is weightedminimal and non-horizontal if and only if h(xn+1) = x2n+1/2+ cxn+1 +b, where b ∈ R is a constant.

If h is monotone, any hyperplane is not weighted minimal and a Bernstein type theorem does not exist for this case. Sinceweighted minimality of a hypersurface in Gn+1 is equivalent to that in Gn

× (R, e−h), where h(xn+1) = x2n+1/2 + cxn+1 + b,the Bernstein type theorem proved by Ecker–Huisken and Wang [9,10] for self-similar shrinkers is adapted for weightedminimal graphs in the later case. The following example shows that in Gn

× (R+, e−h), where R+ = {x ∈ R : x ≥ 0} andh(z) = z2 − ln

√1 + 4z, there exist both hyperplanar and non-hyperplanar entire weighted minimal graphs.

Consider the graph of the function z = u(x, y) = x2 over G2 in G2× (R, e−h). A direct computation yields

H(f+h) =1

(1 + 4z)3/2−

2z + h′(z)

2√1 + 4z

= 0.

Moreover, the horizontal plane z = (1 +√17)/8 is also weighted minimal.

3.2. Entire weighted minimal graph in Gn× R

This subsection considers the casewhereh = const. andwe can assume thath = 1. The spaceGn×R is justRn+1

= Rn×R

endowed with the Euclidean–Gaussian density

e−f= (2π)−

n2 e−

|x|22 .

Denote by:

• Bn+1(p, R) the (n + 1)-ball in Gn× R with center p and radius R,

• Bn(p, R) the n-ball in Gn with center p and radius R,• Sn(p, R) the n-sphere in Gn

× R with center p and radius R,• Sn−1(p, R) the (n − 1)-sphere in Gn with center p and radius R,• Sn

+(p, R) the upper half of Sn(p, R),

• Bn+1+ (p, R) the upper half of Bn+1(p, R).

Let Σ be the weighted minimal graph of a function u(x1, x2, . . . , xn) = xn+1 over Gn and let p be the intersection point ofΣ and xn+1-axis, then we have the following area estimates.

Lemma 2.

Volf (Σ ∩ Bn+1(p, R)) ≤ Volf (Bn(O, R)) + n(2π)−n/2e−R2/2CnRn, (2)

where Cn = Vol Bn(O, 1).

Proof. Let n be a unit normal field of Σ and consider the smooth extension of n by the translation along xn+1-axis, alsodenoted by n.

Consider the n-differential form defined by

w(X1, X2, . . . , Xn) = det(X1, X2, . . . , Xn,n), (3)

where Xi, i = 1, 2, . . . , n are smooth vector fields. It is not hard to see that:

1. |w(X1, X2, . . . , Xn)| ≤ 1, for every normal vector field Xi, i = 1, 2, . . . , n and the equality holds if and only ifX1, X2, . . . , Xn are tangent to Σ .

2. d(e−f w) = 0, because Σ and all its translations along the xn+1-axis are weighted minimal.

90 D.T. Hieu, T.L. Nam / Journal of Geometry and Physics 81 (2014) 87–91

Such a differential form is called a weighted calibration that calibrates Σ . A simple proof by using Stokes’ Theorem provesthat Σ is weighted area-minimizing, i.e. any compact portion of Σ has least area among all surfaces in its homology class(see [15]).

Now let Σ ∩ Bn+1(p, R) := ΣR. Since ΣR is weighted area-minimizing and ∂ΣR ⊂ Sn(p, R), we have the followingestimate:

Volf (ΣR) ≤12Volf Sn(p, R).

Note that the density does not depend on the last coordinate, therefore

12Volf Sn(p, R) =

12Volf Sn(O, R) = Volf Sn+(O, R).

Let η be the volume form of Sn+(O, R) associated with the outward unit normal defined as in (3). Its extension by thetranslation along xn+1-axis in the cylinder C = Bn(O, R) × R is also denoted by η. By applying Stokes’ Theorem togetherwith choosing suitable orientations for objects, we have

Volf Sn+(O, R) =

Sn+

(O,R)e−f η

=

Bn(O,R)

e−f η +

Bn+1+

(O,R)d(e−f η) (by Stokes’ Theorem)

≤ Volf (Bn(O, R)) +

Bn(O,R)×[0,R]

d(e−f η) (|η| ≤ 1 and Bn+1+

(O, R) ⊂ C)

= Volf (Bn(O, R)) +

Sn−1(O,R)×[0,R]

e−f η (by Stokes’ Theorem)

≤ Volf (Bn(O, R)) + (2π)−n/2e−R2/2 Vol(Sn−1(O, R) × [0, R])

= Volf (Bn(O, R)) + n(2π)−n/2e−R2/2CnRn. �

Corollary 3.

Volf (Σ) ≤ 1.

Proof. Taking the limit of both sides of (2) as R goes to infinity. �

Theorem 4. The graph Σ of a function u(x1, x2, . . . , xn) = xn+1 over Gn is weighted minimal if and only if it is a hyperplanexn+1 = a, i.e. u is constant.

Proof. Of course, a horizontal hyperplane in Gn× R is weighted minimal. Now suppose that Σ is weighted minimal. Let

dV = dx1 ∧ dx2 ∧ · · · ∧ dxn. We have

1 ≥ Volf (Σ) =

Gn

e−f1 + |∇u|2dV ≥

Gn

e−f dV = Volf (Gn) = 1.

The equality holds if and only if |∇u|2 = 0, i.e. u is constant. �

Acknowledgments

WethankProfessor FrankMorgan aswell as the referee for reading carefully the first draft of the paper and for corrections.The authors are supported in part by the National Foundation for Science and Technology Development, Vietnam (Grant No.101.01-2011.26).

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