¹

1 Instruction↩︎

In this paper, we mainly concern geometric characterizations of uniform/inner uniform domains in metric spaces. Recall that

If the condition [con2] in Definition 1 is replaced by the weaker inequality \[\label{eq:def32for32inner32quasiconvexity} \ell_d(\gamma)\leq c\,\sigma_\Omega( z_{1}, z_{2}),\tag{1}\] where \(\sigma_\Omega\) is the inner distance in \(\Omega\) defined by \[\sigma_\Omega(z_1,z_2)=\inf \{\ell(\alpha):\; \alpha\subset \Omega\; is a rectifiable curve joining\; z_1\; and\; z_2 \},\] then \(\Omega\) is said to be \(c\)-inner uniform and the corresponding curve \(\gamma\) is called a \(c\)-inner uniform curve. When the context is clear, we often drop the subscript \(\Omega\) from \(\sigma_{\Omega}\) and simply write \(\sigma\).

If \(\Omega\) only satisfies the condition [con1] in Definition 1, then it is said to be a \(c\)-John domain, and the corresponding curve \(\gamma\) is called a \(c\)-John curve or a double \(c\)-cone curve. It is immediate from the definition that \[\{c\text{-uniform domains}\} \subset \{c\text{-inner uniform domains}\}\subset \{c\text{-John domains}\}.\] It is not difficult to construct examples to see that the above inclusions are strict.

The class of John domains in \(\mathbb{R}^n\) was initially introduced by F. John in his study of elasticity [1] and the name was coined by Martio and Sarvas in [2], where they also introduced the class of uniform domains. These classes of domains are central in modern geometric function theory and have wide connections with many other mathematical subjects related to analysis and geometry; see for instance [3]–[14].

In a seminal work [15], Bonk-Heinonen-Koskela established a rich uniformization theory for Gromov hyperbolic spaces, using the class of uniform domains as model domains: \[\{\text{some class of Gromov hyperbolic spaces}\}=\mathcal{F}\left(\{\text{bounded locally compact uniform spaces}\}\right),\] where \(\mathcal{F}\) is a class of good homeomorphisms (i.e. quasiisometries). To introduce the class of Gromov hyperbolic domains, we first recall the notion of Gromov hyperbolic spaces introduced by M. Gromov in his celebrated work [16].

The Gromov hyperbolicity is a large-scale property, which generalizes the metric properties of classical hyperbolic geometry and of trees, and it turns out to be very useful in geometric group theory and metric geometry [17]–[20]. For more about the Gromov hyperbolicity and its connection to geometric function theory, see for instance [15], [21]–[25].

Next, we recall the definition of quasihyperbolic metric, which was initially introduced by Gehring and Palka [5] for domains in \({\mathbb{R}}^n\), and then has been extensively studied in [26]. The quasihyperbolic length of a rectifiable arc \(\gamma\) in a proper domain \(\Omega\subsetneq (X,d)\) is defined as \[\ell_{k}(\gamma):=\int_{\gamma}\frac{|dz|}{d_\Omega(z)}.\] For any \(z_1\), \(z_2\) in \(\Omega\), the quasihyperbolic distance \(k_\Omega(z_1,z_2)\) between \(z_1\) and \(z_2\) is set to be \[k_\Omega(z_1,z_2)=\inf_{\gamma}\{\ell_k(\gamma)\},\] where the infimum is taken over all rectifiable arcs \(\gamma\) joining \(z_1\) and \(z_2\) in \(\Omega\). An arc \(\gamma\) from \(z_1\) to \(z_2\) is called a quasihyperbolic geodesic if \(\ell_k(\gamma)=k_\Omega(z_1,z_2)\). Clearly, each subarc of a quasihyperbolic geodesic is a quasihyperbolic geodesic.

In their second main result [15], Bonk, Heinonen and Koskela obtained the following useful characterization of uniformity/inner uniformity. To record their result, recall that

With these definitions at hand, the result of Bonk-Heinonen-Koskela can be stated as follows.

By a seminal result of Balogh-Buckley [22], the class of Gromov hyperbolic domains in \(\mathbb{R}^n\) can be characterized by the Gehring-Hayman inequality and the ball separation condition. Very recently, Guo-Huang-Wang [27] proved that Gromov hyperbolic domains in \(Q\)-doubling length spaces (and thus particularly in \(\mathbb{R}^n\)) are indeed completely characterized by the geometric ball separation condition. Consequently, Theorem 1 holds if the \(\delta\)-Gromov hyperbolicity gets replaced by the ball separation condition. Recall that

A simple volume comparison implies that \({\mathbb{R}}^n\) (equipped with the standard Euclidean distance) is \(2^n\)-doubling, according to Definition 6.

A central question asked in [28] is to characterize quasihyperbolic John domains in metric spaces. Recall that

Addressing the above central question, it is important to study the geometry of quasihyperbolic geodesics, as it will be the natural candidate of uniform/inner uniform curves [7], [15], [26], [29]. In [28], the authors proved that if \(\Omega\) is \(\delta\)-Gromov hyperbolic and \(c_0\)-John, then it is quasihyperbolic \(c\)-John, quantitatively. In a recent work, as an application of a deep improved geometric characterization of Gromov hyperbolicity, the authors proved the following geometric characterization of quasihyperbolic John domains and inner uniform domains, which in particular improves the second assertion in Theorem 1.

Note that the assumption of \((\Omega,k)\) being geodesic is quite natural as it is part of the definition of a quasihyperbolic John domain and also the definition of the ball separation condition. If \(X\) is a locally compact length space, then the identity map \(id:(\Omega,d)\to (\Omega,\sigma_{\Omega})\) is a homeomorphism, and so, by [15], \((\Omega,k)\) is geodesic. This property remains valid for nice domains in certain infinite dimensional spaces as well; see for instance the work of Martio and Väisälä [30]. Theorem 2 substantially improves [28] above, as it gives a complete characterization of quasihyperbolic John property in terms of inner uniformity or in terms of two geometric condtions (LLC-2 and the ball separation condition). It also provides a concrete answer to the above central question of Zhou and Ponnusamy in the setting of metric doubling spaces.

As an application of [28], the authors obtained in [28] the following characterization of uniformity in Ahlfors regular metric measure spaces: Let \(\Omega\) be minimally nice and Ahlfors \(Q\)-regular. Then \(\Omega\) is uniform if and only if it is quasiconvex, John and roughly starlike Gromov hyperbolic. We shall not give precise definitions of minimally nice spaces or Ahlfors \(Q\)-regular spaces/measures, but refers to [28] for details. Here, we would like to point out that the requirement of an Ahlfors regular measure is not natural as all the concepts involved are purely metric. It is thus natural to ask

Note that one can not directly extend the proof of Theorem 2 to uniform domains, as there is no obvious way to verify the the quasiconvexity condition (2) in Definition 1. New ideas are necessary to find a satisfied solution of Question 3.

In this paper, we shall develop an elementary approach to prove the following characterization of uniformity, akin to Theorem 2, which addresses Question 3 above.

By [31], there is an LLC/LLC-2 Gromov hyperbolic domain \(\Omega\) in an infinite dimensional Banach space, which fails to be uniform/inner uniform. This suggests that Theorems 8 and 2 would probably fail in infinite dimensions. On the other hand, using the new techniques developed in [32], Guo-Huang-Li-Wang [33] did prove that a domain \(\Omega\) in a general Banach space \(E\) is \(c\)-inner uniform if it is \(\delta\)-Gromov hyperbolic and \(c_0\)-John, quantitatively.

Comparing with [28], our assumptions in Theorem 8 are much weaker, in particular, we do not require a measure. On the other hand, our conclusions are indeed stronger: LLC and ball separation condition are sufficient to achieve uniformity, while in [28], it requires quasiconvexity, John and roughly starlike Gromov hyperbolicity.

The implication “(2) \(\Rightarrow\) (3)" in Theorem 8 has already been proved in [27] (see Lemma 12 below). The main step for the proof of Theorem 8 (or Theorem 2) is thus the reverse implication”(3) \(\Rightarrow\) (1)", as the implication “(1) \(\Rightarrow\) (2)" is rather direct from definition. The main step for this implication lies in Lemma 13, which gives length comparison for quasihyperbolic geodesics with different end points. A version of this result was proved earlier in [32] for Gromov hyperbolic domains in Banach spaces. One can check the proof there does not work in our case. The novelty here is that we found a smart construction of points on a given quasihyperbolic geodesic so that one can obtain effective estimates on the quasihyperblic distance between consecutive points, using merely the metric doubling condition. A similar idea is used in the final proof as well. As a contrast, we would like to point out that all the earlier known proof for such a characterization (as in [15], [28]) relies on the uniformization theory of Bonk-Heinonen-Koskela and thus it requires an Ahlfor regular measure (as well as some other technical assumptions).

Structure. The structure of this paper is as follows. Section 2 contains some elementary results that are necessary for later proofs. In Section 3, we prove our main theorems. As some elementary results from [27] are used in our proofs, we add an appendix to include the proofs of these auxiliary results so that it is self-contained and easier for the readers to understand the complete proof.

Notations. Throughout this paper, \(X\) is always assumed to be a metric space. For a given metric \(d\) on \(X\), \(x\in X\) and \(r>0\), we define \[B_d(x,r):=\{y\in X: d(x,y)<r\}.\] For each proper subdomain \(\Omega\subset X\), we use \(\Lambda_{xy}(\Omega)\) to represent the set of all quasihyperbolic geodesics in \(\Omega\) with end points \(x\) and \(y\), and use \(\gamma_{xy}\) to denote some quasihyperbolic geodesic in \(\Lambda_{xy}(\Omega)\). Meanwhile, we use \(\Gamma_{xy}(\Omega)\) to represent the set of all curves in \(\Omega\) with end points \(x\) and \(y\).

2 Some elementary estimates↩︎

For any \(z_1\), \(z_2\) in \(\Omega\), let \(\gamma\subset \Omega\) be an arc with end points \(z_1\) and \(z_2\). Then we have the following elementary estimates (see for instance [34]) \[\begin{align} \label{40246141} \ell_{k}(\gamma)\geq \log\Big(1+\frac{\ell(\gamma)}{\min\{d_\Omega(z_1), d_\Omega(z_2)\}}\Big) \end{align}\tag{3}\] and \[\begin{align} \label{40246241} \begin{aligned} k_{\Omega}(z_1, z_2) &\geq \log\Big(1+\frac{\sigma_\Omega(z_1,z_2)}{\min\{d_\Omega(z_1), d_\Omega(z_2)\}}\Big) \\ &\geq \log\Big(1+\frac{|z_1-z_2|}{\min\{d_\Omega(z_1), d_\Omega(z_2)\}}\Big) \geq \Big|\log \frac{d_\Omega(z_2)}{d_\Omega(z_1)}\Big|. \end{aligned} \end{align}\tag{4}\]

The follow two elementary lemmas were proved in [27]. For convenience, we also include the proofs in the appendix.

We shall repeatedly use the following elementary estimate in \(Q\)-doubling metric spaces.

3 Proofs of the main results↩︎

In this section, we shall prove Theorem 8 and Theorem 2. It is clear that for Theorem 8, it suffices to show that

• For \(x,y \in \Omega\), \(\gamma_{xy}\in \Lambda_{xy}(\Omega)\), it holds \[\label{H25-0520-1}\ell(\gamma_{xy})\leq e^{19\kappa_0\kappa_1\kappa_2}d(x,y).\tag{5}\]

• For each \(z\in\gamma_{xy}\), it holds \[\label{H25-0520-2}\min\left\{\ell(\gamma_{xy}[x,z]),\ell(\gamma_{xy}[y,z])\right\}\leq \kappa_0 d_{\Omega}(z),\tag{6}\] where \(\kappa_0=(28ec_0c_1)^2([Q^{2\log_2 70ec_0c_1}]+1)\), \(\kappa_1=\kappa_0([Q^{\log_2 15\kappa_0c_0}]+1)\) and \(\kappa_2=\kappa_1^{4\kappa_1^2+1}\).

Then it follows from i) and ii) that for each \(x,y\in \Omega\), each \(\gamma_{xy}\in \Lambda_{xy}(\Omega)\) is a \(C\)-uniform curve and in particular \(\Omega\) is \(C\)-uniform with \(C=e^{19\kappa_0\kappa_1\kappa_2}\).

The quasihyperbolic John condition ii) as above has been proved in [27]. For the convenience of readers, we shall repeat its proof in the appendix.

3.1 A technical lemma↩︎

For the proof of statement i), we need the following technical lemma. A version of this lemma was proved in [32] for Gromov hyperbolic domains (in Banach spaces).

Proof. . It follows from Lemma 10 and the assumption “\(\min\{k_{\Omega}(x,y), k_{\Omega}(x,z)\}\geq 1\)" that \[\label{H25-0525-1} d(x,y)\geq \frac{9}{19}\max\{d_{\Omega}(x),d_{\Omega}(y)\}\; and \; d(x,z)\geq \frac{9}{19}\max\{d_{\Omega}(x),d_{\Omega}(z)\}.\tag{7}\]

Without loss of generality, we may assume that \[\min\{\ell(\gamma_{xy}),\ell(\gamma_{xz})\}=\ell(\gamma_{xy}).\] Note that by 4 , we have \[\max\left\{\log\Big(1+\frac{\sigma(y,z)}{\min\{d_{\Omega}(y),d_{\Omega}(z)\}}\Big),\;\log \frac{\max\{d_{\Omega}(y),d_{\Omega}(z)\}}{\min\{d_{\Omega}(y),d_{\Omega}(z)\}}\right\}\leq k_{\Omega}(y,z)\leq 1.\] It follows that \[\label{H25-0514-1}\sigma(y,z)\leq (e-1)\min\{d_{\Omega}(y),d_{\Omega}(z)\}\tag{8}\] and \[\label{H25-0514-2} \max\{d_{\Omega}(y),d_{\Omega}(z)\}\leq e \min\{d_{\Omega}(y),d_{\Omega}(z)\}.\tag{9}\]

Take \(y_0\in\gamma_{xy}\) and \(z_0\in\gamma_{xz}\) such that \[\label{H25-0514-3}\ell(\gamma_{xy}[x,y_0])=\frac{1}{2}\ell(\gamma_{xy})\; and \;\ell(\gamma_{xz}[z,z_0])=\frac{1}{2}\ell(\gamma_{xz}).\tag{10}\] We consider the following two cases.

Case I: \(\ell(\gamma_{xy})\leq e^{\kappa_2}\min\{d_{\Omega}(y),d_{\Omega}(x)\}\).

In this case, we have \[\label{eq:lemma326462} \log \Big(1+\frac{\ell(\gamma_{xz})}{\min\{d_{\Omega}(z),d_{\Omega}(x)\}}\Big)\stackrel{\eqref{40246141}}{\leq} k_{\Omega}(x,z)\leq k_{\Omega}(x,y)+k(y,z)\leq k_{\Omega}(x,y_0)+k_{\Omega}(y,y_0)+1.\tag{11}\] Note that by Lemma 12 and the choice of \(y_0\), it holds \[\ell(\gamma_{xy}[x,y_0])\leq \kappa_0 d_{\Omega}(y_0) \quad \text{and}\quad \ell(\gamma_{xy}[y,y_0])\leq \kappa_0 d_{\Omega}(y_0).\] Thus it follows from Lemma 9 that \[k_{\Omega}(x,y_0)\leq 4\kappa_0\log \Big(1+\frac{\ell(\gamma_{xy}[x,y_0])}{d_{\Omega}(x)}\Big)\] and \[k_{\Omega}(y,y_0)\leq 4\kappa_0\log \Big(1+\frac{\ell(\gamma_{xy}[y,y_0])}{d_{\Omega}(y)}\Big).\] Substituting these estimates in 11 , we obtain that \[\begin{align} \begin{aligned} \log \Big(1+\frac{\ell(\gamma_{xz})}{\min\{d_{\Omega}(z),d_{\Omega}(x)\}}\Big) &\leq 4\kappa_0\left(\log \Big(1+\frac{\ell(\gamma_{xy}[x,y_0])}{d_{\Omega}(x)}\Big)+\log \Big(1+\frac{\ell(\gamma_{xy}[y,y_0])}{d_{\Omega}(y)}\Big)\right)+1 \\ &< 8\kappa_0\kappa_2 \stackrel{\eqref{H25-0525-1}}{<} \frac{20\kappa_0\kappa_2\ell(\gamma_{xy})}{\min\{d_{\Omega}(x),d_{\Omega}(y)\}} \stackrel{\eqref{H25-0514-2}}{\le}\frac{20e\kappa_0\kappa_2\ell(\gamma_{xy})}{\min\{d_{\Omega}(x),d_{\Omega}(z)\}}. \end{aligned} \end{align}\] Since \(\log(1+t)\geq a_1t\) for \(t\leq a_1^{-1}-1\) and \(a_1>0\), it follows from the above inequality that \[{ \frac{e^{-8\kappa_0\kappa_2}\ell(\gamma_{xz})}{\min\{d_{\Omega}(x),d_{\Omega}(z)\}}\leq \log \Big(1+\frac{\ell(\gamma_{xz})}{\min\{d_{\Omega}(z),d_{\Omega}(x)\}}\Big)}\leq 20e\kappa_0\kappa_2\frac{\ell(\gamma_{xy})}{\min\{d_{\Omega}(x),d_{\Omega}(z)\}}\] and so \[\label{H25-0515-2} \ell(\gamma_{xz})\leq 20\kappa_0\kappa_2e^{1+8\kappa_0\kappa_2}\ell(\gamma_{xy})\leq e^{9\kappa_0\kappa_2}\min\{\ell(\gamma_{xy}),\ell(\gamma_{xz})\}.\tag{12}\] This completes the proof for this case.

Case II: \(\ell(\gamma_{xy})> e^{\kappa_2}\min\{d_{\Omega}(x),d_{\Omega}(y)\}\).

In this case, select \(m_1\geq[\kappa_2]-1\) such that \[e^{m_1+1}d_{\Omega}(y)<{\ell(\gamma_{xy})}\leq e^{m_1+2}d_{\Omega}(y).\] No loss of generality, by (9 ), we may assume that \[\min\{d_{\Omega}(x),d_{\Omega}(y)\}=d_{\Omega}(y),\] since the proof for the other case is similar.

Set \(y_1=y\) and \(z_1=z\). Choose \(y_{m_1+1} \in\gamma_{xy_1}[y_1,y_0]\), \(y_{1,m_1+1} \in\gamma_{xy_1}[x,y_0]\), \(z_{m_1+1} \in \gamma_{xz_1}[z_1,z_0]\) and \(z_{1,m_1+1} \in \gamma_{xz_1}[x,z_0]\) such that \[\begin{align} \label{5-14-1} \begin{aligned} \ell (\gamma_{xy_1}[y_1,y_{m_1+1}]) &=\ell (\gamma_{xz_1}[z_1,z_{m_1+1}])=\ell (\gamma_{xy_1}[x,y_{1,m_1+1}])=\ell (\gamma_{xz_1}[x,z_{1,m_1+1}])\\ &=e^{m_1}d_{\Omega}(y_1). \end{aligned} \end{align}\tag{13}\]

Next, we claim that

Suppose on the contrary that \[\max \{k_{\Omega}(y_{m_1+1},z_{m_1+1}), k_{\Omega}(y_{1,m_1+1},z_{1,m_1+1})\} > 2\kappa_2.\] Without loss of generality, we may assume that \[\begin{align} \label{5-14-3} k_{\Omega}(y_{m_1+1},z_{m_1+1}) >2\kappa_2, \end{align}\tag{14}\] as the proof for the other case is similar.

For each \(i \in \{2,\cdots,m_1+1\}\), let \(y_i \in \gamma_{xy_1}[y_1 ,y_0 ]\) and \(z_i \in \gamma_{xz_1}[z_1 ,z_0 ]\) be such that \[\begin{align} \label{5-14-4} \ell (\gamma_{xy_1}[y_1 ,y_i ]) =\ell (\gamma_{xz_1}[z_1 ,z_i ])=e^{i-1} d_{\Omega}(y_{1}). \end{align}\tag{15}\] Then, for each \(i \in \{1,\cdots,m_1 \}\), we have \[\ell(\gamma_{xy_1}[y_{i+1},y_i])=\ell(\gamma_{xy_1}[y_1,y_{i+1}])-\ell(\gamma_{xy_1}[y_1,y_{i}])\stackrel{\eqref{5-14-4}}{=}(e-1)e^{i-1}d_{\Omega}(y_1).\] By Lemma 12 and our choice of \(y_{i+1}\) in 15 , it holds \[\label{H25-0527-1} \ell(\gamma_{xy_1}[y_1,y_{i+1}])=e^{i} d_{\Omega}(y_{1})\leq \kappa_0 d_{\Omega}(y_{i+1}).\tag{16}\] Thus, combining the above two estimates gives \[\ell(\gamma_{xy_1}[y_{i+1}, y_i])\leq (e-1)\kappa_0 d_{\Omega}(y_i)\] and \[\ell(\gamma_{xy_1}[y_{i+1}, y_i])\leq\frac{e-1}{e}\kappa_0 d_{\Omega}(y_{i+1}).\] For each \(i \in \{1,\cdots,m_1 \}\), it follows from the above inequality and Lemma 9 that \[\begin{align} \label{5-14-7} k_{\Omega}(y_{i},y_{i+1})\leq 4\kappa_0 \log\Big(1+\frac{\ell (\gamma_{xy_1}[y_i,y_{i+1}])}{d_{\Omega}(y_{i})}\Big)\leq 4\kappa_0 \log(1+e{\kappa_0}). \end{align}\tag{17}\] Similarly, one can prove that \[\label{H-25-0515-7} k_{\Omega}(z_{i},z_{i+1})\leq 4\kappa_0 \log(1+e{\kappa_0}).\tag{18}\]

Based on 14 , we take \(t\in\{2,\cdots, {m_1+1}\}\) to be the smallest integer such that \[k_{\Omega}(y_t,z_t)\geq \kappa_2.\] Then, it follows from 17 and 18 that \[\kappa_2\leq k_{\Omega}(y_t,z_t)\leq \sum_{i=1}^{t-1}\left(k_{\Omega}(y_i,y_{i+1})+k_{\Omega}(z_i,z_{i+1})\right)+k_{\Omega}(y,z)\leq8(t-1)\kappa_0\log(1+e{\kappa_0})+1.\] Thus, we obtain \[\label{H25-0519-1} t> \frac{\kappa_2}{8\kappa_0\log(1+e{\kappa_0})}.\tag{19}\]

Now, let \(s_1=t\), and based on 19 , for each \(i\in\{2,\cdots,[\kappa_1^2]\}\), set \(s_i=s_{i-1}-[\kappa_1^{2i}]\). For each \(i\in\{2,\cdots, [\kappa_1^2]\}\), by our choices of \(\kappa_0\), \(\kappa_1\) and \(\kappa_2\), we have \[\ell(\gamma_{xy_1}[y_1,y_{s_i}])\geq d_{\Omega}(y_{s_i})-d_{\Omega}(y_1)\stackrel{\eqref{H25-0527-1}}{\geq} (1-\kappa_0e^{1-s_i})d_{\Omega}(y_{s_i})>\frac{1}{2}d_{\Omega}(y_{s_i}).\] Then \[\label{H25-0519-0} {\frac{1}{2}e^{[\kappa_1^{2i}]}d_{\Omega}(y_{s_i})\leq e^{[\kappa_1^{2i}]}\ell(\gamma_{xy_1}[y_1,y_{s_i}])}\stackrel{\eqref{5-14-4}}{=}\ell(\gamma_{xy_1}[y_1,y_{s_{i-1}}])\stackrel{\text{Lemma }\ref{Theorem-1468}}{\leq} \kappa_0d_{\Omega}(y_{s_{i-1}})\tag{20}\] and \[d_{\Omega}(y_{s_{i-1}})\leq d_{\Omega}(y_1)+\ell(\gamma_{xy_1}[y_1,y_{s_{i-1}}])\stackrel{\eqref{5-14-4}}{<}2e^{[\kappa_1^{2i}]}\ell(\gamma_{xy_1}[y_1,y_{s_i}])\stackrel{\text{Lemma }\ref{Theorem-1468}}{\leq} 2\kappa_0e^{[\kappa_1^{2i}]}d_{\Omega}(y_{s_i}).\] It follows from these two estimates that \[\label{H25-0519-2} \frac{1}{2\kappa_0}e^{[\kappa_1^{2i}]}d_{\Omega}(y_{s_i})\leq d_{\Omega}(y_{s_{i-1}})\leq 2\kappa_0e^{[\kappa_1^{2i}]}d_{\Omega}(y_{s_i}).\tag{21}\] Similarly, one can prove \[\label{H25-0519-3} \frac{1}{2}d_{\Omega}(y_{s_i})\leq\ell(\gamma_{xy_1}[y_1,y_{s_i}])=\ell(\gamma_{xz_1}[z_1,z_{s_i}])\leq \kappa_0d_{\Omega}(z_{s_i}).\tag{22}\]

For each \(i\in\{2,\cdots, [\kappa_1^2]\}\) and \(\gamma_{y_{s_i}z_{s_i}}\in \Lambda_{y_{s_i}z_{s_i}}\), let \(x_{0,i}\) bisect the length of \(\gamma_{y_{s_i}z_{s_i}}\). Then we have \[\begin{align} k_{\Omega}(y_{s_i}, z_{s_i})&\geq k_{\Omega}(y_t,z_t)-k_{\Omega}(y_t,y_{s_i})-k_{\Omega}(z_t,z_{s_i})\\ &\geq k_{\Omega}(y_t,z_t)-\sum_{l=s_i}^{t-1}\left(k_{\Omega}(y_{l+1},y_l)+k_{\Omega}(z_{l+1},z_l)\right)\\ &\stackrel{\eqref{5-14-7}+\eqref{H-25-0515-7}}{\geq} \kappa_2-8\kappa_0{(t-s_i-1)}\log(1+e\kappa_0) \\ &> \kappa_2-\kappa_1^{2[\kappa_1^2]+2}, \end{align}\] where in the last inequality, we used the fact that \(t-s_i=\sum\limits_{l=2}^i [\kappa_1^{2l}]<2\kappa_1^{2i}\).

On the other hand, it follows from Lemmas 9 and 12, and 22 that \[\begin{align} \begin{aligned} k_{\Omega}(y_{s_i}, z_{s_i})&= k_{\Omega}(y_{s_i}, x_{0,i})+k_{\Omega}(x_{0,i}, z_{s_i})\\ &\leq 4\kappa_0\left(\log \left(1+\frac{\ell(\gamma_{y_{s_i}z_{s_i}})}{2d_{\Omega}(y_{s_i})}\right)+\log \left(1+\frac{\ell(\gamma_{y_{s_i}z_{s_i}})}{2d_{\Omega}(z_{s_i})}\right)\right) \\ &\leq 4\kappa_0\left(\log (1+\frac{\ell(\gamma_{y_{s_i}z_{s_i}})}{2d_{\Omega}(y_{s_i})})+\log \left(1+\frac{\kappa_0\ell(\gamma_{y_{s_i}z_{s_i}})}{d_{\Omega}(y_{s_i})}\right)\right). \end{aligned} \end{align}\] Combining the above two estimates with Lemma 12 gives \[\label{H25-0519-4} \ell(\gamma_{y_{s_i}z_{s_i}})>e^{\frac{\kappa_2}{9\kappa_0}} d_{\Omega}(y_{s_i})\geq \frac{1}{\kappa_0}e^{\frac{\kappa_2}{9\kappa_0}}\ell(\gamma_{xy_1}[y_1,y_{s_i}])\geq \frac{1}{\kappa_0}e^{\frac{\kappa_2}{9\kappa_0}-{\color{blue}2}\kappa_1^{2i}}\ell(\gamma_{xy_1}[y_1,y_{s_1}]).\tag{23}\]

For each \(i\in\{2,\cdots, [\kappa_1^2]\}\), based on 23 , we may select \(y_{1,i}\in\gamma_{y_{s_i}z_{s_i}}[y_{s_i},x_{0,i}]\) and \(z_{1,i}\in\gamma_{y_{s_i}z_{s_i}}[z_{s_i},x_{0,i}]\) so that \[\label{H25-0519-5} \ell(\gamma_{y_{s_i}z_{s_i}}[y_{s_i},y_{1,i}])=\ell(\gamma_{y_{s_i}z_{s_i}}[z_{s_i},z_{1,i}])=\ell(\gamma_{xy_1}[y_1,y_{s_1}]).\tag{24}\] Note that \[\label{H25-0519-7} k_{\Omega}(z_{s_1}, z_{s_{i-1}})\leq \sum_{l=s_{i-1}}^{s_1-1}k_{\Omega}(z_{l+1}, z_{l})\stackrel{\eqref{H-25-0515-7}}{\leq}8\kappa_0\kappa_1^{2(i-1)}\log(1+e{\kappa_0})\tag{25}\] and \[\label{H25-0519-7-2} k_{\Omega}(y_{s_1}, y_{s_{i-1}})\leq \sum_{l=s_{i-1}}^{s_1-1}k_{\Omega}(y_{l+1}, y_{l})\stackrel{\eqref{5-14-7}}{\leq}8\kappa_0\kappa_1^{2(i-1)}\log(1+e{\kappa_0})\tag{26}\] Thus, we obtain from Lemmas 9 and 12, 16 and 24 that \[\begin{align} k_{\Omega}(y_{s_{i-1}}, y_{1,i-1})&\leq 4\kappa_0\log \Big(1+\frac{\ell(\gamma_{y_{s_{i-1}}z_{s_{i-1}}}[y_{s_{i-1}},y_{1,i-1}])}{d_{\Omega}(y_{s_{i-1}})} \Big) \\ &= 4\kappa_0\log \Big(1+\frac{\ell(\gamma_{xy_1}[y_1,y_{s_1}])}{d_{\Omega}(y_{s_{i-1}})}\Big)< 12\kappa_0\kappa_1^{2(i-1)}. \end{align}\] On the other hand, \[\begin{align}k_{\Omega}(y_{s_i}, y_{1,i})&\stackrel{\eqref{40246141}}\geq \log \frac{\ell(\gamma_{y_{s_i}z_{s_i}}[y_{s_i},y_{1,i}])}{d_{\Omega}(y_{s_i})}\stackrel{\eqref{H25-0519-3}+\eqref{H25-0519-5}}\geq \log \frac{\ell(\gamma_{xy_1}[y_1,y_{s_1}])}{2\ell(\gamma_{xy_1}[y_1,y_{s_i}])}\\&\stackrel{\eqref{5-14-4}}>\kappa_1^{2i}. \end{align}\] The above two inequalities yield that \[\label{H25-0519-18}k_{\Omega}(y_{s_i}, y_{1,i})>k_{\Omega}(y_{s_{i-1}}, y_{1,i-1})+(\kappa_1-12\kappa_0)\kappa_1^{2i-1}.\tag{27}\] A similar discussion as in (27 ) shows that \[\label{H25-0519-19}k_{\Omega}(z_{s_i}, z_{1,i})>k_{\Omega}(z_{s_{i-1}}, z_{1,i-1})+(\kappa_1-12\kappa_0)\kappa_1^{2i-1}.\tag{28}\]

Let \(B_0=B(y_1, 3\ell(\gamma_{xy_1}[y_1,y_{s_1}]))\). For each \(t \in \{2, \dots, [\kappa_1^2]\}\), we take \[B_{t}^1=B\left(y_{1,t}, \frac{1}{2\kappa_0}\ell(\gamma_{xy_1}[y_1,y_{s_1}])\right)\;\quadand\;\quad B_{1,t}^1=B\left(z_{1,t}, \frac{1}{2\kappa_0}\ell(\gamma_{xy_1}[y_1,y_{s_1}])\right).\] For each \(u \in (\overline{B_t^1}\cup \overline{B_{1,t}^1})\), by (24 ), \[d(y_1, u) < 3\ell(\gamma_{xy_1}[y_1,y_{s_1}]),\] and so \[(\overline{B_t^1}\cup \overline{B_{1,t}^1})\subset B_0.\]

Next, we observe that there exist \(m=[Q^{\log_2 6\kappa_0}]+1\) balls \(B_{1_j}^1\) in \(\bigcup\limits_{t=1}^{[\kappa_1^2]}\{B_t^1\}\) such that for each \(i\not=j\in\{2,\cdots,m\}\), it holds \[\label{H25-10-15-1} B_{1_i}^1\cap B_{1_j}^1\not=\emptyset.\tag{29}\] To get 29 we come to prove the following claim.

We will prove this claim by a contradiction. Suppose on the contrary that for each \(s\in\{t_1,\cdots,t_2\}\), there exist at most \(s_0\leq [\frac{t_2-t_1}{m^2}]\) balls \(B_{p_j}^1\) such that for each \(j\in\{1,\cdots,s_0\}\), \[\label{H25-12-21-3}p_j>s\,and\, B_{p_j}^1\cap B_{s}^1\not=\emptyset.\tag{30}\] Then (30 ) implies that there exist at least \([(1-\frac{1}{m^2})(t_2-t_1)]\) balls \(B_{t_{1,j}}^1\) which are disjoint with \(B_{t_1}^1\). We take \(B_{t_{0,1}}^1=B_{t_1}^1\). Similarly, by (30 ), there exist at least \([(1-\frac{2}{m^2})(t_2-t_1)]\) balls \(B_{t_{2,j}}^1\subset \cup_{j=1}^{[(1-\frac{1}{m})(t_2-t_1)]}B_{t_{1,j}}^1\) which are disjoint with \(B_{t_{1,1}}^1\). Repeating the procedure \(m\) times we shall get \(m\) balls \(B_{t_{i,1}}^1\) which are disjoint from each other, which contradicts with Lemma 11. Hence, Claim 2 holds.

For each \(i\in\{1,\cdots, [\kappa_1^2]\}\), it follows from Claim 2 that there exist some \(1_1\in\{1,\cdots,[\kappa_1^2]\}\) such that there are \(n_1\geq [\frac{[\kappa_1^2]}{m^2}]\) balls \(B_{q_{1,j}}^1\) which intersect with \(B_{1_1}^1\). Similarly, there exist some \(B_{1_2}^1\subset \cup_{j=1}^{n_2}B_{q_{1,j}}^1\) such that there are at least \(n_2\geq [\frac{n_1}{m^2}]\) balls \(B_{q_{2,j}}^1\) which intersect with \(B_{1_2}^1\). Repeating the procedure \(m\) times we shall get 29 .

It follows from Lemma 11 and 29 that there must exist \(p<q\in\{2,\cdots, [\kappa_1^2]\}\) such that \[\label{H25-10-15-2} B(y_{1,p}, r_2)\cap B(y_{1,q}, r_2)\not=\emptyset\quad\;and\;B(z_{1,p}, r_2)\cap B(z_{1,q}, r_2)\not=\emptyset.\tag{31}\]

Let \(y_{1,p}^1\in [y_{1,p},y_{1,q}]\cap (B_{p}^1\cap B_q^1)\) and \(z_{1,q}^1\in [z_{1,p},z_{1,q}]\cap (B_{1,p}^1\cap B_{1,q}^1)\). Then, based on Lemma 12, 24 , and 31 , we have \[d(y_{1,p}^1,y_{1,p})\leq\frac{1}{2\kappa_0}\ell(\gamma_{xy_1}[y_1,y_{s_1}])=\frac{1}{2\kappa_0}\ell(\gamma_{y_{s_p}z_{s_p}}[y_{s_p},y_{1,p}])\leq\frac{1}{2}d_{\Omega}(y_{1,p}).\] Similarly, \[d(y_{1,p}^1,y_{1,q})\leq\frac{1}{2}d_{\Omega}(y_{1,q}),\quad\; d(z_{1,p}^1,z_{1,p})\leq\frac{1}{2}d_{\Omega}(z_{1,p})\quad\;and\quad\;d(z_{1,p}^1,z_{1,q})\leq\frac{1}{2}d_{\Omega}(z_{1,q}).\] Thus we obtain from Lemma 10 that \[\begin{align} &\quad\max\{k_{\Omega}(y_{1,p},y_{1,q}), k_{\Omega}(z_{1,p},z_{1,q})\}\\ &\leq \max\{k_{\Omega}(y_{1,p},y_{1,p}^1)+k_{\Omega}(y_{1,q},y_{1,p}^1), k_{\Omega}(z_{1,p},z_{1,p}^1)+k_{\Omega}(z_{1,q},z_{1,p}^1)\}\leq\frac{20}{9}, \end{align}\] which, together with 27 and 28 , yields \[\begin{align} k_{\Omega}(y_{s_q}, z_{s_q})\nonumber&=&k_{\Omega}(y_{s_q}, y_{1,q})+k_{\Omega}(y_{1,q}, z_{1,q})+k_{\Omega}(z_{1,q}, z_{s_q}) \\ \nonumber&>&k_{\Omega}(y_{s_p}, y_{1,p})+k_{\Omega}(y_{1,q}, z_{1,q})+k_{\Omega}(z_{1,p}, z_{s_p})+2(\kappa_1-12\kappa_0)\kappa_1^{2q-1} \\ \nonumber&\geq&k_{\Omega}(y_{s_p}, y_{1,p})+k_{\Omega}(y_{1,p}, z_{1,p})+k_{\Omega}(z_{1,p}, z_{s_p})+2(\kappa_1-12\kappa_0)\kappa_1^{2q-1}-\frac{40}{9} \\ \nonumber&=&k_{\Omega}(y_{s_p},z_{s_p})+2(\kappa_1-12\kappa_0)\kappa_1^{2q-1}-\frac{40}{9}, \end{align}\] and so (25 ) shows that \[\begin{align} k_{\Omega}(y_{s_q}, z_{s_q})\nonumber&>&k_{\Omega}(y_{s_p},z_{s_p})+k_{\Omega}(y_{s_1},y_{s_p})+k_{\Omega}(z_{s_1},z_{s_p})\\ \nonumber&+& 2(\kappa_1-12\kappa_0)\kappa_1^{2q-1}-16\kappa_0\kappa_1^{2p}\log(1+e{\kappa_0})-\frac{40}{9} \\ \nonumber&>& k_{\Omega}(y_{s_1},z_{s_1})=k_{\Omega}(y_t,z_t), \end{align}\] which contradicts with the construction of \(y_t\). Hence Claim 1 holds.

By Claim 1, we have \[\begin{align} \nonumber\label{5-14-2} k_{\Omega}(z_{1,m_1+1},z_{m_1+1}) &\leq k_{\Omega}(y_{m_1+1},y_{1,m_1+1})+k_{\Omega}(y_{1,m_1+1},z_{1,m_1+1})+k_{\Omega}(y_{m_1+1},z_{m_1+1})\\ &\leq k_{\Omega}(y_{m_1+1},y_{1,m_1+1})+4\kappa_2. \end{align}\tag{32}\] Meanwhile, we know from Lemma 12, Lemma 9, 10 and 15 that \[\begin{align} \nonumber k_{\Omega}(y_{m_1+1},y_{1,m_1+1}) &= k_{\Omega}(y_{m_1+1},y_{0})+k_{\Omega}(y_{1,m_1+1},y_{0}) \\ \nonumber &\leq 4\kappa_0 \log\Big(1+\frac{\ell (\gamma_{xy_1}[y_{m_1+1},y_0])}{d_{\Omega}(y_{m_1+1})}\Big)+4\kappa_0 \log\Big(1+\frac{\ell (\gamma_{xy_1}[y_{1,m_1+1},y_0])}{d_{\Omega}(y_{1,m_1+1})}\Big)\\ \nonumber &< 8 \kappa_0 \log(1+e^2\kappa_0), \end{align}\] which, together with 3 and 32 , shows that \[\begin{align} \log\Big(1+\frac{\ell (\gamma_{xz_1}[z_{1,m_1+1},z_{m_1+1}])}{d_{\Omega}(z_{m_1+1})}\Big) \leq k_{\Omega}(z_{1,m_1+1},z_{m_1+1})<5\kappa_2. \end{align}\] Hence, by 9 and the assumption of Case II, it holds \[\begin{align} \nonumber \ell (\gamma_{xz_1}[z_{1,m_1+1},z_{m_1+1}]) &< e^{5\kappa_2}d_{\Omega}(z_{m_1+1})\\ \nonumber &<e^{5\kappa_2}(d_{\Omega}(z_{1})+\ell (\gamma_{xy_1}[y_1,y_{m_1+1}]))<e^{5\kappa_2}\ell (\gamma_{xy}). \end{align}\] Finally, we get from (13 ) that \[\begin{align} \nonumber \ell (\gamma_{xz_1})&=\ell (\gamma_{xz_1}[x,z_{1,m_1+1}])+\ell (\gamma_{xz_1}[z_{1,m_1+1},z_{m_1+1}])+\ell (\gamma_{xz_1}[y_{1},z_{m_1+1}])\\ \nonumber &< (e^{5\kappa_2}+1) \ell (\gamma_{xy}). \end{align}\] The proof of Lemma 13 is thus complete. 0◻

3.2 Proofs of Theorems 8 and 2↩︎

The proof of Theorem 8 extends to Theorem 2 with only minor differences.

4 Proofs of auxiliary results↩︎

Proof. Proof of Lemma 9. We first show that for each \(x\in \alpha\), it holds \[\label{eq:basic32on32k} d_\Omega(x)\geq \max \left\{\frac{2\ell(\alpha[u,x])+d_\Omega(x)}{4c},\frac{1}{2c}d_\Omega(u)\right\}.\tag{44}\]

Indeed, if \(\ell(\alpha[u,x])\geq \frac{1}{2}d_{\Omega}(u)\), then \[d_\Omega(u)\leq 2\ell(\alpha[u,x])\leq 2c d_\Omega(x)\] and so \[\max \left\{\frac{2\ell(\alpha[u,x])+d_\Omega(u)}{4c},\frac{1}{2c}d_\Omega(u)\right\}=\frac{2\ell(\alpha[u,x])+d_\Omega(u)}{4c}\leq d_\Omega(x).\]

If \(\ell(\alpha[u,x])< \frac{1}{2}d_{\Omega}(u)\), then \[d_\Omega(x)\geq d_\Omega(u)-\ell(\alpha[u,x])>\frac{1}{2}d_\Omega(u).\] Since \(c\geq 1\), we infer from the above estimate that \[d_\Omega(x)\geq \frac{1}{2c}d_\Omega(u)=\max \left\{\frac{2\ell(\alpha[u,x])+d_\Omega(u)}{4c},\frac{1}{2c}d_\Omega(u)\right\}.\]

In either case, we proved 44 .

For each \(w\in \alpha\), by 44 , we have \[\begin{align} \begin{aligned} k_{\Omega}(u,w)&\leq \ell_k(\alpha[u,w])=\int_{x\in\alpha[u,w]}\frac{ds}{d_{\Omega}(x)}\leq 4c\int_{x\in\alpha[u,w]}\frac{ds}{2\ell(\alpha[u,x])+d_\Omega(u)}\\ &\leq 2c \log\Big(1+\frac{2\ell(\alpha[u,w])}{d_\Omega(u)}\Big)\leq 4c \log\Big(1+\frac{\ell(\alpha[u,w])}{d_\Omega(u)}\Big). \end{aligned} \end{align}\] 0◻

Proof. Proof of Lemma 12. Select \(x_0 \in \gamma\) such that \[\begin{align} \label{222} \min \left\{{\operatorname{diam}}( \gamma[x_1, x_0]), {\operatorname{diam}}( \gamma[x_2, x_0])\right\} \geq \frac{1}{3} {\operatorname{diam}}(\gamma), \end{align}\tag{47}\]

We first prove that for each \(i \in \{1,2\}\) and \(x\in \gamma[x_i, x_0]\), \[\label{H25-0729-1} {\operatorname{diam}}( \gamma[x_i ,x])\leq 10c_0 c_1d_{\Omega}(x).\tag{48}\] Without loss of generality, we only consider the case \(x\in \gamma[x_1,x_0]\) and shall prove that \[\begin{align} \label{333} {\operatorname{diam}}(\gamma[x_1, x]) \leq 10c_0 c_1d_{\Omega}(x). \end{align}\tag{49}\]

Let \(z_1\in \gamma([x_1,x])\cap \Omega \backslash\overline{B \left(x,\frac{1}{9}{\operatorname{diam}}( \gamma[x_1, x])\right)}\). It follows from \({\operatorname{diam}}(\gamma[x_2,x])\geq \frac{1}{3}{\operatorname{diam}}(\gamma)\) that there exists \(z_2 \in \gamma[x_2,x]\) that belongs to \(\Omega \backslash\overline{B \left(x,\frac{1}{9}{\operatorname{diam}}( \gamma[x_1, x])\right)}\).

Since \(\Omega\) is \(c_0\)-LLC-2, there exists some curve \[\beta \subset \Omega\backslash \overline{B\left(x,\frac{1}{9c_0}{\operatorname{diam}}( \gamma[x_1, x])\right)}\] joining \(z_1\) and \(z_2\), and since \((\Omega,\sigma)\) satisfies the \(c_1\)-ball separation condition, we have \[\begin{align} B_{\sigma}(x,c_1d_{\Omega}(x)) \cap \beta \neq \emptyset, \end{align}\] from which it follows \[\begin{align} {\operatorname{diam}}(\gamma[x_1, x]) \leq 9c_0 c_1d_{\Omega}(x). \end{align}\] This establishes 49 and thus completes the proof of 48 .

Next, our aim is to prove that for each \(i=1,2\) and \(x \in \gamma[x_i, x_0]\), it holds \[\begin{align} \label{444} \ell (\gamma[x_i, x])\leq (28ec_0c_1)^2([Q^{2\log_2 70ec_0c_1}]+1) d_{\Omega}(x). \end{align}\tag{50}\]

Without loss of generality, by (48 ), we only consider the case \(x\in \gamma[x_1,x_0]\). Take \(y_1 \in \gamma[x_1, x]\) such that \[d_{\Omega}(y_1)\geq \frac{1}{2}\sup_{y \in \gamma[x_1, x]}\{d_{\Omega}(y)\}.\] Then we have \[\begin{align} \label{555} d_{\Omega}(y_1) \leq d_{\Omega}(x)+ d(y_1 ,x)\leq d_{\Omega}(x)+{\operatorname{diam}}(\gamma[x_1, x]) \stackrel{\eqref{333}}{\leq} (1+10c_0 c_1)d_{\Omega}(x). \end{align}\tag{51}\] Note that by 49 , for each \(z\in \gamma[x_1 ,x])\), it holds \[\begin{align} d(x_1,z) \leq {\operatorname{diam}}( \gamma[x_1, z]) \leq 10c_0 c_1 d_{\Omega}(z). \end{align}\] Thus we infer from [27] (with \(\beta=\gamma[x_1, x]\), \(\mu_1=10c_0c_1\) and \(x_0=y_1\)) that \[\begin{align} \ell(\gamma[x_1 ,x]) \leq {25}e^3c_0c_1([Q^{2\log_2 70ec_0c_1}]+1)d_{\Omega}(y_1) \stackrel{\eqref{555}}{\leq} ({28}ec_0c_1)^2([Q^{2\log_2 70ec_0c_1}]+1)d_{\Omega}(x). \end{align}\] This establishes 50 as desired. The proof of Lemma 12 is thus complete. 0◻

Acknowledgment. We would like to thank Prof. Pekka Koskela, Dr. Abhishek Pandey, Yi Xuan, Zhiqiang Yang and Linhong Wang for numerous helpful discussions and comments.

References↩︎