On $n$-superharmonic functions and some geometric applications

In this paper we study asymptotic behavior of $n$-superharmonic functions at isolated singularity using the Wolff potential and $n$-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro in 2 dimensions. To study $n$-superharmonic functions we use a new notion of $n$-thinness by $n$-capacity motivated by a type of Wiener criterion in Arsove-Huber's paper. To extend Taliaferro's work, we employ the Adams-Moser-Trudinger inequality for the Wolff potential, which is inspired by the one used by Brezis-Merle. For geometric applications, we study the asymptotic end behavior of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. In both geometric applications the strong $n$-capacity lower bound estimate of Gehring in 1961 is brilliantly used. These geometric applications seem to elevate the importance of $n$-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.


INTRODUCTION
In this paper we will develop some understanding of isolated singularities of n-superharmonic functions in n dimensions and apply it to study some geometric problems. Recall that the n-Laplace (1.1) ∆ n u = div(|∇u| n−2 ∇u) 1 The author is the corresponding author and partially supported by NSFC 11571185 2 The author is partially supported by NSF DMS-1608782 1 is a quasilinear, possibly degenerate, elliptic operator that agrees with the Laplace operator in 2 dimensions.
The theory of n-Laplace equations is as fundamental as that of classic Laplace equations since it is also in the center of the interplay of several important fields of mathematics including calculus of variations, partial differential equations, nonlinear potential theory, and mathematical physics. Obviously the theory of n-Laplace equations is more interesting as well as more challenging, because the principle of superposition is no longer available, instead, understanding of interactions is indispensable. We would like to develop higher dimensional extensions to what have been done for the theory of subharmonic functions in [AH73,HK76,Tal06] (references therein) regarding asymptotic behavior and their applications in differential geometry. Our research in this paper seems to elevate the importance of n-Laplace equations and makes a closer tie to the classic analysis developed in conformal geometry.
The first goal for us is to study the behavior of n-superharmonic functions at isolated singularity. The first main theorem in general dimensions is inspired by and extends the work of Arsove-Huber in [AH73, Theorem 1.3].
Then there is a set E ⊂ R n , which is n-thin by capacity at the origin, such that The definition of n-superharmonic functions is given in Definition 2.3. The definition of a set to be n-thin by capacity is given in Definition 3.1, which is inspired by and extends the definition of thinness in [AH73] (see the discussion on the comparison of different notions of thinness in Section 2.2). Thanks to [BV89, Proposition 1.1], the n-superharmonic function in Theorem 1.1 satisfies (1.4) −∆ n w = g + βδ for some g ∈ L 1 (B(0, 1)) and β ≥ 0. The proof of Theorem 1.1 combines the blow-down argument from [KV86] and the nonlinear potential theory [AM72, HKM93, KM94, L06, PV08] for n-Laplace equations, particularly the use of the Wolff potential and n-capacity estimates.
The proof of Theorem 1.1 consists of four major steps. The first is to use nonlinear potential theory, particularly [KM94, Theorem 1.6 and Lemma 3.9] on the Wolff potential and n-capacity estimates (cf. (3.10)) to show that, the blow-down quotient (1.5) w r (ξ) = w(rξ) log 1 |r| is bounded outside a subsetÊ that is n-thin by capacity. The second step is to use a very clever cut-off technique from [DHM97] to modify and cut off the unbounded part in order to take sequential limit for the blow-down quotients as r → 0. Based on Liouville Theorem of [Se64,Re66,HKM93], one knows that the sequential limits are all constants. In the third step, we use comparison principle (cf. [Tol83,Lemma 3.1] and [KV86,KV87]) to conclude that all sequential limits have to be the same as m in the Theorem 1.1. In the final step, based on the uniqueness of sequential limits, we re-run the proof in the first step to extract a subset E that is n-thin by capacity and finish the proof of Theorem 1.1. The first run of the argument in the first step is to get bounds; while the second run is to get uniform convergences. It is essential and very interesting to see how the classic Paul du Bois-Reymond Theorem (cf. [R1873] and [B1908, (5) Page 40]) for infinite series helps to re-enforce the argument in the first run in applying [KM94, Theorem 1.6 and Lemma 3.9] (cf. (3.32)) to get the uniform convergence.
for the Wolff potential induced by a nonnegative function in L 1 (Ω), where D is the diameter of Ω and δ ∈ (0, 1). (1.10) is stated in Proposition 4.1 and extends the one discovered by Brezis-Merle in 2 dimensions (cf. [BM91,FM11] and references therein). This Adams-Moser-Trudinger inequality (1.10) for the Wolff potential helps control any possible concentration and rule out any possible nontrivial n-thin subset E in Theorem 1.1. As stated in Remark 4.1, the critical growth condition may be described as (1.11) 0 ≤ f (x, w, ∇w) ≤ C|∇w| p e αw for any 0 < p < n and α > 0 to be more general. Our proof of Theorem 1.2 is a streamlined one from [Tal99,Tal01,Tal06] with the help of the Adams-Moser-Trudinger inequality (1.10) for the Wolff potential.
As applications we first want to study asymptotic behavior at the end of complete locally conformally flat manifolds with nonnegative Ricci. After the classification theorems of [Zhu94,CH06], we want to focus on complete metrics e 2φ |dx| 2 on R n . One may calculate and find that (1.12) −∆ n φ = Ric g (∇ g φ)|∇φ| n−2 e 2φ , where Ric g (∇φ) is the Ricci curvature of the metric g = e 2φ |dx| 2 in ∇ g φ direction.
(1.12) is clearly a generalization of Gauss curvature equations in higher dimensions.

Moreover,
• m = 0 if and only if g = e 2φ |dx| 2 is flat; • if Ric g is bounded in addition, then This theorem gives some precise description of asymptotic end behavior. More importantly it also includes a rigidity result that does not assume Ricci is bounded. The rigidity result in this theorem should be compared with [Cd97, Theorem 0.3] and [BKN89,CZ02,CH06]. It is particularly desirable to compare the blow-down approaches here and those in [BKN89,Cd97,CH06]. The proof invokes Theorem 1.1 and Theorem 1.2. But it is not straightforward at all to calculate m and derive the rigidity, especially when Ricci is not assumed to be bounded. Our argument relies on the ingenious construction of exhausting family of domains to perform integrations (please see Ω ± ε,t in the proof of Theorem 5.1 in Section 5). The exhausting family of domains Ω ± ε,t are good because of the strong n-capacity lower bound estimate in Lemma 5. Our second application is to study the asymptotic behavior at the end of properly embedded complete hypersurfaces with nonnegative Ricci curvature in hyperbolic space. It was shown in [BMQ17, Main Theorem] that such hypersurfaces have at most two ends, and are equidistant hypersurfaces if with two ends. Based on Theorem 1.1, we are able to improve the theorems on asymptotic at infinity in [AC90,AC93] assuming only Ricci to be nonnegative.
Theorem 1.4. Suppose that Σ n is a properly embedded, complete hypersurface with nonnegative Ricci curvature and one single end in hyperbolic space H n+1 . Then it is a global graph of ρ = ρ(x) in Busemann coordinates and it is asymptotically rotationally symmetric in the sense that there is a number m ∈ [0, 1] such that Moreover, m = 0 implies that the hypersurface is a horosphere. In any case, the hypersurface Σ always stays inside a horosphere and is supported by some equidistant hypersurface.
The proof follows from the one in [AC90,AC93], and in fact is simpler than the one in [AC90, AC93], because of Theorem 1.1. The use of n-subharmonic functions is more suitable than the use of subharmonic functions restricted to each 2-plane (cf. [AC90,AC93] The rest of the paper is organized as follows: In Section 2, we present definitions and basic facts that are useful. We describe what have been done in 2 dimensions to motivate our study in this paper. We also explain the opportunity for the use of n-superharmonic functions in geometric problems. In Section 3, we define n-thinness by n-capacity and prove Theorem 1.1. In Section 4, we establish the Adams-Moser-Trudinger inequality for the Wolff potential and prove Theorem 1.2. In Section 5, we introduce the classification of complete locally conformally flat manifolds with nonnegative Ricci curvature and prove Theorem 1.3. Finally, in Section 6, we recall the classification of complete properly embedded hypersurfaces with nonnegative Ricci curvature and prove Theorem 1.4.

PRELIMINARIES AND BACKGROUND
In this section, after adopting the definitions of n-harmonic functions and n-superharmonic functions from [KM94, Section 2], we would like to first present a review of what have been done in 2 dimensions to motivate what we want to do in general dimensions. Then we would like to introduce some background and tools from the theory of quasilinear elliptic equations and nonlinear potential theory that are useful to us. We also introduce the geometric problems that we expect to use n-superharmonic functions to study in this paper. For a domain Ω ⊂ R n , a function u ∈ W 1,n loc (Ω) is said to be weakly n-harmonic in Ω if |∇u| n−2 ∇u · ∇φ = 0 for all φ ∈ C ∞ c (Ω). A weak n-harmonic function u ∈ W 1,n loc (Ω) is said to be n-harmonic if it is continuous in Ω.
We know from [L06,Theorem 2.19] that any weak n-harmonic function is always continuous and therefore n-harmonic. For further regularity of n-harmonic functions we referred readers to [L06] and references therein. For the definitions of n-superharmonic functions, we first recall Definition 2.2. ([L06, Definition 2.12]) For a domain Ω ⊂ R n and a function u ∈ W 1,n loc (Ω) satisfying is called a weakly n-superharmonic function in Ω. A function u is said to be weakly nsubharmonic if −u is weakly n-superharmonic.
In the mean time, the following definition for n-superharmonic functions is often used in nonlinear potential theory. To avoid confusions, we quote the following definition for nsuperharmonic functions.
Fortunately, the relations between these two definitions has been clarified very well in [HK88,KM94]. For instance, we have • If u is a weakly n-superharmonic in Ω ⊂ R n , then there is an n-superharmonic function v such that v = u a.e. in Ω; • If u is n-superharmonic in Ω and u ∈ W 1,n loc (Ω), then u is weakly n-superharmonic; • If u is n-superharmonic and locally bounded, then u ∈ W 1,n loc (Ω) and weakly n-super -harmonic.
Clearly, when functions are C 2 or better, these two definitions agree, we will simply refer them n-superharmonic with no confusion. For n-superharmonic functions, one still has integrability of the gradient as shown in [L06,Theorem 5.15].
Lemma 2.2. ([L06, Theorem 5.15]) Suppose that u is an n-superharmonic function in Ω. Let D ⊂⊂D ⊂ Ω be a bounded subdomain and 0 < q < n. Then there is a constant C > 0 such that Therefore, if u is n-superharmonic or weakly n-superharmonic function, then µ = −∆ n u may be considered to be a nonnegative Radon measure on Ω (cf. [L06] and [KM92, Theorem 2.1]). And, by a simple approximation argument, |∇u| n−2 ∇u · ∇φ = φdµ for any testing function φ ∈ W 1,n 0 (D), if u ∈ W 1,n (D) and D ⊂ Ω. It is also helpful to mention the following weak comparison principle from Theorem 2.15 and the remark right after the proof in [L06]. For more basic properties of n-superharmonic functions, we refer readers to [HK88, KM88, HKM93, KM94, L06].
2.2. The story in 2 dimensions. Thanks to the seminal paper [Hu57] of Huber in 1957 (see also [CV35,F41,BF42,Ho52]), to explore the connection between geometric properties of surfaces and potential theory based on Gauss curvature equations has been the major part of the theory of surfaces. The Gauss curvature equation in an isothermal coordinates on a surface is where K is the Gauss curvature of the surface metric e 2u |dx| 2 . Let us focus on one thread of developments on this subject: local behavior of superharmonic functions near an isolated singular point or equivalently asymptotic behavior at infinity of superharmonic functions on the entire plane.
A function that is subharmonic on the entire plane is representable as a function of potential type for z, ξ ∈ C the complex plane, where µ is a positive mass distribution and vanishes in a neighborhood of the origin for our purposes. To describe the asymptotic behavior of the function v at infinity one aims to understand the limit In this regard, notions of thinness play the natural and important role. Notions of thinness at a point was considered by Brelot in [B40] in 1940, where a subset E in C is said to be thin at a point z 0 if either z 0 / ∈Ē or there exists a subharmonic function v in a neighborhood of z 0 such that which we will refer it as thinness by Cartan property (cf. [AH73]). This notion of thinness at a point is for potential functions with no point charge at the point.
In [AH73, (1.8)], a subset E of C is said to be thin at infinity if either it is bounded or there exists a function that is subharmonic on the entire complex plane C such that At the end of [AH73], there was a discussion about the correlation of these two notions of thinness. For a function v of potential type, one may take an inversion and consider the subharmonic function |z| on the punctured plane with no charge at the origin, where M is the total mass of the potential function v. Then (2.6) is equivalent to lim sup z∈Ẽ and z→0 This is to say that the thinness defined in [AH73, (1.8)] is the one for potential functions with point charge. It was then pointed out in [AH73] that E is thin at infinity by (2.6) if and only ifẼ is thin at the origin by (2.5) thanks to [B44, Theorem 2]. We remark here that, to us, it is a question whether these two types of thinness are still equivalent for a nonlinear potential theory.
In the geometric viewpoint, more interestingly, an equivalent criterion for a set to be thin at infinity using log-capacity was established as a Wiener type criterion in [AH73].  It is known that m = 1 2π C Ke 2u dz and m ∈ [0, 1] due to [CV35,Hu57], where m = 0 implies u is a constant. The proof of the above result in [BMQ16] relies on two important ingredients that are deep in geometric analysis and partial differential equation. One is the non-collapsing result of Croke-Karcher [CK88, Theorem A] in 1988 for complete surfaces with nonnegative Gauss curvature; the other is asymptotic estimates for nonnegative solutions to Gauss curvature type equations of Taliaferro in [Tal06, Theorem 2.1] (see also his previous work [Tal99,Tal01]). One of the key analytic ingredients in [Tal99,Tal01,Tal06] is the Brezis-Merle inequality of Moser-Trudinger type Taliaferro's estimates in [Tal99,Tal01,Tal06] are the major work in the theory of local behavior of a class of subharmonic functions near an isolated singular point in 2 dimensions. And, in the spirit of Huber that was reflected in [Hu57], on geometric side, it was a very successful story that the above theorem of sharp local behavior (cf. [BMQ16, Lemma 4.2]) turns out to be essential to the proof of [BMQ16, Main Theorem] in 2 dimensions that a complete, nonnegatively curved, immersed surface in hyperbolic 3-space is necessarily properly embedded, except coverings of equidistant surfaces, which was conjectured by Epstein and Alexander-Currier in [AC90, AC93, E86, Enote, E87] around 1990.
2.3. Isolated singularity for nonnegative n-superharmonic functions. There have been significant developments of the study on local and global behaviors for solutions to (degenerate) quasilinear elliptic equations that include the study of n-Laplace equations, for example, [KV86,BV89,V17] and references therein. The following result on the isolated singularities of nonnegative n-superharmonic functions is particularly useful to us.
Suppose that w is a nonnegative n-superharmonic function on the punctured ball B(0, r) \ {0} for some r > 0. Assume that w is continuous and |∇w| n is locally integrable in the punctured ball. Furthermore, assume that −∆ n w is also locally integrable in the punctured ball. Then, if lim x→0 w(x) = ∞, then there are a function g ∈ L 1 (B(0, r)) and a number β ≥ 0 such that in the distributional sense, where δ 0 is the Dirac function at the origin.
Remark 2.1. We remark here that the function w in the above theorem is in fact an n-superharmonic in the ball B(0, 1) as the potential of the nonnegative Radon measure that is induced from g+βδ.
The other important contribution in the study of isolated singularity of n-harmonic functions is the following result in [KV86,KV87], which is based on previous works in [Se64,Se65,Tol83].
The idea of the proof of this theorem in [KV86] is particularly helpful to us. In fact, in some sense, what we would like to have is the extension of this theorem to cover n-superharmonic functions. Our approach combines that in [KV86] and the use of the nonlinear potential theory [HKM93, KM94, PV08, AM72, L06].
2.4. Non-linear potential theory for n-Laplace equations. The nonlinear potential theory itself is a vast and profound subject in Mathematics. We certainly do not intend to give an comprehensive introduction here. Instead we will collect useful facts in a cohesive way that we perceive. To study n-Laplace equations where µ is a nonnegative Radon measure representing the mass distribution, there is the nonlinear potential theory developed to replace the principle of superposition (cf. [KM94, HKM93, HK88, PV08, AM72, L06]). The fundamental tool is the Wolff potential The Wolff potential plays the same role in the nonlinear potential theory as the Riesz potential plays in the linear one. And the foundational estimates in the nonlinear potential for the equation (2.14) is as follows: Suppose that w is a nonnegative n-superharmonic function satisfying (2.14) for a nonnegative Radon measure µ in B(x 0 , 3r). Then It is easily seen that the study of n-Laplace equations is intimately related to n-capacity since solutions to n-Laplace equations are critical points for the functional |∇u| n dx. We therefore recall the definition of n-capacity from [KM94, Section 3].
Definition 2.4. For a compact subset K of a domain Ω in Euclidean space R n , we define n-capacity is clearly invariant under conformal transformations, and therefore is also called the conformal capacity (cf. [Ge61], for example). The notions of n-thinness in the potential theory are important in the study of n-superharmonic functions. Notions of n-thinness were first considered in [AM72], and readers are referred to [AM72, HK88, KM94] for more background and references. One notion of n-thinness is defined via Wiener integral given in [AM72,KM94], which we will refer to as thinness by Wiener integral. One of the major achievements in [KM94] is to establish the complete equivalence between the thinness by Wiener integral and the one by Cartan property (2.5) in general dimensions, based on [KM94, Theorem 1.6] and early works [AM72,HK88]. But, these notions of thinness at a point are for potential functions with no point charge, which is only known to be the same as the notion of thinness for potential functions with point charge in 2 dimensions ([B44, Theorem 2] and [AH73]) . In higher dimensions, inspired by [AH73, Theorem 1.3], we will introduce a notion of thinness using n-capacity and study its relation to the Cartan property (2.6) for n-subharmonic functions at isolated singular point (cf. Definition 3.1 and Theorem 3.1 in next section).
2.5. n-Laplace equations in differential geometry. What can we do in higher dimensions following the approach in [Hu57] by Huber? We have seen successful efforts in [SY88, Zhu94, CQY00, CEOY08, CH02, CH06] to explored higher dimensional counterparts of Gauss curvature equations (2.4) such as the scalar curvature equations where R andR are scalar curvature of the metrics g andḡ = u 4 n−2 g respectively in dimensions n ≥ 3; and the higher order analogue: Q-curvature equations, where P n = (−∆) n + lower order is the so-called Paneitz type operator and Q n ,Q n are socalled Q-curvature of the metrics g andḡ = e 2w g respectively in dimensions 2n ≥ 2. We have also seen remarkable successes in using fully nonlinear equations of Weyl-Schouten curvature, as replacements of Gauss curvature equations, in [CGY02, CHY04, GLW05, G05]. The above mentioned seem to represent major developments in conformal geometry and conformally invariant partial differential equations following the approach in [Hu57] by Huber.

n-Laplace equations in conformal geometry. Recall the change of Ricci curvature under conformal change of metrics is
where R ij ,R ij are Ricci curvature tensors for the metrics g andḡ = e 2φ g respectively in n dimensions. Contracting with φ i and φ j on both sides of the above equation, one gets that . Therefore one arrives at another generalization of Gauss curvature equations in higher dimensions, Particularly, when g is Ricci-flat, we have In this paper we want to explore properties of n-superharmonic functions and the geometric consequences. Following the approach in [Hu57] by Huber we want to extend the success in 2 dimensions to higher dimensions and complement contemporary developments in conformal geometry and conformally invariant partial differential equations.
2.5.2. Hypersurfaces in hyperbolic space. Apparently, the first use of n-subharmonic functions in differential geometry was in [BMQ17]  It is perhaps worth to mention, for immersed hypersurfaces Σ n ⊂ H n+1 with appropriate orientation, the following successively stronger pointwise convexity conditions on the principal curvatures κ 1 , . . . , κ n : In this paper we will use the properties of n-superharmonic functions to derive asymptotic behaviors for hypersurfaces in herperbolic space with nonnegative Ricci and improve the asymptotic results in [AC90,AC93].

HIGHER DIMENSIONAL ARSOVE-HUBER'S THEOREM
In this section our goal is to extend Theorem 2.2 and (2.9) (cf. [AH73, Theorem 1.3]) in general dimensions. First we define a notion of thinness by capacity inspired by that in 2 dimensions in [AH73, Theorem 1.3] for potential functions with point charge.
And we set Again, E is trivially n-thin at ∞ if E is bounded.
Clearly the inversion x |x| 2 of R n takes a subset E ⊂ R n that is n-thin by capacity at infinity to a subsetẼ that is n-thin by capacity at the origin.
Then there is a set E ⊂ R n , which is n-thin by capacity at the origin, such that Moreover, if (B(0, 2) \ {0}, e 2w |dx| 2 ) is complete at the origin, then m ≥ 1.
First of all, based on Theorem 2.3 (cf. [BV89, Proposition 1.1]), there is a nonnegative number β and g ∈ L 1 (B(0, 1)) such that where the nonnegative Radon measure is generated by g + βδ.
To study the local behavior for a nonnegative n-superharmonic function w on the punctured ball, we follow the idea from [KV86] to consider the blow-down 3.1. The first step in the proof of Theorem 3.1. The first we need is that the quotient w(x) log 1 |x| is mostly uniformly bounded. Therefore the following proposition is the first key step to prove Theorem 3.1.
Proposition 3.1. Assume the same assumptions as in Theorem 3.1. Then, there is a setÊ, which is n-thin by capacity at the origin, and a constantĈ such that The proof of Proposition 3.1 starts with the following simple fact observed in [KM94, Lemma 3.9].
Lemma 3.1. ([KM94, Lemma 3.9]) Suppose that u ∈ W 1,n 0 (Ω) is an n-superharmonic function satisfying −∆ n u = µ for a nonnegative Radon measure µ. Then, for λ > 0, The proof of Lemma 3.1 is to use min{u,λ} λ as a test function and get which is easily seen to imply the above n-capacity estimate (3.4). The next fact we need to prove Proposition 3.
To make use of the fundamental estimates (2.16) in Theorem 2.5(cf. [KM94, Theorem 1.6]), we also need the following estimates on the infimum.
Proof. We will rely on some estimates from [DHM97, Section 7] to derive this lemma. Readers are referred to [DHM97] for definitions and notations. Particularly, in the light of [DHM97, Lemma 14 and 15], we know that ). Meanwhile, from [L06, Theorem 5.11], for instance, we know w L n (B(0, 1 2 ) is finite. Therefore w BM O(B(0, 1 4 ),B(0, 1 2 )) is finite. Here we remark that in [DHM97, Section 7], the assumption that the right hand side in L 1 can be generalized to a nonnegative Radon measure easily. And the assumption that u ∈ W 1,n is not essential, because, if not, we can substitute u by min{u, k}, which belongs to W 1,n , and use Fatou's lemma to prove (3.6) for u.
Suppose otherwise that (3.5) were not true. Then we have a sequence We let where the finite numberw is the average of w on B(0, 1 4 ). Clearly, at least for i large, (3.8) because (3.7). On the other hand, from [JN61, Lemma 1], we know that, there are B and b, such that which is a contradiction with (3.8). Thus the proof is completed.
For convenience and simplicity, we use Now we are ready to start the proof of Proposition 3.1.
The proof of Prposition 3.1. It is obvious that w(y) log 1 |y| ≥ 0. We are going to prove that outside some setÊ, which is thin by capacity at the origin, the quotient w(y) log 1 |y| has upper bound.
We cover ω 0 with finite number of balls {B 0 1 , · · · , B 0 m }, where the center of B 0 j lies in ω 0 , the concentric ball 4B 0 j ⊂ Ω 0 for j = 1, · · · , m, and m depends only on the dimension n. For i ≥ 0, we denote B i j = 2 −i B 0 j . It's obvious that {B i j : j = 1, · · · , m} cover ω i and each 4B i j lie in Ω i . We let r ij be the radius of B i j . Clearly r ij = 2 −i r 0j .
For any y ∈ B i j , from [KM94, Theorem 1.6] and Lemma 3.3, we have Since |y| ∼ r ij ∼ 2 −i and |W µ 1,n (y, We arrive at (3.9) w(y) ≤ C(log 1 |y| + W µ 1,n (y, To estimate W µ 1,n (y, 1 3 r ij ), we use Lemma 3.2 and solve the following The advantage is that, from [KM94, Lemma 3.9], we know that Now, using [KM94, Theorem 1.6] again, we have Then we have Thus, from (3.9), there is a constantĈ > 0 such that, outsideÊ, which is thin by capacity according to Definition 3.1, (3.3) holds. The proof is completed.
3.2. The second step in the proof of Theorem 3.1. The second key step in the proof of Theorem 3.1, for the sake of the blow-down argument as the one used in [KV86], is to modify the function w(rξ) log 1 r to accommodate the lack of boundedness. We use the trick from [DHM97] and consider the cut-off function where α is to be fixed asĈ + 1 throughout this paper, whereĈ is the one in (3.3). One may calculate that Now we are to carry out the blow-down argument as in [KV86]. For each r > 0 and small, we consider the modified blow-down (3.16)ŵ r (ξ) = a α (w r (ξ)) = a α ( w(rξ) log 1 r ).
Clearly, we have To summarize, we state the following lemma to use the above calculations.
Lemma 3.4. Assume the same assumptions as in Theorem 3.1. Then the modified blow-down w r (ξ) is a nonnegative and bounded n-superharmonic function satisfying More importantly, for any fixed R > 1, whereÊ is the subset given in Proposition 3.1, which is thin by capacity at the origin.
We want to say that Lemma 3.4 implies that there is no concentration other than that possibly at the origin. Therefore, at least, for sequences r k → 0, one may manage to show thatŵ r k (ξ) converges to a bounded n-harmonic function on the entire space R n except possibly the origin, which can only be a constant due to [Re66] because the origin is a removable singularity by [Se64]. To be more precise, we need the following convergence lemma. Proof. For the convenience of readers, we present proof here. First we prove Similar to the argument in [L06,Theorem 5.15], based on Lemma 2.1 (cf. [HK88] and [HK76, Proposition 2.7]), we simply use the testing functions ζ n (u i + 1) −1 , where .
This obviously implies (3.22). Next, to prove (3.21) by (3.22), we derive Hence there is u ∈ W 1,n (D) such that u i ⇀ u in W 1,n (D), at least for a subsequence. In the light of , it suffices to prove that (3.24) ζ n |∇u| n u −1−α dx ≤ C(n, α) u n−1−α |∇ζ| n dx for any α ∈ (0, n − 1] and any cut-off function as in the above proof. The right hand side of (3.24) is finite by [L06,Theorem 5.11]. This remark is useful to handle the second term on the right side of (3.20).
3.3. The third step in the proof of Theorem 3.1. The third key step in the proof of Theorem 3.1 is to show the uniqueness of possible limits of all blow-down sequences. We continue to use the approach used as in [KV86]. One of the key tool is the following weak comparison principle as a consequence of [ For any blow-down sequenceŵ r i (ξ) with r i → 0, there is ξ r i with |ξ r i | = 1 and Because, Lemma 3.6 implies that the quotient min |x|=r w(x) log 1 |x| is non-increasing as r → 0, since the infimum is always achieved at the inner sphere of the annulus B(0, r 0 ) \ B(0, s) for r 0 < 1 fixed while s arbitrarily small. Notice that we may assume if necessary. Because, one may deal with w+ǫ for arbitrarily small ǫ instead. We will present the proof of the uniqueness of all blow-down limits based on Lemma 3.6 in the proof of Theorem 3.1 in next section.
3.4. The last step of the proof of Theorem 3.1. With all the preparation we finally are ready to prove Theorem 3.1. At this point, we have cleared almost everything except that the convergences of each blow-down sequenceŵ r k to a constant is weaker than the pointwise one. This in principle is caused by the fact that the density function is just a Radon measure µ = g + δ for g ∈ L 1 . Our main goal here, after presenting a proof of the uniqueness of the sequential blow-down limits, is to extract a possible bad set E, which is again n-thin by capacity so that outside E the limit of the quotient w(x) The proof of Theorem 3.1. To recap, first, from Proposition 3.1 in Section 3.1, we know that, outside the thin setÊ, w(x) log 1 |x| ≤Ĉ.
Then, based on the discussion in Section 3.2, we consider the modified blow-down functionŝ w r (ξ) by (3.16) for α = 1 +Ĉ. From Lemma 3.5 and Lemma 3.4, for a sequence r i → 0, we may assume thatŵ r i (ξ), converges to a bounded n-harmonic functionŵ(ξ) in A(0, ∞) = R n \ {0} (for some subsequence if necessary). When appying Lemma 3.5 and verifying f i → 0 in L 1 , one needs to use (3.20) and Remark 3.1. Thanks to Liouville type theorem of Reshetnyak [Re66], 0 and ∞ are removable singularities ofŵ(ξ) andŵ(ξ) =ŵ is a constant. Finally, one would like to use Lemma 3.6 in Section 3.3 to derive (3.27)ŵ = γ − = lim inf r→0 w(x) log 1 |x| for any sequence r i → 0. The remaining issue is that all the sequential convergences are only the one weak in W 1,n and strong in W 1,p for any 1 ≤ p < n, which does not yet imply pointwise convergence as desired. Now let us start with a proof of the uniqueness ofŵ (i.e. (3.27)). Recall from (3.26) for any sequence r i → 0. Sinceŵ r i (ξ) converges toŵ strongly in W 1,p (A(r 0 , 1 r 0 )), 1 ≤ p < n for any fixed small r 0 > 0, we know that for any 0 < q < ∞. By the way, w r (ξ) ≥ γ − due to the definition of γ − and Lemma 3.6. By invoking the weak Harnack inequality [HKM93, Theorem 3.51]), we knoŵ for any ξ ∈ B 1 4 (1−r 0 ) (ξ r i ) and some 0 < q < ∞. Clearly this would be a contradiction if w = γ − . So this finishes the proof of the uniqueness for sequential blow-down limits.
In the following, what we need to do is to refine the argument in the proof of Proposition 3.1 to show that, outside an n-thin set, the quotient w(x) log 1 |x| is not just bounded but actually convergent at the origin pointwisely. We will use the same notations and follow the same process. But we are in a better position than that we were in the proof of Proposition 3.1. First, we have the following improved (3.5) in Lemma 3.3 This is because, from the uniqueness of all blow-down limits, we know lim r→0ŵ r (ξ) = γ − almost everywhere in A r 0 , 1 r 0 and thatŵ r and w r only differ at the setẼ that is n-thin by capacity at the origin. In fact we have the following, which is even more useful.  ∈ (0, 1).
Secondly, we apply [KM94, Theorem 1.6] to w(y) − inf B(y, 3 4 |y|) w in B(y, 3 4 |y|) and obtain Hence, inf B(y, 1 4 |y|) w log 1 |y| + C 3 W µ 1,n (y, 1 2 |y|) log 1 |y| which implies, by (3.29) in Lemma 3.7, Thirdly, regarding the Wolff potential term in (3.31), we will also need an improved (3.10). For this purpose we first consider the convergent infinite series 3µ(B(0, 1)) < ∞ and use Paul du Bois-Reymond Theorem [B1908, (5) Page 40] (cf. [R1873]) to find a sequence for all y ∈ A 0,1 . From the similar argument as in the proof of (3.10), we have, which says that E is n-thin by capacity and This can be viewed as the improvement of [AH73, Theorem 1.3], having no thin subset where the asymptotic behavior may differ from (4.1). Our next goal is to establish the higher dimensional analogue of [Tal06, Theorem 2.1] as follows: Theorem 4.1. Let w ∈ C 2 (B(0, 1) \ {0}) be nonnegative and satisfy in a punctured neighborhood of the origin in R n and that lim   1 n−1 dt t associated with a Radon measure µ, and a Radon measure µ f that is induced from a function f ∈ L 1 (Ω) Proposition 4.1. Let Ω ⊂ R n be a bounded domain with the diameter D. And let f ∈ L 1 (Ω) be nonnegative. Then, for δ ∈ (0, 1), Proof. The proof is more or less standard in harmonic analysis. For the convenience of readers, we present a proof here. To start, we let p > n − 1 and be the Hardy-Littlewood maximal function of f . Hence almost everywhere, that is to say, µ(B(x, t)) 1 p log t| D 0 = α log D almost everywhere. Therefore, by Jensen's inequality If α < n p , then So we have, for λ ≥ 2, |{x ∈ Ω : exp(W µ 1,n (x, D)) ≥ λ}| ≤ |{x ∈ Ω : thanks to the weak type Hardy-Littlewood maximal inequality. For 0 < q < p, This finishes the proof. Proof. We prove Lemma 4.1 by contradiction. Assume otherwise, there is a sequence {y k } inside the punctured ball such that w(y k ) log 1 |y k | → ∞ as |y k | → 0.
We thus get which contradicts with (4.9) in the light of (4.12). So Lemma 4.1 is proved.
4.3. The proof of Theorem 4.1. Lemma 4.1 enables us to proceed with blow-down argument without going through Sections 3.1 and 3.2. We are now ready to prove Theorem 4.1.
The proof of Theorem 4.1. We again consider the blow-down w r (ξ) = w(rξ) log 1 r and calculate that, from Lemma 4.1, From here, similar to the approach of the proof of Theorem 3.1 in the previous section, one may complete the proof of Theorem 4.1. To do so, we continue to use the notation First, as in the previous section, one may prove that on R n \{0}, w r (ξ) converges to γ − in W 1,n loc weakly and W 1,p loc strongly for any p < n, which implies that w r (ξ) converges to γ − pointwisely almost everywhere. This heavily relies on the uniqueness of sequential blow-down limits established in the proof Theorem 3.1 in Section 3.4. Hence we want to improve from here that w(x) log 1 |x| converges to γ − pointwisely as we did in the proof Theorem 3.1 in Section 3.4.
In the light of (3.34) and (3.31), we need to show (3.33) with no thin set E excluded, i.e. To prove (4.13), we recall that From (4.8) and [BV89, Proposition 1.1], we know that µ(B(0, 1)) < ∞ and that µ(B(y, s)) → 0 as s → 0 for s ≤ 1 2 |y|. But that is not enough, particularly when s is very small in calculating the Wolff potential. Therefore we recall [HK76, Theorem 1.6] Finally, we invoke L p bound for the gradient of the n-superharmonic function w for p = n− 1 2 < n and get 2 ) , (4.14) Here we are indifferent to constants except maybe those from [HK76, Theorem 1.6]. Therefore, going back to estimate the Wolff potential, we have and o(1) is with respect to y → 0, which implies (4.13). So (4.4) is established. It is then easily seen that 1 0 e w dr = ∞ implies m ≥ 1 from (4.4). Thus the proof is completed.

LOCALLY CONFORMALLY FLAT MANIFOLDS
In this section we are going to use the property of n-superharmonic functions to study the asymptotic behavior at the end of a complete locally conformally flat manifold (M n , g). Based on the injectivity of the development maps of [SY88,Theorem 4.5], in [Zhu94, Theorem 1] and later in [CH06], the following classification result was shown.
Theorem. ([Zhu94, Theorem 1] [CH06]) Let (M n , g) be a complete conformally flat manifold of dimension n ≥ 3 with nonnegative Ricci curvature. Then, exactly one of the following holds: • The universal cover of (M n , g) is globally conformally equivalent to the flat Euclidean space; • The universal cover of (M n , g) is globally conformally equivalent to around sphere (S n , g S n ); • (M n , g) is locally isometric to the standard cylinder R × S n−1 .
We confine ourselves to the first case in the above classification theorem. Recall that, on (R n , e 2φ |dx| 2 ), in the light of (2.18), where Ric g (∇ g φ) is the Ricci curvature of the conformal metric g = e 2φ |dx| 2 in the ∇ g φ direction. As a consequence of Theorem 3.1 and Theorem 4.1, for a globally conformally flat manifold (R n , e 2φ |dx| 2 ), we therefore are able to deduce the following: Theorem 5.1. Suppose that (R n , e 2φ |dx| 2 ) is complete with nonnegative Ricci (n ≥ 3). Then there is a subset E ⊂ R n , which is n-thin by capacity at infinity, such that for some constant C, where (5.2) m|m| n−2 = 1 w n−1 R n Ric g (∇ g φ)|∇φ| n−2 e 2φ dx.

Moreover,
• m ∈ [0, 1] and m = 0 if and only if g is flat, i.e. φ(x) is a constant function; • if Ric g is bounded in addition, then We remark that Theorem 5.1 should be compared with [BKN89,Cd97,CZ02]. In [BKN89] it was proved that, a complete noncompact manifold (M n , g) satisfying Ric ≥ 0 vol(B(0, r)) ≥ γr n for some γ > 1 2 w n−1 |Rm| ≤ Cr −2 and in addition, is actually isometric to the Euclidean space. The assumption of γ > 1 2 w n−1 is essential, in the light of Eguchi-Hanson metrics. In [Cd97], Colding proved remarkably that a complete manifold with nonnegative Ricci curvature is isometric to the Euclidean space, if one tangent cone at infinity is the Euclidean space. In [CZ02], on the other hand, it was proved, a complete noncompact conformally flat manifold with nonnegative Ricci and satisfying 1 vol (B(x 0 , r) where the scalar curvature R is bounded, is actually isometric to the Euclidean space. The comparison of Theorem 5.1 to the rigidity results in [BKN89,Cd97,CZ02] would be more direct if the intrinsic distance function r on the manifold with |x| in Euclidean space as the background metric are equivalent, which seems to require something stronger than (5.3).
Hence, from Theorem 3.1, we know there are a number m 1 ≥ 1 and a set E 1 , which is n-thin by capacity at 0 such that, where m = 2 − m 1 ≤ 1 and E = {x; x |x| 2 ∈ E 1 }. Moreover, from Definition 3.1, we know E is n-thin by capacity at infinity. So (5.1) is proved.
If, in addition, Ricci curvature is bounded, then Ric g (∇ g φ)|∇w| n−2 e 2w ≤ C|∇ y w| n−2 e 2w and (5.3) follows from Theorem 4.1. Assume (5.3) holds. Then it is obvious that m ∈ [0, 1]. And, when m = 0, then φ is n-harmonic in the entire space with growth o(log |x|). So φ has to be a constant in this case due to [HKM93, 6.2 Theorem and 6.11 Corollary].
To finish the proof of Theorem 5.1, it suffices to prove (5.2). To do so, we are going to integrate To avoid relying on sharp gradient estimates for φ on the boundary of any exhausting family of domains Ω in R n , we will work with chosen exhausting families of domains. Our construction of the exhausting families of domains is ingenious and turns out to be very natural and very desirable. Let us define, for a positive small number ε and a positive large number t, } that includes the origin. Claim. For a fixed ε > 0, there is a sequence of positive number t k → ∞ such that the collection {Ω + ε,t k } is an exhausting family of smooth and bounded domains for R n . Similarly, for a fixed ε > 0, there also exists a sequence of positive number s k → ∞ such that the collection {Ω − ε,s k } is an exhausting family of smooth and bounded domains for R n . Proof of Claim. Let us first consider Ω + ε,t . Smoothness is not a problem, one can always perturb and get the smooth ones. From the definition, it is easily seen that, for any fixed R, whenever t is sufficiently large. Hence Ω + ε,t can exhaust the entire space. Meanwhile, for each fixed ε and t, Ω + ε,t is bounded in the light of (5.4).
Let us turn to Ω − ε,t . The only issue different is the boundedness for Ω − ε,t when ε and t are arbitrarily fixed. It is easily seen that each Ω − ε,t \ E is bounded, because of (5.5). Then Ω − ε,t is the connected component that includes the origin and n-thin by capacity at infinity. Recall: (2) B includes the origin and has a point with length L. Then cap n (A, B) ≥ c n (log(1 + 1 L )) n−1 To see connected Ω − ε,t is bounded from the above lemma, one just need to realize that each Ω − ε,t ∩ ω i is connected and includes points on the both components of the boundary ∂ω i and therefore cap n (Ω − ε,t ∩ ω i , Ω i ) has a uniform lower bound. So the proof of this claim is finished.
Here in the last step, we use the fact that for t very large, whereG + ε,t = −(m + ε) log |x| + t which agrees with G + ε,t outside the unit ball.
Similarly, using G − ε,t and Ω − ε,t , we have Thus, by the exhaustion property of the chosen families of domains, (5.2) follows. The proof of Theorem 5.1 is completed.

HYPERSURFACES IN HYPERBOLIC SPACE
In this section we want to use Theorem 3.1 and Theorem 4.1 to study the asymptotic end structure of embedded hypersurfaces in hyperbolic space with nonnegative Ricci. Our work here is inspired by and improves the results in [AC90,AC93]. In the light of [BMQ17, Main Theorem], in this paper, we focus on the study of end structure at infinity for these hypersurfaces in hyperbolic space with nonnegative Ricci and one single end. We refer readers to Section 2.5.2 for a very brief introduction of complete and globally strictly convex hypersurfaces in hyperbolic space (cf. [AC90, AC93, BMQ16, BMQ17]). For convenience of readers, we first remind us what is Busemann coordinates in hyperbolic space. We start with half space model for hyperbolic space R n+1 + = {(x 1 , x 2 , · · · , x n , x n+1 ) : (x 1 , x 2 , · · · , x n ) ∈ R n and x n+1 > 0} with the hyperbolic metric We use the notation that ∂ ∞ H n+1 = R n {p ∞ } in this half space model. A vertical graph in hyperbolic space is the hypersurface given by The Busemann coordinates is (x, ρ) ∈ R n × R such that ρ = log x n+1 .
Therefore the height function for a vertical graph in Busemann coordinates is It is worth to mention that, in such coordinates, an equidistant hypersurface with one end at p ∞ is represented by ρ = log |x − x 0 | + C for the other end at some point x 0 ∈ R n ⊂ ∂ ∞ H n+1 and a constant C.
To see the use of inscribed radially symmetric graphsρ, similar to what was observed when hypersurfaces were assumed to be nonnegatively curved in [AC90,AC93], we first observe: Lemma 6.1. Suppose that the graph ρ = ρ(x) over Ω ⊂ R n in Busemann coordinates in hyperbolic space is complete and with nonnegative Ricci and one single end at p ∞ . Then Ω = R n and there is an equidistant hypersurface ρ = log |x| + C such that Proof. First of all, we know the hypersurface is globally and strictly convex. LetΣ be the inscribed radially symmetric hypersurface as the graph ofρ to the hypersurface as the graph of ρ.
It is easy to see that ∂ ∞Σ = {p ∞ }.
First, from [HK76, Page 66],ρ is non-decreasing and convex in log r. Henceρ is continuous and differentiable except at countably many points. Moreover, at a singular point a, ρ ′ − (a) <ρ ′ + (a). Whenρ is differentiable for r ∈ (a, b), the corresponding portion ofΣ has nonnegative Ricci curvature. Because, for any fixed r ∈ (a, b),Σ is supported by Σ at least at some point x with |x| = r. By the comparison of principal curvatures, one may easily derive that Ricci ofΣ is nonnegative from that the Ricci of Σ is nonnegative. ThereforeΣ is with Ricci curvature nonnegative everywhere on the regular part ofΣ. Now, assume without loss of any generality that 0 ∈ Ω. Let R be the radius of the maximal ball B(0, R) ⊂ Ω. For any fixed r 0 < R, we take C sufficiently large such that ρ(r 0 ) < log r 0 + C.
Here ρ = log |x|+C is the equidistant hypersurface about the vertical geodesic line γ connecting p ∞ and 0 ∈ R n ⊂ ∂ ∞ H n+1 . Then we claim (6.2)ρ(r) ≤ log r + C for all r ∈ (r 0 , R). Assume otherwise, there is some interval [r 1 , r 2 ] ⊂ (r 0 , R), such that ρ(r 1 ) = log r 1 + C andρ(r) > log r + C for r ∈ (r 1 , r 2 ]. Then there has to be some ξ ∈ (r 1 , r 2 ) whereρ is differentiable andρ ′ (ξ) > 1/ξ. This implies, the horizontal spherical section ofΣ at r = ξ is with negative definite second fundamental form, in contrast to the equidistant hypersurface, whose horizontal spherical sections are totally geodesic. Because thatΣ is with nonnegatively Ricci when it is differentiable, and that the mean curvature of the spherical section only drops at singular point, one may derive thatΣ can only be compact, which clearly is a contradiction. So we proved (6.2). To see R = ∞, we assume otherwise. Then, from the fact that Σ is complete and has only one end at p ∞ , lim r→Rρ (r) = ∞, which contrdicts with (6.2). Hence Ω = R n and (6.2) holds for all r ≥ r 0 . Thus the proof of the lemma is completed.
Based on Theorem 3.1 and Theorem 4.1, we are able to improve the results on asymptotic behavior of global vertical graph of nonnegative sectional curvature in [AC90,AC93]. Namely, Theorem 6.1. Suppose that Σ is a properly embedded, complete hypersurface with nonnegative Ricci and single end. Then it is a global graph of ρ = ρ(x) in Busemann coordinates and it is asymptotically rotationally symmetric in the sense that there is a number m ∈ [0, 1] such that m log |x| + o(log |x|) ≤ ρ(x) ≤ m log |x| + C as x → ∞ in R n . Moreover, m = 0 implies that the hypersurface is a horosphere. In any case, the hypersurface Σ always stays inside a horosphere and is supported by some equidistant hypersurface.
Proof. As the first step, to use Theorem 3.1, we first want to change a coordinatees in hyperbolic space, that is, to choose a different point at infinity ∂ ∞ H n+1 for the half space model. Then, based on Lemma 6.1, we know a priori that the hypersurface Σ is above a horosphere and below an equidistant hypersurface at least near the end at p ∞ . Hence, in the new Busemann coordinates, the hypersurface Σ is no longer a global graph of the height function, rather, a graph of the height function over a punctured ball, without loss of generality, we may assume the ball is the unit ball at the origin of the new Busemann coordinates(y, τ ) (in other words, we may put the end at infinity of Σ at the origin of the new Busemann coordinates). Therefore we are looking at the part of the hypersurface Σ that is parametrized as the graph of the height function τ = τ (y), which is a n-subharmonic function in B(0, 1)\{0} with lim y→0 τ (y) = −∞.
Thus, we may apply Theorem 3.1 to −τ and obtain x n+1 = 1 − |Z| 2 |Z + e n+1 | 2 for Z = (z, z n+1 ) ∈ B(0, 1) ⊂ R n+1 , Y = (y, y + n + 1) ∈ R n+1 + and X = (x, x n+1 ) ∈ R n+1 + , where e n+1 = (0, 1) is the north pole of the unit sphere in R n+1 . In Y coordinates it takes the north pole to infinity and the south pole to the origin; while in X coordinates it takes the south pole to infinity and the north pole to the origin. Hence the coordinate change between X and Y is the inversion with respect to the unit sphere centered at the origin: We may assume from the beginning that the hypersurface Σ has its end at p ∞ = −e n+1 . Therefore 1 |y| = |x| · (1 + x 2 n+1 |x| 2 ) and τ = log y n+1 = ρ − 2 log |x| − log(1 + x 2 n+1 |x| 2 ).
So we may translate (6.3) into ρ(x) ≤ m log |x| + C for all x ∈ R n (6.4) ρ(x) ≥ m log |x| + o(log |x|) for all x / ∈ E and x → ∞ (6.5) for some set E that is n-thin by capacity at infinity and some m = 2 − m 1 ≤ 1. Here we use (6.1) from Lemma 6.1 to control x 2 n+1 /|x| 2 .
Next step is to improve (6.5) and eliminate any nontrivial n-thin set E. Our approach here is to use the strict and global convexity of the hypersurface Σ to rule out the nontrivial n-thin set E, which is close to that in [AC90,AC93] in 2 dimensions but more straightforward. Assume otherwise, (6.5) is not true on a set E, which is n-thin by capacity and non-compact. Hence, there is a positive number ǫ 0 and a sequence point p k = (s k θ k , τ (s k θ k )) ∈ Σ such that (6.6) y Σ n+1 (s k θ k ) = e τ (s k θ k ) < s m 1 +ǫ 0 k and s k → 0. We have, in the light of (5.7) in Lemma 5.1 and Definition 3.1, for each s k θ k ∈ E, there always existsŝ k θ k / ∈ E forŝ k ∈ (s k (1 − s l k ), s k ) for any fixed large l ≥ 1. This can be proved by contradiction. Assume otherwise, one derives from (5.7) that, for each i, (6.7) cap n (E ∩ ω i , Ω i ) ≥ c n (2i(l + 1) log 2) n−1 , by the scaling invariance, which is impossible by Definition 3.1. On the other hand, there is δ 0 such that y Σ n+1 (sθ) = e τ (sθ) ≥ s m 1 + 1 2 ǫ 0 for all sθ / ∈ E and 0 < s < δ 0 . In particular (6.8) y Σ n+1 (ŝ k θ k ) ≥ŝ m 1 + 1 2 ǫ 0 k ≥ a 0 s m 1 + 1 2 ǫ 0 k for some positive a 0 , at least when k is large. Let us assume the following is the equation for the semi-circle that is inside the hyperplane tangent to Σ at the point overŝ k θ k and in the 2-plane for the fixed θ k ∈ S n−1 |s − c k | 2 + y 2 n+1 = r 2 k = |ŝ k − c k | 2 + (y Σ n+1 (ŝ k θ k )) 2 where (c k , 0) is the center of the semi-circle and 0 < c k < s k due to the fact that y Σ n+1 (ŝ k θ k ) < y Σ n+1 (s k θ k ). We may estimate the height of this semi-circle at s = s k : y 2 n+1 | s=s k ≥ a 2 0 s 2m 1 +ǫ 0 i + |ŝ k − c k | 2 − |s k − c k | 2 ≥ a 2 0 s 2m 1 +ǫ 0 k + |ŝ k − s k | 2 − 2(s k −ŝ k ) · (s k − c k ) ≥ a 2 0 s 2m 1 +ǫ 0 k − c 0 s 2m 1 +1+2ǫ 0 k > (y Σ n+1 (s k θ k )) 2 (6.9) for some uniform a 0 and c 0 and some appropriately large l, in the light of (6.6), which means the point p k on Σ falls under the hyperplane and violates the strict and global convexity of Σ, at least when k is large enough.
So far we have shown that m log |x| + o(log |x|) ≤ ρ(x) ≤ m log |x| + C as x → ∞ in R n with m ≤ 1. In the last step, we prove that m ∈ [0, 1] and Σ is a horosphere when m = 0. We at this point go back to the Busemann coordinates (x, ρ), use the similar argument in the last step of the proof of Theorem 5.1 (even easier, because that there is no bad thin set), and obtain |m| n−2 m = 1 w n−1 R n (∆ n ρ)dx ≥ 0.
Therefore, when m = 0, ρ in fact is an n-harmonic function and ρ = o(log |x|). In the light of Liouville Theorem in [HKM93, 6.2 Theorem and 6.11 Corollary], ρ is a constant, i.e. Σ is a horosphere.
At last, it is easily seen that Σ stays inside a horosphere, when m > 0 or m = 0, that is, there is some constant C such that ρ(x 1 , · · · , x n ) ≥ C.
The fact that Σ is supported by some equidistant hypersurface is proved in Lemma 6.1. The proof of Theorem 6.1 is complete.