Some results on slices and entire graphs in certain weighted warped products

SOME RESULTS ON SLICES AND ENTIRE GRAPHS IN CERTAIN WEIGHTED WARPED PRODUCTS Nguyen Thi My Duyen Department of Mathematics College of Education, Hue University 32 Le Loi, Hue, Vietnam e-

SOME RESULTS ON SLICES AND ENTIRE GRAPHS IN CERTAIN WEIGHTED

WARPED PRODUCTS Nguyen Thi My Duyen

Department of Mathematics College of Education, Hue University

32 Le Loi, Hue, Vietnam e-mail: ntmyduyen2909@gmail.com

Abstract

We study the area-minimizing property of slices in the weighted warped product manifold (R+× f Rn , e −ϕ), assuming that the density function

e −ϕand the warping functionf satisfy some additional conditions Based

on a calibration argument, a slice {t0} × G n is proved weighted

area-minimizing in the class of all entire graphs satisfying a volume balance condition and some Bernstein type theorems inR+× fGnandG+× fGn ,

whenf is constant, are obtained.

Recently, the study of weighted minimal submanifolds, and in particular weighted minimal hypersurfaces had attracted many researchers (see, for in-stance, [2], [4], [5], [7]) A weighted manifold (also called a manifold with

density) is a Riemannian manifold endowed with a positive function e −ϕ ,

called the density, used to weight both volume and perimeter elements The

weighted area of a hypersurface Σ in an (n + 1)-dimensional weighted

man-ifold is Areaϕ(Σ) = 

Σe −ϕ dA and the weighted volume of a region Ω is

Volϕ(Ω) = 

Ωe −ϕ dV, where dA and dV are the n-dimensional Riemannian

area and (n + 1)-dimensional Riemannian volume elements, respectively A

typical example of such manifolds is Gauss spaceGn+1 , Rn+1 with Gaussian

Key words: Manifold with density, weighted warped product manifold, calibration.

2010 AMS Mathematics Classification: 53C25, 53C38; Secondary: 53A10, 53A07.

76

density (2π) − n+1

2 e − r2

2 , which is appeared in probability and statistics The

hypersurface Σ inRn+1 is said to be weighted minimal or ϕ-minimal if

H ϕ (Σ) := H(Σ) +1

n ∇ϕ, N = 0,

where H(Σ) and N are the classical mean curvature and the unit normal vector field of Σ, respectively H ϕ (Σ) is called the weighted mean curvature of Σ.

A theme widely approached in recent years is problems concerning to hy-persurfaces in a warped product manifold of the typeR+× f M, where R+=

[0, + ∞), (M, g) is an n-dimensional Riemannian manifold and f is a positive

smooth function defined onR+(see [8]) Note that with these ingredients, the

product manifoldR+× f M is endowed with the Riemannian metric

¯

g = π ∗

R +(dt2) + f(π

R +)2π ∗

M (g),

where πR + and π M denote the projections ontoR+ and M, respectively.

InRn , let P be a part of a slice, viewed as a graph over a domain D and

let Σ be a graph of a function u over D It is clear that

Area(Σ) =



D



1 +|∇u|2dA ≥



D

dA = Area(P ).

However, in general, the above inequality doesn’t always hold if the ambient space is a weighted manifold For instance, consider R2 with radial density

e −1(x2+y2 ) Let R be a positive real number, P = {(x, 0) ∈ R2: −R ≤ x ≤ R}

and Σ be the half circle defined by x2+ y2= R2, y ≥ 0 The weighted length

of P, L ϕ (P ), and the weighted length of Σ, L ϕ (Σ), are

L ϕ (P ) =

 R

−R

e −1x2

dx,

and

L ϕ(Σ) =

 π

0

e −1R2

R dt = e −1R2

Rπ.

A simple computation shows that



2π(1 − e −1R2

)≤ L ϕ (P ) ≤π(1 − e −R2) When R = 2, we have L ϕ (P ) ≥ L ϕ (Σ).

As another example, we considerR2with density e y Let

P =



x, − ln cos π

3



∈ R2: − π

3 ≤ x ≤ π

3

and Σ be the graph of function y = − ln cos x, x ∈ − π

3,

π

3

It’s not hard to

check that L ϕ (P ) ≥ L ϕ (Σ).

Hence, the area-minimizing property of slices in weighted warped product manifolds is not a trivial matter In this paper, using the same method as in

[2] we prove that if (log f)  (t) ≤ 0, then the slice is weighted area-minimizing

under a volume balance condition In particular, when f is constant we get

some Bernstein type theorems inR+× fGn andG+× fGn .

Consider the warped productR+× f Rn with density e −ϕ , where ϕ = ϕ(t, x).

Let u ∈ C2(Rn ), and Σ = {(u(x), x) : x ∈ R n } be the entire graph defined by

u A unit normal vector field of Σ is

N =

f(u)



f(u)2+|Du|2, − 1

f(u)

f(u)2+|Du|2Du ,

where Du is the gradient of u inRn , and |Du|2 = Du, Du The curvature

function (relative to N ) is H = 1

n trace(A), where A is the shape operator A

direct computation gives (see [8, Section 5])

nH(u) = div

Du f(u)

f2+|Du|2 − f  (u)

f(u)2+|Du|2



n − |Du|2 f(u)2



.

Thus,

nH ϕ (u) = f(u)1 div

Du



f(u)2+|Du|2 − nf  (u)

f(u)2+|Du|2 +

f(u)



f(u)2+|Du|2ϕ t

f(u)

f(u)2+|Du|2Du, Dϕ.

It is easy to see that the mean curvature as well as the weighted mean curvature

of slice are constants

H(t0) := H(t0, x) = −(log f)  (t

0),

and

H ϕ (t0) := H ϕ (t0, x) = −(log f)  (t

0) + ϕ t (t0, x).

Furthermore, if ϕ = ϕ(x), x ∈ R n (i.e., the weighted function e −ϕ does not

depend on the parameter t ∈ R+), H

ϕ (t0) =−(log f)  (t

0).

Let Σ and N as above Consider the smooth extension of N by the trans-lation along t-axis, also denoted by N and the n-differential form defined by

φ(t, x) = f(t) n ω(x),

where ω(X1, , X n ) = det(X1, , X n , N ), X i , i = 1, 2, , n, are smooth vector

fields on Σ It is clear that f(t) n |ω(X1, , X n)| ≤ 1, for all orthonormal vector

fields X i , i = 1, 2, , n and f(t) n |ω(X1, , X n)| = 1 if and only if X1, , X n

are tangent to Σ Therefore, φ(t, x) represents the weighted volume element of

Σ in (R+× fRn , e −ϕ ) We have

divN = −nH − f 

f2+|Du|2



n − |Du|2

f2



 |Du|2

(f2+|Du|2)3.

Note that dω = div(N ) dVR +×R n , thus

dφ = d(f n ω) = div(f n N ) dV

R +×R n = f n divN dV

R +×R n + nf n−1 f  ∂ t , N  dVR+×R n

= divN dVR +× fRn + n f



f ∂ t , N  dVR+× fRn

=

−nH + f  |Du|2

f2

f2+|Du|2+

f  |Du|2

(f2+|Du|2)3 dVR +× fRn

Since

d(e −ϕ φ) = d(e −ϕ f n ω) = e −ϕ f n divN dV

R +×R n+∇(e −ϕ f n ), N  dVR+×R n

= e −ϕ dφ − e −ϕ f n ∇ϕ, N dVR+×R n

= e −ϕ



−nH + f  |Du|2

f2

f2+|Du|2+

f  |Du|2

(f2+|Du|2)3 − ∇ϕ, N



dVR +× fRn

= e −ϕ



 |Du|2

f2

f2+|Du|2 +

f  |Du|2

(f2+|Du|2)3



dVR +× fRn ,

we have

d ϕ φ = e ϕ d(e −ϕ φ) =

 |Du|2

f2

f2+|Du|2+

f  |Du|2

(f2+|Du|2)3 dVR +× fRn

When Σ is a slice, d ϕ φ = −nH ϕ dVR +× fRn

3.1 The results on slices

ConsiderR+× fRn with density e −ϕ , ϕ = ϕ(t, x) Suppose that D is a domain

inRn such that D, the closure of D, is compact Let P D={t0} × D and Σ D

be the graph of a function t = u(x), x ∈ D, such that P D and ΣD have the

same boundary, i.e., ∂P D = ∂Σ D Let E1 ={(t, x) ∈ R+× D : t ≤ u(x)} and

E2 = {(t, x) ∈ R+× D : t ≤ t0} The following theorem shows that P D has least weighted area in the class of hypersurfaces with the same boundary

Theorem 3.1 If Vol ϕ (E1) = Volϕ (E2) and (log f)  (t) ≤ 0, then Area ϕ (P D)≤

Areaϕ(ΣD ).

Proof Denote by φ the volume form ofRn By Stokes’ Theorem and the

suit-able orientations for objects (see Figure 1), we get

Areaϕ (D) − Area ϕ(ΣD)≤



D

e −ϕ φ −



ΣD

e −ϕ φ =

D−Σ D

e −ϕ φ

=



E1

e −ϕ d

ϕ φ =



E1\E2

e −ϕ d

ϕ φ +



E1∩E2

e −ϕ d

ϕ φ,

Areaϕ (P D)− Area ϕ (D) ≤



P D

e −ϕ φ −



D

e −ϕ φ =

P D −D

e −ϕ φ

=−



E2

e −ϕ d

ϕ φ = −



E2\E2

e −ϕ d

ϕ φ −



E1∩E2

e −ϕ d

ϕ φ.

Therefore,

Areaϕ (P D)− Area ϕ(ΣD)≤



E1\E2

e −ϕ d

ϕ φ −



E2\E2

e −ϕ d

ϕ φ

=−



E1\ E2

e −ϕ nH

ϕ (t) dV +



E2\E2

e −ϕ nH

ϕ (t) dV.

The condition (log f)  (t) ≤ 0 means that H ϕ is non-decreasing along t-axis.

Figure 1: A part of slice and graph have the same boundary

Therefore,

H ϕ (t0)≤ H ϕ (t), ∀(t, x) ∈ E1\ E2; H ϕ (t) ≤ H ϕ (t0), ∀(t, x) ∈ E2\ E1.

Areaϕ (P D)− Area ϕ(ΣD)≤ −nH ϕ (t0)



E1\E2

e −ϕ dV −



E2\E1

e −ϕ dV

=−nH ϕ (t0)(V ol ϕ (E1\ E2)− Vol ϕ (E2\ E1)) = 0,

because Volϕ (E1) = Volϕ (E2) Thus, Area ϕ (P D)≤ Area ϕ(ΣD ). 2

In the case of Rn is the Gauss space G n , consider R+× f Gn , i.e., R+× f Rn

with density e −ϕ = (2π) −n/2 e − |x|22 In this space, slices are proved to be global

weighted area-minimizing

Theorem 3.2 If (log f)  (t) ≤ 0, then a slice is weighted area-minimizing in the class of all entire graphs satisfying Vol ϕ (E1) = Volϕ (E2).

Proof Let P be the slice {t0} × G n and Σ be the graph of a function t = u(x)

overGn Let S n−1

R be the (n − 1)-sphere with center O and radius R in G n and

C R=R × S n−1

R be the n-dimensional cylinder Let E1={(t, x) ∈ R+× G n :

t ≤ u(x)} and E2 ={(t, x) ∈ R+× G n : t ≤ t0} Let A = E1\ E2 ∪ E2\ E1.

The parts of P, Σ, E1, and E2, bounded by C R , are denoted by P R , Σ R , E1R ,

and E2R , respectively.

Denote by φ the volume form ofGn Let R be large enough such that C

R

meets both E1\ E2 and E2\ E1 (see Figure 2) In a similar way to the proof

of Theorem 3.1, we have

Areaϕ(Gn

R)− Area ϕ(ΣR) +



C R ∩E1

e −ϕ φ ≤



Gn R

e −ϕ φ −



ΣR

e −ϕ φ +

C R ∩E1

e −ϕ φ

=



E 1R

e −ϕ d

ϕ φ =



E 1R \E 2R

e −ϕ d

ϕ φ +



E 1R ∩E 2R

e −ϕ d

ϕ φ,

Areaϕ (P R)− Area ϕ(Gn

R) +



C R ∩E2

e −ϕ φ ≤



P R

e −ϕ φ −



Gn R

e −ϕ φ +

C R ∩E2

e −ϕ φ

=−



E 2R

e −ϕ d

ϕ φ = −



E 2R \E 1R

e −ϕ d

ϕ φ −



E 2R ∩E 1R

e −ϕ d

ϕ φ.

Therefore,

Areaϕ (P R)− Area ϕ(ΣR) +



C R ∩A

e −ϕ φ ≤



E 1R \E 2R

e −ϕ d

ϕ φ −



E 2R \E 2R

e −ϕ d

ϕ φ

=



E 2R \E 1R

e −ϕ nH

ϕ (t) dV −



E 1R \ E 2R

e −ϕ nH

ϕ (t) dV (3.1)

Figure 2: The slice P, entire graph Σ and G nin R +× fGn

Since (log f)  (t) ≤ 0,

H ϕ (t0)≤ H ϕ (t), ∀(t, x) ∈ E1R \ E2R and H ϕ (t) ≤ H ϕ (t0), ∀(t, x) ∈ E2R \ E1R

Thus,

Areaϕ (P R)− Area ϕ(ΣR) +



C R e ∩A

−ϕ φ ≤ nH ϕ (t

0 )  Volϕ (E2R \ E1 R)− Vol ϕ (E1R \ E2 R) 

.

(3.2)

Moreover, it is easy to see that limR→∞

C R ∩A e −ϕ φ = lim R→∞ e −cR

2

C R ∩A φ =

0.

By the assumption Volϕ (E1) = Volϕ (E2), we have

lim

R→∞Volϕ (E1R \ E2R) = lim

R→∞Volϕ (E2R \ E1R ).

Hence, taking the limit of both sides of (3.2) as R goes to infinity, we obtain

3.2 Some Bernstein type results

3.2.1 A Bernstein type result in R+× aGn

Consider the weighted warped product manifoldR+× aGn with density e −ϕ=

(2π) −n/2 e − |x|22 , where a is a positive constant Let P, Σ, E1, E2, A, C R , P R ,

ΣR , E1R , E2R be defined as in the proof of Theorem 3.2 If u is bounded, then

Volϕ (E1), Vol ϕ (E2) and Volϕ (A) are finite Since the weighted mean curvature

of Σ on the region A, H ϕ , does not change along any vertical line, we get the

following results:

Theorem 3.3 If H ϕ (Σ) and u are bounded and Vol ϕ (E1) = Volϕ (E2), then

Areaϕ(Σ)≤ Area ϕ (P ) + 1

2n(M − m) Vol ϕ (A),

where m = inf H ϕ (Σ) and M = sup H ϕ (Σ).

Proof Denote by φ the volume form of Σ In this case, d ϕ φ = −nH ϕ dV Let

R be large enough such that C R meets both E1\ E2 and E2\ E1 (see Figure 2) By changing ΣR and P R together in (3.1), we have

Areaϕ(ΣR)− Area ϕ (P R) +



C R ∩A e

−ϕ φ ≤

E 1R \E e 2R

−ϕ nH ϕ (Σ) dV −

E 2R \E e 1R

−ϕ nH ϕ (Σ) dV

≤ nM Vol ϕ (E1R \ E2 R)− nm Vol ϕ (E2R \ E1 R ).

(3.3)

By the assumption Volϕ (E1) = Volϕ (E2), taking the limit of both sides of (3.3)

as R goes to infinity, we get Area ϕ(Σ)≤ Area ϕ (P ) +12n(M − m) Vol ϕ (A). 2

Corollary 3.4 (Bernstein type theorem in R+× aGn ) A bounded entire

constant mean curvature graph must be a slice and therefore, is minimal Proof Assume that Σ is an entire constant mean curvature graph of a bounded

function u Since Vol ϕ (E1) is finite, there exists a slice P such that Vol ϕ (E1) =

Volϕ (E2) Because m = M, by Theorem 3.3, it follows that Area ϕ(Σ) ≤

Areaϕ (P ) Moreover,

Areaϕ(Σ) =



Gn

e −ϕ

a4+ a2|Du|2dA ≥



Gn

e −ϕ √

a4dA = Area ϕ (P ).

Therefore, Areaϕ(Σ) = Areaϕ (P ) and Du = 0, i.e., u is constant It is not

hard to see that Σ = P and therefore, is minimal 2

3.2.2 A Bernstein type result in G+× aGn

Now, consider the weighted warped product manifoldG+× aGn with density

e −ϕ = (2π) −(n+1)/2 e − r2

2 , and let Σ be an entire graph of a function u(x) over

Gn , since

∇ϕ(u(x) + Δt, x), N(u(x) + Δt, x) − ∇ϕ(u(x), x), N(u(x), x)

=(u(x) + Δt, x) − (u(x), x), N = (Δt, 0), N ≥ 0, for Δt ≥ 0,

the weighted mean curvature of Σ is increasing along any vertical line We have

Lemma 3.5.

Areaϕ(Gn)≤ Area ϕ (Σ).

Proof Denote by φ the volume form ofGn Replacing C

R by S R , the n-sphere

with center O and radius R, in Subsection 3.2.1 Let R be large enough such that S R meets Σ (see Figure 3), we get

Areaϕ(Gn

R)− Area ϕ(ΣR) +



S R ∩E1

e −ϕ φ ≤ −



E 1R

e −ϕ nH

ϕ(Gn ) dV = 0.

Figure 3: An entire graph Σ and GninG +× aGn

Theorem 3.6 (Bernstein type theorem in G+× aGn ) The only entire

weighted minimal graph inG+× aGn isGn .

Proof Denote by φ the volume form of Σ (see Figure 3), we have

Areaϕ(ΣR)− Area ϕ(Gn

R) +



S R ∩E1

e −ϕ φ ≤



E 1R

e −ϕ nH

ϕ (Σ) dV = 0. (3.4)

Taking the limit of both sides of (3.4) as R goes to infinity, we get

Areaϕ(Σ)≤ Area ϕ(Gn ).

Hence, it follows from Lemma 3.5 that

Areaϕ(Σ) = Areaϕ(Gn ). (3.5)

Since Volϕ(G+ × a Gn ) is finite, there exists a slice P such that Vol

ϕ (E1) =

Volϕ (E2) Using the similar arguments as in the proof of Theorem 3.3 (see

Figure 4), we get

Figure 4: The slice P and entire graph Σ in G+× aGn

Areaϕ(ΣR)− Area ϕ (P R) +



S R e ∩A

−ϕ φ ≤

E 1R e \E 2R

−ϕ nH ϕ (Σ) dV −

E 2R e \E 1R

−ϕ nH ϕ (Σ) dV = 0,

because Σ is a weighted minimal graph Therefore, Areaϕ(Σ)≤ Area ϕ (P ) By

Theorem 3.2, it follows that

Areaϕ(Σ) = Areaϕ (P ). (3.6)

Thus, it follows from (3.5) and (3.6) that

Areaϕ (P ) = Area ϕ(Gn ).

Hence, P =Gn and Vol

ϕ (E1) = Volϕ (E2) = 0, i.e., Σ =Gn . 2

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Hence, taking the limit of both sides of (3.2) as R goes to in? ??nity, we obtain

3.2 Some Bernstein type results< /b>

3.2.1 A Bernstein type result in R+×... (A) are finite Since the weighted mean curvature

of Σ on the region A, H ϕ , does not change along any vertical line, we get the

following results:

Theorem... aGn

Theorem 3.6 (Bernstein type theorem in G+× aGn ) The only entire< /b>

weighted minimal graph in< /i>G+×