The area minimizing problem in conformal cones: II

In this paper we continue to study the area minimizing problem with prescribed boundary in a class of conformal cones similar to [14]. A conformal cone in this paper is defined as follows.

Let $\psi(x)$ be a $C^{1}$ function on $\partial\Omega$ and $\Gamma$ be its graph in $\partial\Omega{\mkern-1.0mu\times\mkern-1.0mu}\mathbb{R}$. The area minimizing problem in a conformal cone $Q_{\phi}$ is to find a $n$-integer multiplicity current in $\bar{Q}_{\phi}$, the closure of $Q_{\phi}$, to realize

where $\mathbb{M}$ is the mass of integer multiplicity currents in $M_{\phi}$, and $\mathcal{G}$ denotes the set of n-integer multiplicity currents with compact support in $\bar{Q}_{\phi}$, i.e. for any $T\in\mathcal{G}$, its support $spt(T)$ is contained in $\bar{\Omega}{\mkern-1.0mu\times\mkern-1.0mu}[a,b]$ for some finite numbers $a<b$. See subsection 4.1 for more details.
The main reason to study the conformal product manifold $M_{\phi}$ in Definition 1.1 is that hyperbolic space is a special case of $M_{\phi}$ (see remark 6.5). With this model in [22, Theorem 4.1], Hardt-Lin showed that there is a unique local area minimizing $n$-integer multiplicity current for any prescribed infinity $C^{1}$ star-shaped boundary. Moreover it is a radial minimal graph over $S^{n}_{+}$ in hyperbolic spaces. On the other hand in [25, Section 4.1], Lin described the solution to the area minimizing problem (1.2) with $C^{1}$ graphical boundary in a bounded $C^{2}$ cylinder via a minimizing problem of an area functional of bounded variation (BV) functions. Motivated by Lin’s idea in [14] we studied the area minimizing problem (1.2) in a conformal product manifold $M_{h}=\{N{\mkern-1.0mu\times\mkern-1.0mu}\mathbb{R},h^{2}(r)(\sigma+dr^{2})\}$. Note that there $h(r)$ only depends on $r\in\mathbb{R}$. Based on these preceding results, it is natural to consider the corresponding area minimizing problem (1.2) in conformal cones of $M_{\phi}$. In particular it is desirable to know how many phenomena in hyperbolic sapce does not really rely on the hyperbolic structures. We refer readers to [2] for some history remarks on such kinds of area minimizing problems.
For any open set $W$ we denote the set of all bounded variation functions on $W$ by $BV(W)$. A key concept for our study of the problem (1.2) is an area functional in $BV(W)$ defined as follows.

where $dvol$ and $div$ are the volume form and the divergence of $N$ respectively, $C_{0}(W)$ and $T_{0}(W)$ denote the set of smooth functions and vector fields with compact support in $W$ respectively. Note that when $u\in C^{1}(W)$, $\mathfrak{F}_{\phi}(u,W)$ is the area of the graph of $u(x)$ in $M_{\phi}$.
Let $\Omega$ be the $C^{2}$ domain in Definition 1.1. Suppose $\Omega^{\prime}$ is a $C^{2}$ domain in $N$ such that $\Omega\subset\subset\Omega^{\prime}$. That is, the closure of $\Omega$ is a compact set in $\Omega^{\prime}$. Suppose $\psi(x)\in C^{1}(\Omega^{\prime}\backslash\Omega)$. The following minimizing problem

also plays an important role to solve (1.2).
In fact the key idea to solve (1.2) is to establish the connection among the problem (1.2), the area functional minimizing problem (1.4) and the Dirichlet problem of minimal surface equations in $M_{\phi}$. This can be easily seen when $\Sigma$ is a minimal graph of $u(x)$ in $M_{\phi}$ over $\Omega$ with $C^{1}$ boundary $\psi(x)$ on $\partial\Omega$. From Theorem 2.3 one has (1) $u(x)$ should satisfy

with $u(x)=\psi(x)$ on $\partial\Omega$; (2) the area of $\Sigma$ is less than that of any compact $C^{2}$ surface $S$ with $\partial S$ as the graph of $\psi(x)$ containing in $\bar{\Omega}{\mkern-1.0mu\times\mkern-1.0mu}\mathbb{R}$; (3) $u(x)$ realizes the minimum of (1.4) if requiring all $v(x)\in C^{2}(\Omega)\cap BV(\Omega^{\prime})\cap C(\bar{\Omega})$.
In $M_{\phi}$ a similar connection between the problem (1.2) and the problem (1.4) is obtained in Theorem 4.1. It says that if $u(x)$ is the solution to the problem (1.4), then $T=\partial[[U]]|_{\bar{Q}_{\phi}}$ solves the problem (1.2) in $M_{\phi}$ where $U$ is the subgraph of $u(x)$ and $[[U]]$ is the corresponding integer multiplicity current. This generalizes Lin’s result [25] into conformal cones in $M_{\phi}$. Its proof is based on the following three observations: (1) by Theorem 3.9, $\mathfrak{F}_{\phi}(u,.)$ is just the perimeter of its subgraph $U$; (2) if $T$ solves (1.2) and $\Gamma=(x,\psi(x))$ for $\psi(x)\in C^{1}(\partial\Omega)$, then $T$ is a boundary of a Caccioppoli set (see Lemma 4.6); (3) by Theorem 3.10, for any Caccioppoli set $F$ in $Q\subset M_{\phi}$ with $\partial F\subset\Omega{\mkern-1.0mu\times\mkern-1.0mu}[a,b]$, there is a $u(x)\in BV(\Omega)$ such that $\mathfrak{F}_{\phi}(u,\Omega)$ is less than the perimeter of $F$. Note that the observation (3) is a very general phenomenon in area-type functionals [29], [26], [30], [18],[15] and [7] (see remark 3.11). We refer it as the Miranda’s observation.
A direct application of Theorem 4.1 is the Dirichlet problem of minimal surface equations in $M_{\phi}$. In Theorem 5.3 we show that if $\Omega$ is $\phi$-mean convex, then the Dirichlet problem (1.5) with continuous boundary data has a unique solution in $C^{2}(\Omega)\cap C(\bar{\Omega})$. Here $\Omega$ is $\phi$-mean convex if the mean curvature of $\partial\Omega$ satisfies that $H_{\partial\Omega}+n\langle\vec{\gamma},D\log\phi\rangle\geq 0\text{ on }\partial\Omega$ where $\vec{\gamma}$ is the outward normal vector of $\partial\Omega$ and $H_{\partial\Omega}=div(\vec{\gamma})$. A more general form of the Dirichlet problem (1.5) was already considered by Casteras-Heinonen-Holopainen [8]. To obtain the $C^{0}$ esimate they relied on a lower bound of the Ricci curvature $\Omega$ and the positive mean curvature of $\partial\Omega$ (see remark 5.4) . But out result in Theorem 5.3 is independent of the curvature of $\Omega$ and also can not be obtained by the classical continuous method in [16, Chapter 11, 17.2] (see remark 5.5). A consequence of Theorem 5.3 is that if $\Omega$ is $\phi$-mean convex, the area minimizing integer multiplicity current in (1.2) is unique as the graph of a $C^{2}$ function to solve the Dirichlet problem (see Theorem 5.6).
At last we consider the existence and unqiueness of local area minimizing integer multiplicity current in $M_{\phi}$ with infinity boundary $\Gamma$ when $\phi(x)$ can be written as $\phi(d(x,\partial N))$ which goes to $+\infty$ as $d(x,\partial N)$ goes to zero in $N$. Here $N$ is a compact Riemannian manifold with $C^{2}$ boundary and $d$ is the distance function in $N$. Let $N_{r}$ be the set $\{x\in N:d(x,\partial N)>r\}$. In Theorem 6.4 we obtain that if there is $r_{1}$ such that for any $r\in(0,r_{1})$ $N_{r}$ is $\phi$-mean convex, then for any $\psi(x)\in C(\partial N)$ and $\Gamma=(x,\psi(x))$ there is a unique local area minimizing integer multiplicity current $T$ with infinity boundary $\Gamma$. Moreover $T$ is a minimal graph in $M_{\phi}$ over $N$. This illustrates that Theorem 4.1 in [22] does not depend on the hyperbolic structure (see remark 6.5 ). Unlike the case in hyperbolic spaces our existence in Theorem 6.4 is from the Dirichlet problem of minimal surface equations in Theorem 5.3, not from the results in [5, 6] by Anderson via geometric measure theory.
For the idea of radial graphs in hyperbolic space we refer to [19, 9, 27, 30, 36, 31, 20, 21, 26] etc. For the Dirichlet problem of minimal surface equations in Riemannian manifolds, we refer to [8], [4], [35], [1], [3] and references therein. As for the variational method to study the Dirichlet problem of minimal surface equations, we refer to [18, 17, 7], [41] and references therein.
Our paper is organized as follows. In section 2 we show three properties of $C^{2}$ minimal graphs in $M_{\phi}$ and collect preliminary facts on BV functions. In section 3 we show the $C^{\infty}$ approximation theorem of $\mathfrak{F}_{\phi}(u,\Omega)$ (Theorem 3.8) and the Miranda’s observation (Theorem 3.9 and Theorem 3.10). In section 4 we consider the connection between the problem (1.2) and the problem (1.4) (Theorem 4.1). In section 5 we discuss the Dirichlet problem of minimal surface equations in $M_{\phi}$ on a $\phi$-mean convex domain (Theorem 5.3). As an application we obtain the uniqueness of the problem (1.2) under the $\phi$-mean convex assumption (Theorem 5.6). In section 6 we discuss the existence and uniqueness of local area minimizing $n$-integer multiplicity current with infinity graphical boundary (Theorem 6.4). In Appendix A we give a proof of the interior estimate of minimal surface equations in (1.5) (Theorem A.3) following from an idea of Eichmair [11].
The first author is supported by the National Natural Science Foundation of China, No. 11771456. The second author is supported by the National Natural Science Foundation of China, No. 11801046 and by the Fundamental Research Funds for the Central Universities, China, No.2019CDXYST0015.

Throughout this section we assume $N$ is a manifold with a metric $\sigma$, $\phi(x)>0$ is a positive $C^{3}$ function defined on $N$ and $M_{\phi}$ is the conformal product manifold $N{\mkern-1.0mu\times\mkern-1.0mu}\mathbb{R}$ with the metric $\phi^{2}(x)(g+dr^{2})$. We show three properties of $C^{2}$ minimal graphs in $M_{\phi}$ and collect some results on BV functions in Riemannian manifolds for later use.

In this section we define a new area functional $\mathfrak{F}_{\phi}(u,\Omega)$ for BV functions to generalize the area of the graph of $C^{1}$ functions in $M_{\phi}$ (see (2.5)). Suppose $N$ is a fixed Riemannian manifold with metric $\sigma$ and $\phi(x)$ is a $C^{3}$ positive function on $N$. Here $M_{\phi}$ is the set $N{\mkern-1.0mu\times\mkern-1.0mu}\mathbb{R}$ equipped with $\phi^{2}(x)(\sigma+dr^{2})$. We shall establish the $C^{\infty}$ approximation of $\mathfrak{F}_{\phi}(u,\Omega)$ and the Miranda’s observation mentioned in the introduction.
Recall that $\Omega$ is a bounded domain in $N$, $T_{0}(\Omega)$ and $C_{0}(\Omega)$ denote the set of all smooth vector fields and smooth functions with compact supports in $\Omega$ respectively. Let $dvol$ denote the volume form of $N$.