EXISTENCE AND NON-EXISTENCE OF AREA-MINIMIZING HYPERSURFACES IN MANIFOLDS OF NON-NEGATIVE RICCI CURVATURE

HYPERSURFACES IN MANIFOLDS OF NON-NEGATIVE RICCI CURVATURE

QI DING, J. JOST, AND Y.L. XIN

Abstract. We study minimal hypersurfaces in manifolds of non-negative Ricci curva- ture, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature de- cay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.

1. Introduction

Bernstein’s theorem says that an entire minimal graph inR³ has to be a plane. This is a classical theorem, and several proofs have been found for it. The original proofs were strictly two-dimensional, making essential use of conformal coordinates, but the statement itself is certainly meaningful in any dimension. Therefore, it was asked whether it also holds in higher dimensions. By using and developing tools from geometric measure theory, higher dimensional generalizations of the Bernstein theorem were achieved by successive efforts of W. Fleming [13], E. De Giorgi [9], F. J. Almgren [1] and J. Simons [29] up to dimension seven within the framework of geometric measure theory. In 1969, Bombieri- De Giorgi-Giusti [4] then provided a counterexample by constructing a nontrivial entire minimal graph in Rⁿ⁺¹ with n > 7 whose tangent cone at infinity had been described earlier by Simons.

Clearly, the Bernstein problem can be further generalized. We can not only increase the dimension of the ambient space, but also allow for more general Riemannian geometries than the Euclidean one. In order to see what might happen then, we observe that minimal graphs in Euclidean space are automatically area minimizing. Thus, the Bernstein problem is essentially about the (non-)existence of a particular class of complete area-minimizing hypersurfaces. Therefore, the challenge of the Bernstein problem consists in finding sharp conditions for the existence or non-existence of complete area-minimizing hypersurfaces in curved ambient manifolds.

Let us therefore review the previous results in this direction. Schoen-Simon-Yau [26]

obtained L^p−estimates for the squared norm of the second fundamental form for sta- ble minimal hypersurfaces in certain curved ambient manifolds. As a consequence, they showed that any stable minimal hypersurface with Euclidean volume growth in a flatNⁿ⁺¹ withn≤5 has to be totally geodesic. Later, Fischer-Colbrie and Schoen [12] proved that

The first author is supported partially by Natural Science Foundation of Shanghai (Grant No.

15ZR1402200). The second author is supported by the ERC Advanced Grant FP7-267087. The third author is supported partially by NSFC. He is also grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and continuous support.

1

there are no stable minimal surfaces in 3-dimensional manifolds with positive Ricci cur- vature. Shen-Zhu [27] proved certain rigidity results for stable minimal hypersurfaces in N⁴ or N⁵. On the other hand, P. Nabonnand [23] constructed a complete manifold Nⁿ⁺¹ with positive Ricci curvature which admits area-minimizing hypersurfaces. M. An- derson [3] proved a non-existence result for area-minimizing hypersurfaces in complete non-compact simply connected manifolds Nⁿ⁺¹ of non-negative sectional curvature with diameter growth conditions. For rotationally symmetric spaces with conical singularities, some explicit results were obtained by F. Morgan in [22]. These results will provide us with important model spaces for the general theory.

In the present paper we will study minimal hypersurfaces in complete Riemannian manifolds that satisfy three conditions:

C1) non-negative Ricci curvature;

C2) Euclidean volume growth;

C3) quadratic decay of the curvature tensor.

Such manifolds can be much more complicated than Euclidean space, but on the other hand, this class of manifolds possesses certain topological and analytical properties [24],[8]

that constrain their geometry. They admit tangent cones at infinity over a smooth compact manifold in the Gromov-Hausdorff sense. These cones may be not unique, but they have certain nice properties, proved by Cheeger-Colding [5]. Another important fact is that their Green functions have a well controlled asymptotic behavior. In particular, the Hessian of such a Green function converges to the metric tensor (up to a constant factor 2) point- wisely at infinity, as shown by Colding-Minicozzi [8]. The precise results will be described in section 4.

While our non-existence results are quite general, the existence results that we devel- op here, mainly for the purpose of showing that our non-existence results are sharp, are more explicit and depend on special constructions. Essentially, for these constructions, we consider ambient manifolds of the form Σ×R where Σ is an n-dimensional Riemannian manifold with a conformally flat metric whose conformal factor depends only on the ra- dius. This class will include a capped spherical cone with opening angle 2πκ, denoted by M CSκ. Its tangent cone at infinity is the uncapped spherical cone CSκ, or equivalently, the Euclidean cone over a sphere of radiusκ. These cones will be on one hand our main ex- amples for existence results and on the other hand our model spaces for the non-existence results. The border between those two phenomena, existence vs. non-existence, will be sharp. Existence takes place forκ≥ _n²√

n−1, non-existence else. The intuitive geometric reason is simply that for larger values of κ, in order to minimize area, it is most efficient to go through the vertex of the cone, whereas for smaller values of κ, it is better to avoid the vertex and go around the cone. This had already been observed by F. Morgan in [22].

As a by-product we can answer some questions raised by M. Anderson in [3].

Whereas the existence examples are specific, our non-existence results will be general.

Essentially, the idea consists in reducing them to the model cases by taking cones at infinity. For this, we need some heavier machinery, including the theory of Gromov- Hausdorff limits [17, 18, 25, 16] and the theory of currents in metric spaces developed by Ambrosio-Kirchheim [2]. In order to apply those tools, we shall analyze the Green function at infinity of the ambient space and minimal hypersurfaces with Euclidean volume growth, in order to carry the stability inequality for minimal hypersurfaces over to the asymptotic limit. The corresponding results may be of interest in themselves, see Theorem 5.1.

Our main results thus are general non-existence results for stable minimal hypersurfaces in (n+ 1)−manifoldsN with conditions C1), C2) and C3) under an additional growth con- dition on the non-radial Ricci curvature involving a constantκ⁰. For the capped spherical conesM CSκ, this constantκ⁰ can be expressed in terms of the constantκ. More precisely, we show that N admits no complete stable minimal hypersurface with at most Euclidean volume growth if the above constantκ⁰ > ⁽ⁿ⁻²⁾₄ ², see Theorem 5.5. The existence result of Theorem 3.4 then tells us that our condition on the asymptotic non-radial Ricci curvature is optimal.

Acknowledgments. The authors would like to thank referees for insightful comments which improved the paper.

2. Preliminaries

Let Σ be ann-dimensional Riemannian manifold with metric ds² =σ_ijdx_idx_j in local coordinates. LetDbe the corresponding Levi-Civita connection on Σ. For a subset Ω⊂Σ let M be a graph in the product manifold Ω×Rwith smooth defining function u on Σ, i.e.,

(2.1) M ={(x, u(x))∈Ω×R|x∈Ω}.

Since N = Σ×Rhas the product metricds² =σijdxidxj+dt², then the induced metric on M is

ds² =g_ijdx_idx_j = (σ_ij+u_iu_j)dx_idx_j, where ui = _∂x^∂u

j and uij = _∂x^∂2u

i∂xj in the sequel. Let (σ^ij) be the inverse metric tensor on Σ. Let Ei and En+1 be the dual vectors of dxi and dt, respectively. Let Γ^k_ij be the Christoffel symbols of Σ with respect to the frame Ei, i.e., DEiEj = P

kΓ^k_ijEk. Set uⁱ =σ^iju_j, |Du|² =σ^iju_iu_j, D_iD_ju =u_ij −Γ^k_iju_k and v = p

1 +|Du|². If f stands for the immersion (2.1) of Σ in M ⊂ N, then X_i = f∗E_i = E_i+u_iE_n+1, i = 1,· · ·, n, are tangent vectors ofM inN. Letν_M and H be the unit normal vector field and the mean curvature of M inN. Then, direct computation yields

ν_M = 1

v(−σ^iju_jE_i+E_n+1), H = div_Σ

Du v

= 1

√detσ_kl∂j

pdetσ_klσ^ijui

v

. M is a minimal graph in Ω×Rif and only if H≡0 and u satisfies (2.2) divΣ

Du p1 +|Du|²

!

= 1

√detσ_kl∂j

pdetσkl

σ^ijui

p1 +|Du|²

!

= 0.

This is the Euler-Lagrangian equation of the volume functional of M in N. Moreover, similar to the Euclidean case [31], any minimal graph on Ω is also an area-minimizing hypersurface in Ω×R, see Lemma 2.1 below.

We introduce an operatorL on a domain Ω⊂Σ by (2.3) LF = 1 +|DF|²³₂

div_Σ DF

p1 +|DF|²

!

= 1 +|DF|²

∆_ΣF−F_i,jFⁱF^j,

whereFⁱ=σ^ikF_k, andF_i,j =F_ij−Γ^k_ijF_kis the covariant derivative. Clearly,{(x, F(x))|x∈ Ω} is a minimal graph on Σ if and only if LF = 0 on Ω. We call F L-subharmonic (L- superharmonic) if LF ≥0 (LF ≤0).

Lemma 2.1. Let Ω be a bounded domain in Σ and M be a minimal graph on Ω as in (2.1) with volume element dµ_M. For any hypersurface W ⊂Ω×R with ∂M =∂W, one has

(2.4)

Z

M

dµ_M ≤ Z

W

dµ_W, with equality if and only if W =M.

Proof. LetU be the domain inN enclosed byM andW. Recall thatν_M is a unit normal vector field on M. Viewing ui and v as functions on Σ, we define a vector field Y such that for every (x, t) ∈ U, Y is just ν_M at M ∩({x} ×R) up to a translation along the En+1 axis. Namely,

Y(x, t) =−

n

X

i=1

σ^ij(x)uj(x)

v(x) E_i(x) + 1

v(x)E_n+1. From the minimal surface equation (2.2) we have

div(Y) =−X

i

√ 1 detσkl

∂_xi √

detσ_klσ^iju_j v

= 0,

where div stands for the divergence operator onN. LetνM, νW be the unit outside normal vectors ofM, W respectively. Observe thatY|_M =ν_M. Then by Green’s formula,

0 = Z

U

div(Y) = Z

M

hY, ν_Midµ_M − Z

W

hY, ν_Widµ_W

≥ Z

M

dµM − Z

W

dµW.

Obviously, equality holds if and only if M =W.

The index form from the second variational formula for the volume functional for a two-sided minimal hypersurfaceM in N is (see Chapter 6 of [31])

(2.5) I(φ, φ) =

Z

M

|∇φ|²− |A|¯²φ²−Ric_N(ν_M, ν_M)φ² dµ_M,

for any φ∈C_c²(N), where∇and ¯A are the Levi-Civita connection and the second funda- mental form of M, respectively.

LetSκ be ann−sphere inRⁿ⁺¹ with radius 0< κ≤1, namely, S_κ ={(x₁,· · ·, x_n+1)∈Rⁿ⁺¹|x²₁+· · ·+x²_n+1=κ²}.

If {θ_i}ⁿ_i=1 is an orthonormal basis of S_κ, then the sectional curvature of S_κ is KS(θi, θj) = 1

κ² fori6=j.

Let CSκ =R⁺×_ρSκ be the cone overSκ with vertex o, which has the metric σ_C =dρ²+κ²ρ²dθ²,

where dθ² is the standard metric onSⁿ(1).

Let{e_α}ⁿ_α=1S

{_∂ρ^∂} be an orthonormal basis at the considered point ofCS_κ away from the vertex, then the sectional curvature and Ricci curvature of CS_κ are

(2.6)

K_CSκ ∂

∂ρ, e_α

= 0, K_CSκ(e_α, e_β) = 1 ρ²

1 κ² −1

, RicCSκ

∂

∂ρ, ∂

∂ρ

=RicCSκ

∂

∂ρ, eα

= 0, RicCSκ(eα, eβ) = n−1 ρ²

1 κ² −1

δαβ. Setρ=r^κ, thenσC can be rewritten as a conformally flat metric

(2.7) σ_C =κ²r^2κ−2dr²+κ²r^2κdθ² =κ²r^2κ−2

n+1

X

i=1

dx²_i =e^{2 log}κ−2(1−κ) logr n+1

X

i=1

dx²_i, where r² =P

ix²_i.

LetY be an (n−1)−dimensional minimal hypersurface in S_κ with the second funda- mental form A and CY be the cone over Y in CSκ with vertex o. For any 0 < < 1 denote

CY={tx∈Sκ×R|x∈Y, t∈[,1]}.

Clearly,Y is a minimal hypersurface inSκ if and only ifCY is minimal inCSκ. Moreover, let ¯A be the second fundamental form ofCY in CS_κ, then

|A|¯² = 1 ρ²|A|².

At any considered point, we can suppose thatθ_n is the unit normal vector ofY ⊂S_κ and {θ_i}ⁿ⁻¹_i=1 is the orthonormal basis of T Y. Let ν = ¹_ρθn be the unit normal vector ofCY. Let dµand dµY be the volume element of CY andY, respectively.

Now, from (2.5), the index form ofCY inCSκ becomes (2.8) I(φ, φ) =

Z

CY

−φ∆_CYφ− |A|¯²φ²−Ric_CSκ×R(ν, ν)φ² dµ for any φ∈C_c²(CY \ {o}). Note Ric_Sκ(θ_i, θ_j) = ⁿ⁻¹_κ2 δ_ij and

Ric_CSk(ν, ν) = 1

ρ²Ric_Sk(θ_n, θ_n)−n−1

ρ² = n−1 ρ²

1 κ² −1

. Whenφis written as φ(x, ρ)∈C²(Y ×R), a simple calculation implies

(2.9) ∆CYφ= 1

ρ²∆Yφ+ n−1 ρ

∂φ

∂ρ +∂²φ

∂ρ², then

(2.10)

I(φ, φ) = Z 1

Z

Y

−∆Yφ− |A|²φ−n−1

κ² φ+ (n−1)φ

−(n−1)ρ∂φ

∂ρ −ρ²∂²φ

∂ρ²

φ dµ_Y

ρⁿ⁻³dρ.

Whenκ= 1 and Y is the Clifford minimal hypersurface in the unit 7−sphere Y =S³

√2 2

!

×S³

√2 2

! ,

then, CY is Simons’ cone, proved to be stable in [29] (see also Chapter 6 of [31]).

3. Constructions of area-minimizing hypersurfaces Let Σ be the Euclidean spaceRⁿ⁺¹ with a conformally flat metric

ds² =e^φ(r)

n+1

X

i=1

dx²_i, where r = |x| =q

x²₁+· · ·+dx²_n+1 and φ(|x|) is smooth in Rⁿ⁺¹. Let F be a function on Rⁿ⁺¹. Let Ei = {_∂x^∂

i} be a standard basis of Rⁿ⁺¹ and Fi = _∂x^∂

iF be the ordinary derivative inRⁿ⁺¹. Moreover,

Γ^k_ij = φ⁰ 2

δ_ikx_j

r +δ_jkx_i

r −δ_ijx_k r

. Denote|∂F|² =P

iF_i². Let ∆ be the standard Laplacian ofRⁿ⁺¹, then (3.1)

∆_ΣF =σ^ijF_i,j =e^−φδ_ij

F_ij−φ⁰ 2

δ_ikx_j

r +δ_jkx_i

r −δ_ijx_k r

F_k

=e^−φ

∆F +n−1 2 φ⁰F_ix_i

r

.

By (2.3) we can computeLF in the conformal flat metric as follows.

(3.2)

LF =e^−φ

1 +e^−φ|∂F|²

∆F +n−1 2 φ⁰Fi

xi

r

−e^−2φ

FijFiFj−|∂F|² 2 φ⁰Fi

xi

r

=e^−φ 1 +e^−φ|∂F|²

∆F−e^−φF_ijF_iF_j +e^−φ

n−1 2 +n

2e^−φ|∂F|²

φ⁰F_ix_i r

=e^−2φ

|∂F|²

∆F+n 2φ⁰Fi

xi

r

−FijFiFj

+e^−φ

∆F +n−1 2 φ⁰Fi

xi

r

. Lemma 3.1. Let F =F(θ, r) be a function with

(3.3) θ= x_n+1

q

x²₁+· · ·+x²_n+1

, r= q

x²₁+· · ·+x²_n+1, on [−1,1]×(0,∞). Then we have

(3.4)

LF =e^−2φ

n

1−θ²F_θ²

r² +F_r² Fr

r +φ⁰

2Fr− θF_θ r²

+ (1−θ²)F_θ² r²

θFθ

r² +Fr

r

+1−θ²

r² F_θ²Frr+F_r²F_θθ−2F_θFrF_rθ

+e^−φ

F_rr+1−θ²

r² F_θθ+n

rF_r−nθ

r²F_θ+n−1 2 φ⁰F_r

.

Proof. For 1≤α≤nwe have

(3.5)

Fα=∂xαF =Fθ·

−xαxn+1

r³

+Fr

xα

r , Fn+1 =∂xn+1F =F_θ·

1

r −x²_n+1 r³

+Fr

xn+1

r =F_θ P

αx²_α r³ +Fr

xn+1

r .

Hence

(3.6) |∂F|² =X

α

F_α²+F_n+1² =F_θ² P

αx²_α

r⁴ +F_r² = 1−θ²F_θ² r² +F_r², and

(3.7)

n+1

X

i=1

x_iF_i =X

α

x_αF_α+x_n+1F_n+1=rF_r.

In polar coordinates,

n+1

X

i=1

dx²_i =dr²+r² dβ²+ cos²β dSⁿ⁻¹ ,

where sinβ =θ∈[−1,1] anddSⁿ⁻¹ is the standard metric in the unit sphereSⁿ⁻¹ ∈Rⁿ. Hence

n+1

X

i=1

dx²_i =dr²+ r²

1−θ²dθ²+r²(1−θ²)dSⁿ⁻¹, and

(3.8)

∆F = 1

rⁿ(1−θ²)ⁿ²⁻¹

∂r

rⁿ(1−θ²)ⁿ²⁻¹Fr

+∂θ

rⁿ(1−θ²)ⁿ²⁻¹1−θ² r² Fθ

=F_rr+n

rF_r+1−θ²

r² F_θθ−nθ r²F_θ. Moreover,

(3.9)

X

1≤i,j≤n+1

F_ijF_iF_j = 1 2

X

i

F_i∂_i|∂F|²

=1 2

X

α

−xαxn+1

r³ Fθ+xα

r Fr −xαxn+1

r³ ∂θ|∂F|²+xα

r ∂r|∂F|² +1

2 P

αx²_α

r³ F_θ+x_n+1 r F_r

P

αx²_α

r³ ∂_θ|∂F|²+ x_n+1

r ∂_r|∂F|²

=1 2

P

αx²_α

r⁴ F_θ∂_θ|∂F|²+1

2F_r∂_r|∂F|²

=1−θ² 2r² Fθ∂θ

1−θ²F_θ² r² +F_r²

+1

2Fr∂r

1−θ²F_θ² r² +F_r²

=−θ(1−θ²)F_θ³

r⁴ + (1−θ²)²F_θ²F_θθ

r⁴ + 2(1−θ²)FθFrFrθ

r²

−(1−θ²)F_θ²Fr

r³ +F_r²F_rr.

Hence by (3.2) we have

(3.10)

LF =e^−2φ 1−θ²F_θ²

r² +F_r² Frr+ n

rFr+1−θ²

r² F_θθ−nθ

r²F_θ+n 2φ⁰Fr

−

−θ(1−θ²)F_θ³

r⁴ + (1−θ²)²F_θ²F_θθ

r⁴ + 2(1−θ²)F_θFrF_rθ

r² −(1−θ²)F_θ²F_r r³ +F_r²F_rr

+e^−φ

F_rr+n

rF_r+1−θ²

r² F_θθ−nθ

r²F_θ+n−1 2 φ⁰F_r

=e^−2φ

n

1−θ²F_θ²

r² +F_r² F_r r +φ⁰

2F_r−θF_θ r²

+ (1−θ²)F_θ² r²

θF_θ r² +F_r

r

+1−θ²

r² F_θ²F_rr+F_r²F_θθ−2F_θF_rF_rθ

+e^−φ

Frr+1−θ²

r² F_θθ+ n

rFr−nθ

r²F_θ+ n−1 2 φ⁰Fr

.

Theorem 3.2. Let Σ be an (n+ 1)−dimensional Euclidean space Rⁿ⁺¹, n≥2, endowed with a smooth conformally flat metric ds² = e^φP

dx²_i, where φ⁰(r) ≥ −2(1−κ)r⁻¹ and

2 n

√n−1≤κ≤1. If

F(θ, r) =Cθr^p=Cxn+1r^p−1,F(x_n+1, r) with any constant C >0 andp= ⁿ₂κ−

qn²κ²

4 −(n−1),then except at the origin we have (3.11) LF(x_n+1, r)

( ≥0 if (x₁,· · ·, x_n)∈Rⁿ, x_n+1 ≥0

≤0 if (x₁,· · ·, x_n)∈Rⁿ, x_n+1 ≤0 .

Proof. Since φ⁰ ≥ −2(1−κ)r⁻¹ for 0< κ≤1 and Fr =Cpθr^p−1. By (3.4) except at the origin we have

(3.12)

θLF ≥θe^−2φ

n

1−θ²F_θ²

r² +F_r² κFr

r −θFθ

r²

+ (1−θ²)F_θ² r²

θF_θ r² +F_r

r

+1−θ²

r² F_θ²F_rr+F_r²F_θθ−2F_θF_rF_rθ

+θe^−φ

F_rr+1−θ²

r² F_θθ+ (n−1)κ+ 1F_r r − n

r²θF_θ

. Furthermore, we take the derivatives ofF and get

(3.13)

θLF ≥C³θe^−2φ

n

1−θ²

r^2p−2+θ²p²r^2p−2

κθpr^p−2−θr^p−2 + (1−θ²)r^2p−2 θr^p−2+θpr^p−2

+1−θ²

r² p(p−1)θr^3p−2−2p²θr^3p−2

+Cθe^−φ

p(p−1)θr^p−2+ (n−1)κ+ 1

pθr^p−2−nθr^p−2

=C³θe^−2φ

n(κp−1) + 1−p²

(1−θ²) +np²(κp−1)θ² θr^3p−4 +Cθe^−φ

p²+ (n−1)κp−n

θr^p−2.

Note

n(κp−1) + 1−p² =− p− nκ

2 2

+n²κ²

4 −(n−1) = 0.

By the definition ofp, we obtain

(3.14)

p=nκ 2 1−

r

1−4(n−1) n²κ²

!

= nκ

2 1−n−2 n

s

1−4(n−1) (n−2)²

1 κ² −1

!

≥nκ 2

1−n−2 n

1−2(n−1) (n−2)²

1 κ² −1

= 1 κ

1 +1−κ² n−2

≥ 1 κ. Hence

(3.15) θLF ≥C³e^−2φnp²(κp−1)θ⁴r^3p−4+Ce^−φ

p²+ (n−1)κp−n θ²r^p−2

≥Ce^−φ(p²−1)θ²r^p−2≥0.

We complete the proof.

Remark 3.3. There are other L-sub(super)harmonic functions on Σ. For instance, for all j > 0, L(jxn+1w^p−1) ≥ 0 on xn+1 ≥ 0 and L(jxn+1w^p−1) ≤ 0 on xn+1 ≤ 0, where w=p

x²₁+· · ·+x²_n.

DenoteB_R={(x₁,· · ·, xn+1)∈Rⁿ⁺¹|x²₁+· · ·+x²_n+1≤R²}.

Theorem 3.4. If n≥3 and

2 n

√n−1≤κ <1,

then any hyperplane through the origin in Σ as described in Theorem 3.2, that is, Rⁿ⁺¹ equipped with a particular conformally flat metric, is area-minimizing.

Proof. We shall show that the hyperplane T = {(x₁,· · · , xn+1) ∈ Rⁿ⁺¹| xn+1 = 0} in Σ with the induced metric is area-minimizing.

Set ˜φ(r) =Rr

0 e^φ(r)2 dr. Let us defineρ = ˜φ(r) and λ(ρ) =rφ˜⁰(r), then the Riemannian metric in Σ can be written in polar coordinates asds²=dρ²+λ²(ρ)dθ², wheredθ² is the standard metric onSⁿ(1). Moreover,

(3.16) dλ dρ = dλ

dr dr dρ =

φ˜⁰+rφ˜⁰⁰ 1

φ˜⁰ = 1 +r(log ˜φ⁰)⁰ = 1 +1

2rφ⁰ ≥1−(1−κ) =κ.

Whenn≥3 andq=p−1 forp as in the statement of Theorem 3.2, letF_j(xn+1, r) = jxn+1r^q forj >0 with r=

q

x²₁+· · ·+x²_n+1. By Theorem 3.2 we obtain (3.17) LF_j(x_n+1, r)

( ≥0 in {(x₁,· · ·, xn+1)∈Rⁿ⁺¹|xn+1 ≥0} \ {0}

≤0 in {(x₁,· · ·, x_n+1)∈Rⁿ⁺¹|x_n+1 ≤0} \ {0} . Combining (3.16) and formula (2.9) in [10], we know that any geodesic sphere ∂Dρ ⊂Σ centered at the origin has positive inward mean curvature H = (n−1)^λ_λ(ρ)^0(ρ) > 0 for any ρ > 0. By the existence theorem for the Dirichlet problem for minimal hypersurfaces in Σ×R, see Theorem 1.5 in [30], for any constant R > 0 and j = 1,2,· · · ,∞, there is a solution uj ∈C^∞(BjR) to the Dirichlet problem

(3.18)

( Lu_j = 0 inB_jR uj =F_j on ∂BjR

.

By symmetry, u_j = 0 on B_R^∗∩T for any fixed R^∗ > 0. In fact, by the definition (3.2) of Lin a conformally flat metric,−u_j(x1,· · ·, xn,−x_n+1) is also a smooth solution to the above Dirichlet problem. Since Lemma 2.1 implies the uniqueness of the minimal graph, we get uj(x1,· · · , xn, xn+1) = −u_j(x1,· · · , xn,−x_n+1), from which it follows that uj = 0 on B_R^∗∩T. Let U =

(x1,· · ·, xn+1)∈Rⁿ⁺¹

xn+1 >0 , then the comparison theorem on B_R^∗\ {0} implies

(3.19) lim

j→∞uj ≥ lim

j→∞F_j = +∞ in B_R^∗∩U and

(3.20) lim

j→∞uj ≤ lim

j→∞F_j =−∞ in BR^∗∩(Rⁿ⁺¹\U).

Let Uj denote the subgraph ofuj inBR^∗×R, namely, U_j =

(x, t)∈B_R^∗×R

t < u_j(x) .

Clearly, its characteristic functionχ_Uj converges inL¹_loc(BR^∗×R) toχ_U×R. By an analogous argument as in Lemma 9.1 in [15] for the Euclidean case, for any compact setE⊂BR^∗×R, that Graph(u_j),{(x, u_j(x))|x∈Rⁿ⁺¹}is an area-minimizing hypersurface implies that (U×R)∩Eis a minimizing set inE. HenceU×Ris a minimizing set inBR^∗×R⊂Σ×R.

By an analogous argument as in Proposition 9.9 in [15] for the Euclidean case, U is a minimizing set in B_R^∗, namely, the hyperplane T minimizes the perimeter in B_R^∗. Since

R^∗ is arbitrary, we complete the proof.

As we showed in the previous section, on the coneCS_κthe usual metric can be rewritten as a conformally flat one. We shall therefore modify the constructions from the cone CSκ. Lemma 3.5. Let Λ be the rotationally symmetric function on Rⁿ⁺¹ defined by

(3.21) Λ(x) =

( ^√_1−κ2

κ

px²₁+· · ·+x²_n on Rⁿ⁺¹\B1

√1−κ² κ

1−²_πR1

|x|(arctanξ(s))ds

on B1

, where ξ(s) =s

e

1

1−s2 −e

. It is a smooth convex function onRⁿ⁺¹.

Proof. In fact,ξ⁰(0) = 0, ξ^(2k)(0) = 0 fork≥0 and ξ^(j)(1) = +∞ forj ≥0. Then onB1

(3.22)

∂_iΛ(x) =2√ 1−κ²

κπ x_i

|x|arctanξ(|x|),

∂_ijΛ(x) =2√ 1−κ²

κπ

δ_ij −xixj

|x|²

arctanξ

|x| +2√ 1−κ²

κπ

ξ⁰ 1 +ξ²

xixj

|x|². Since

arctanξ(√

√ t)

t =

∞

X

k=0

(−1)^k 2k+ 1t^k

e^1−t1 −e2k+1

in [0, ] for small >0,t⁻¹² arctanξ(√

t) is a smooth function for t∈[0,1) and Λ(x) =

√ 1−κ²

κ 1− 1 π

Z 1

|x|²

arctanξ(√

√ t)

t dt

!

is a smooth convex function on B₁. Denote Λ(r) = Λ(|x|), then the radial derivative of Λ at 1 is

r→1lim∂_rΛ(r) = 2√ 1−κ²

κπ arctanξ(1) =

√1−κ² κ ,

and the higher order radial derivative of Λ at 1 is

r→1lim(∂r)^j+1Λ(r) =2√ 1−κ²

κπ (∂r)^jarctanξ(r) r=1

=2√ 1−κ²

κπ (∂_r)^j−1 ξ⁰

1 +ξ² _r=1

= 0 forj≥1.

Hence Λ is a smooth convex function on Rⁿ⁺¹.

Now we suppose thatM CSκ is an (n+ 1)-dimensional smooth entire graphic hyper- surface in Rⁿ⁺² with the defining function Λ. We see that it has non-negative sectional curvature everywhere. In fact,M CSκ is aκ−sphere coneCSκ with a smooth cap, which we shall call the modified κ−sphere cone.

We already showed that the metric of theκ−sphere cone is conformally flat, and we shall now also derive this for M CSκ.

Lemma 3.6. The(n+ 1)−dimensional M CS_κ has a smooth conformally flat metric ds²=e^Φ(r) X

1≤i≤n+1

dx²_i on Rⁿ⁺¹ with −²_r(1−κ)≤Φ⁰ ≤0.

Proof. M CSκ is defined as an entire graph onRⁿ⁺². Its induced metric can also be written in polar coordinates as

(3.23) ds² =dρ²+λ²(ρ)dθ²,

where dθ² is a standard metric onSⁿ(1), and

(3.24) λ(ρ) =

( κ ρ+_κ¹ −ρ0

forρ≥ρ0

ζ(ρ) for 0≤ρ≤ρ0

. Here

1< ρ0 = Z 1

0

p1 + (∂rΛ)²dr < 1 κ, and the inverse function ofζ satisfies

ζ⁻¹(s) = Z s

0

p1 + (∂rΛ)²dr, where Λ is defined in the last lemma. Moreover, κ≤ζ⁰≤1.

Letψ(r) be a function on h

0, _κ¹¹_κ

withψ

1 κ

_κ¹

=ρ₀ and

(3.25) ψ⁰(r) = 1

rζ(ψ(r)) on h 0,

1 κ

¹_κ

. In fact, let ˜ζ(ρ) =Rρ

1 1

ζ(t)dtforρ∈(0, ρ0], then we integrate the above ordinary differential equation and obtain

ζ(ψ(r))˜ −ζ(ρ˜ ₀) = logr+1 κlogκ.

Since ˜ζ is a monotonic function, we can solve this equation for ψ. Sinceκρ≤ζ(ρ)≤ρ on [0, ρ₀], a standard comparison argument implies that

1 κ

−¹_κ

ρ0r≤ψ(r)≤κρ0r^κ on h

0, 1

κ _κ¹

i . In particular, ψ(0) = 0. Since

ψ⁰⁰(r) = ζ⁰ rψ⁰− ζ

r² = ζ

r²(ζ⁰−1), then,

(3.26) κ−1

r ≤ ψ⁰⁰(r)

ψ⁰(r) = ζ⁰−1 r ≤0.

Let

(3.27) ρ= ˜ψ(r) =







r^κ−_κ¹ +ρ₀ forr≥ ¹_κ¹_κ ψ(r) for 0≤r≤ ¹_κ¹_κ

, then ˜ψalso satisfies (3.25) and hence ˜ψ is smooth on [0,∞). Set (3.28) e^Φ(r)=

ψ˜⁰(r) 2

=







κ²r^2κ−2 forr ≥ _κ¹¹_κ (ψ⁰)²(r) for 0≤r ≤ _κ¹¹_κ

, then

(3.29) ds² =e^Φ(r)dr²+e^Φ(r)r²dθ² =e^Φ(r) X

1≤i≤n+1

dx²_i, where r² =P

ix²_i. By (2.7) and (3.26) we have

−2

r(1−κ)≤Φ⁰ ≤0.

Now, Lemma 3.6 and Theorem 3.4 yield the following conclusion.

Theorem 3.7. Let n≥3. If

2 n

√n−1≤κ <1,

then any hyperplane through the origin in M CSκ is area-minimizing.

Remark 3.8. Let {e_α}ⁿ_α=1S

{_∂ρ^∂} be an orthonormal basis at the considered point of M CSκ. Compared with (2.6) we calculate the sectional curvature and Ricci curvature of M CS_κ as follows (see Appendix A in[20] for instance).

(3.30)

KM CSκ

∂

∂ρ, eα

=−λ⁰⁰

λ ≥0, KM CSκ(eα, eβ) = 1−(λ⁰)² λ² ≥0, Ric_{M CSκ}

∂

∂ρ, e_α

= 0, Ric_{M CSκ} ∂

∂ρ, ∂

∂ρ

=−nλ⁰⁰ λ ≥0, RicM CSκ(eα, e_β) =

(n−1)1−(λ⁰)² λ² −λ⁰⁰

λ

δ_αβ ≥0.

In particular, for ρ≥ρ₀ with 1< ρ₀ < _κ¹ we have

(3.31)

K_{M CSκ} ∂

∂ρ, e_α

= 0, K_{M CSκ}(e_α, e_β) = 1−κ² κ²(ρ+_κ¹ −ρ0)², Ric_{M CSκ}

∂

∂ρ, ∂

∂ρ

=Ric_{M CSκ} ∂

∂ρ, e_α

= 0, RicM CSκ(eα, e_β) = (n−1) 1−κ²

κ²(ρ+¹_κ −ρ₀)²δ_αβ.

From the construction above we see that M CSκ is a complete simply connected manifold with non-negative sectional curvature.

Remark 3.9. SinceM CS_κ in Theorem 3.4 cannot split off a Euclidean factor R isomet- rically, the Cheeger-Gromoll splitting theorem [6] implies that it does not contain a line.

Consequently, this gives a negative answer to the question (1) in [3], which is

IfM is a complete area-minimizing hypersurface in a complete simply connected mani- fold N of non-negative curvature, does it follow that N contains a line, that is a complete length-minimizing geodesic?

If we define for eachx∈Rⁿ

Λ(x) =e 2√ 1−κ²

πκ

Z |x|

0

arctansds,

thenΛ is a smooth strictly convex function one Rⁿand the hypersurfaceΣ =e {(x,Λ(x))|e x∈ Rⁿ}is a smooth manifold with positive sectional curvature everywhere. In fact,Σ can bee seen as a Riemannian manifold (Rⁿ,σ) with˜

˜

σ =dρ²+ ˜λ²(ρ)dθ² in polar coordinates, where the inverse function of ˜λsatisfies

˜λ⁻¹(s) = Z s

0

q

1 + (∂rΛ)e ²dr= Z s

0

r

1 +4(1−κ²)

π²κ² (arctanr)²dr.

Hence

1≥˜λ⁰(s) =

1 +4(1−κ²)

π²κ² (arctan ˜λ(s))² −¹₂

> κ, and

λ˜⁰⁰(s) =−

1 +4(1−κ²)

π²κ² (arctan ˜λ(s))^{2 −3}₂

4(1−κ²)

π²κ² arctan ˜λ(s) λ˜⁰(s) 1 + ˜λ²(s). Clearly,

s→∞lim λ(s)˜

s = lim

s→∞

λ˜⁰(s) =κ, and lim

s→∞(s²λ˜⁰⁰(s)) =−2

π(1−κ²).

If{∂_ρ} and {e_α}ⁿ⁻¹_α=1 are an orthonormal basis ofΣ, then the sectional curvature ofe Σe is

0< K(∂_ρ, e_α) =−λ˜⁰⁰

λ˜ ∼ 2(1−κ²)

πκs³ , K(e_α, e_β) = 1−λ˜⁰²

˜λ² ∼ 1−κ² κ²s² . Clearly,

s→0lim

1−λ˜⁰²(s)

˜λ²(s) = 4(1−κ²) π²κ² >0.

Hence Σ =e {(x,Λ(x))|e x∈Rⁿ} has positive sectional curvature everywhere.

Theorem 3.10. Letn≥4andΣ = (e Rⁿ,σ)˜ be a complete manifold with positive sectional curvature as above. If

2 n−1

√n−2≤κ <1,

then any hyperplane through the origin in Σ = (e Rⁿ,σ)˜ is area-minimizing.

Proof. Note κ < λ˜⁰ ≤ 1, then we can rewrite the metric ˜σ similar to (3.27)(3.28)(3.29).

Apply Theorem 3.4 to complete the proof.

Remark 3.11. Our theorem above gives an example for the question (2) in [3], which is If N is a complete manifold of positive sectional curvature, does N ever admit an area-minimizing hypersurface?

Now scaling the manifold M CS_κ yields ²M CS_κ for > 0, which is Rⁿ⁺¹ endowed with the metric

(3.32) σ=dρ²+²λ²ρ

dθ²

in polar coordinates, where λand dθ² as in (3.23) and (3.24). Obviously λ ^ρ

≥κρand λ ^ρ

converges toκρuniformly as→0. Henceσ converges toσ_C as→0 (uniformly away from the origin), where σC is the metric ofCSκ defined in (2.7).

Now we can derive the result of F. Morgan in [22], obtained there by a different method due to G. R. Lawlor [19].

Proposition 3.12. Let n ≥ 3 and κ ≥ _n²√

n−1. Then any hyperplane in (n+ 1)- dimensional CSκ through the origin is area-minimizing.

Proof. LetT denote the hyperplane in²M CS_κ corresponding to T ⊂M CS_κ during the re-scaling procedure. Denote T0 = lim→0T ⊂ lim→0²M CSκ = CSκ in the sense of Hausdorff distance.

Now we consider a bounded domain Ω₀⊂T₀ and a subsetW₀ ⊂CS_κ with∂Ω₀ =∂W₀. View Ω0 as a set Ω⊂ Rⁿ with the induced metric from T0, and W0 as a set W in Rⁿ⁺¹ with the induced metric from CS_κ. Let Ω be the set Ω ⊂ Rⁿ with the induced metric from T, and W be the set W inRⁿ⁺¹ with the induced metric from ²M CSκ. Clearly,

∂Ω=∂W, Ω₀ = lim→0Ω andW₀ = lim→0W in the sense of Hausdorff distance.

LetHⁿ(K) denote n-dimensional Hausdorff measure of K for any set K ⊂CS_κ, and Hⁿ(K⁰) denote n-dimensional Hausdorff measure of K⁰ for any set K⁰ ⊂²M CSκ. Note that λ ^ρ

→κρ uniformly as→0, we have (3.33) Hⁿ(Ω₀) = lim

→0Hⁿ(Ω), lim sup

→0

Hⁿ(W)≤ Hⁿ(W₀).

Since T is area-minimizing in²M CSκ, then

(3.34) Hⁿ(Ω)≤ Hⁿ(W).

Combining (3.33) and (3.34), we obtain Hⁿ(Ω₀) = lim

→0Hⁿ(Ω)≤lim sup

→0

Hⁿ(W)≤ Hⁿ(W₀).

Hence T₀ is an area-minimizing hypersurface in CS_κ.

Actually, here the number _n²√

n−1 is optimal. Namely, if κ < _n²√

n−1 then every hyperplane in CSκ is no more area-minimizing and even not stable. This also has been proved in [22]. Let us show this fact by using the second variation formula for the volume functional.

Theorem 3.13. Let κ ∈(0,1] and n ≥3. Any hyperplane in (n+ 1)-dimensional CSκ

through the origin is area-minimizing if and only if

(3.35) κ≥ 2

n

√ n−1.

Proof. By Proposition 3.12 we only need to prove that if (3.35) fails to hold, any hyper- plane inCS_κ through the origin is not area-minimizing. LetXbe a totally geodesic sphere in Sκ, then X is minimal in Sκ and P ,CX is a hyperplane inCSκ through the origin.

Clearly, P is a minimal hypersurface in CS_κ because it is totally geodesic. The second variation formula is (see (2.10))

(3.36)

I(φ, φ) = Z 1

Z

X

−∆Xφ−n−1

κ² φ+ (n−1)φ

−(n−1)ρ∂φ

∂ρ −ρ²∂²φ

∂ρ²

φ dµ_X

ρⁿ⁻³dρ, where φ(x, t)∈C²(X×_ρR). Define a second order differential operatorLby

L=ρ² ∂²

∂ρ² + (n−1)ρ ∂

∂ρ. If s= logρ, then

L= ∂²

∂s² + (n−2)∂

∂s =e^{−n−22 s} ∂²

∂s²

e^{n−22 s}·

−(n−2)²

4 .

So the k(k≥1)-th eigenvalue of L on [,1] is

(3.37) (n−2)²

4 +

kπ log

2

with the k-th eigenfunction (see [29] or [31] for instance) ρ²⁻ⁿ² sin

kπ loglogρ

.

By the second variation formula (3.36),P is stable if and only if

−n−1

κ² +n−1 +(n−2)² 4 ≥0, i.e.,

κ≥ 2 n

√n−1.

4. A class of manifolds with non-negative Ricci curvature

LetN be an (n+1)-dimensional complete non-compact Riemannian manifold satisfying the following three conditions:

C1) Nonnegative Ricci curvature: Ric≥0;

C2) Euclidean volume growth:

V_N , lim

r→∞

V ol B_r(x) rⁿ⁺¹ >0;

C3) Quadratic decay of the curvature tensor: for sufficiently large ρ = d(x, p), the distance from a fixed point in N,

|R(x)| ≤ c ρ²(x).

By Gromov’s compactness theorem [17], for any sequence ¯_i →0 there is a subsequence {_i} converging to zero such that iN = (N, ig, p) converges to a metric space (N¯ ∞, d∞) with vertex o in the pointed Gromov-Hausdorff sense. It is called the tangent cone at infinity. N∞\ {o} is a smooth manifold with C^1,α Riemannian metric ¯g∞(0 < α < 1) which is compatible with the distance d∞. The precise statements were derived in [16]

and [25] on the basis of the harmonic coordinate constructions of [18]. In fact,N∞\ {o}is aD^1,1-Riemannian manifold (see [16, 25]). For any compact domainK ⊂N∞\ {o}, there exists a diffeomorphism Φ_i: K→Φ_i(K)⊂_iN such that Φ^∗_i(_i¯g) converges as i→ ∞ to

¯

g∞ in theC^1,α-topology onK.

Cheeger-Colding (see Theorem 7.6 in [5]) proved that under the conditions C1) and C2) the cone N∞is a metric cone, namely N∞=CX =R⁺×_ρX for somendimensional smooth compact manifoldX with DiamX ≤π and the metric

¯

g∞=dρ²+ρ²sijdθidθj

where sijdθidθj is the metric of X and sij ∈ C^1,α(X). Let ρi be the distance function from p to the considered point in_iN. Set B_rⁱ(x) be the geodesic ball with radius r and centered at x in (N, i¯g), and B_r(x) be the geodesic ball with radius r and centered at x inN∞. In particular,X =∂B₁(o).

Mok-Siu-Yau [21] showed that if C1) and C2) hold, then there exists the Green function G(p,·) onNⁿ⁺¹ with limr→∞sup_∂Br(p)

Grⁿ⁻¹−1

= 0 and

(4.1) r¹⁻ⁿ≤G(p, x)≤Cr¹⁻ⁿ

for any n≥2,x∈∂B_r(p) and some constantC. SetR=G¹⁻ⁿ¹ , then

(4.2) ∆NR² = 2(n+ 1)|∇R|².

Under the additional condition C3), Colding-Minicozzi (see Corollary 4.11 in [8]) showed that

(4.3) lim sup

r→∞

sup

∂Br

R r −1

+ sup

∂Br

∇R −1

= 0, and

(4.4) lim sup

r→∞

sup

∂Br

|Hess_R2 −2¯g|

= 0,