Abstract
We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of SobolevtoLipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger, which satisfies a related maximality condition.
1 Introduction
1.1 Main result
A celebrated result due to Bonk and Kleiner [3] states that an Ahlfors 2regular metric sphere Z is quasisymetrically equivalent to the standard sphere if and only if it is linearly locally connected. Recently, it was shown in [35] that the quasisymmetric homeomorphism \(u_Z:S^2\rightarrow Z\) may be chosen to be of minimal energy \(E^2_+(u_Z)\), and in this case is unique up to a conformal diffeomorphism of \(S^2\).
The map \(u_Z\) gives rise to a measurable Finsler structure on \(S^2\), defined by the approximate metric differential \({\text {ap md}}u_Z\); cf. [33] and Sect. 6 below. When Z is a smooth Finsler surface, the approximate metric differential carries all the metric information of Z. In the present generality, however, \({\text {ap md}}u_Z\) is defined only almost everywhere and thus does not determine the length of every curve.
Definition 1.1
Let Y and Z be Ahlfors 2regular, linearly locally connected metric spheres. We say that Y and Z are analytically equivalent if there exist energy minimizing parametrizations \(u_Y\) and \(u_Z\) such that
almost everywhere.
By the aformentioned uniqueness result, analytic equivalence defines an equivalence relation on the class of linearly locally connected, Ahlfors 2regular spheres. The main result of this paper states that the equivalence class of such a sphere contains a maximal representative, unique up to an isometry.
Theorem 1.2
Let Z be a linearly locally connected, Ahlfors 2regular sphere. Then there is linearly locally connected, Ahlfors 2regular sphere \(\widehat{Z}\) which is analytically and biLipschitz equivalent to Z and satisfies the following properties.

(1)
SobolevtoLipschitz property: If \(f\in N^{1,2}({\widehat{Z}})\) has weak upper gradient 1, then f has a 1Lipschitz representative.

(2)
Thick geodecity: For arbitrary measurable subsets \(E,F\subset \widehat{Z}\) of positive measure and \(C>1\), one has \({\text {Mod}}_2\Gamma (E,F;C)>0\).

(3)
Maximality: If Y is analytically equivalent to \(\widehat{Z}\), there exists a 1Lipschitz homeomorphism \(f:\widehat{Z}\rightarrow Y\).

(4)
Volume rigidity: If Y is a linearly locally connected, Ahlfors 2regular sphere, and \(f:Y \rightarrow {\widehat{Z}}\) is a 1Lipschitz area preserving map which is moreover celllike, then f is an isometry.
Moreover \(\widehat{Z}\) is characterized uniquely, up to isometry, by any of the listed properties.
In the setting of Ahlfors regular metric spheres, Theorem 1.2 links the essential metric investigated in [1, 7, 8], the concept of volume rigidity studied under curvature bounds in [27,28,29], the SobolevtoLipschitz property arising in the context of RCD spaces in [15, 16], and the notion of thick quasiconvexity whose connection to Poincaré inequalities is investigated in [10,11,12].
More generally, an Ahlfors 2regular disc Z with finite boundary length admits a quasisymmetric parametrization by the standard disc \(\overline{\mathbb {D}}\) if and only if it is linearly locally connected. The parametrization may again be chosen to be of minimal Reshetynak energy and is then unique up to a conformal diffeomorphism, see [35]. A suitable variant of Theorem 1.2 also holds in this setting. In this case in the formulation of volume rigidity one has to assume that f preserves the boundary curves and the restriction \(f:\partial Y\rightarrow \partial Z\) is monotone, instead of assuming that the entire map f is celllike.
Other parametrization results for metric surfaces have been obtained for example in [23, 36, 39, 40, 43]. These results however do not include a uniqueness part. Thus it seems unclear how to define a suitable notion of analytic equivalence in the settings therein.
1.2 Construction of essential metrics
Our construction of essential metrics relies on the pessential infimum over curve families and extends the construction of the essential metric in [1] to the case \(p<\infty \); cf. (1.3) and Definition 5.1. For a metric measure space X, \(p\in [1,\infty ]\), and a family of curves \(\Gamma \) in X, we define its pessential length \(ess \ell _p(\Gamma )\in [0,\infty ]\) as the essential infimum of the length function on \(\Gamma \) with respect to pmodulus. The pessential distance \(d_{p}':X\times X \rightarrow [0,\infty ]\) is defined by
where \(\Gamma (E,F)\) denotes the family of curves in X joining two given measurable subsets \(E,F\subset X\). This quantity does not automatically satisfy the triangle inequality, and we consider instead the largest metric \(d_p\le d_p'\); see the discussion after Definition 5.3. In general, \(d_p\) might take infinite values, and its finiteness is related to an abundance of curves of uniformly bounded length connecting given disjoint sets. The condition of thick quasiconvexity, introduced in [12], quantifies the existence of an abundance of quasiconvex curves.
Definition 1.3
Let \((X,d,\mu )\) be a metric measure space and \(p \in [1,\infty ]\). We say that X is pthick quasiconvex with constant \(C\ge 1\) if, for all measurable subsets \(E,F\subset X\) of positive measure, we have
We say that X is pthick geodesic if X is pthick quasiconvex with constant C for every \(C>1\).
Here \(\Gamma (E,F;C)\) denotes the family of curves \(\gamma :[0,1]\rightarrow X\) joining E and F such that \(\ell (\gamma )\le C\cdot d(\gamma (0),\gamma (1))\). Note that this is equivalent to the original definition in [12] when X is infinitesimally doubling.
Theorem 1.4
Let \((X,d,\mu )\) be an infinitesimally doubling metric measure space which is pthick quasiconvex with constant C and \(1\le p \le \infty \). Then there exists a metric \(d_p\) on X, for which \(X_p:=(X,d_p,\mu )\) is pthick geodesic and \(d\le d_p \le Cd\). Moreover \(d_p\) is minimal among metrics \(\rho \ge d\) for which \(X_\rho :=(X,\rho ,\mu )\) is pthick geodesic.
It is known that doubling pPoincaré spaces are pthick quasiconvex for some constant depending only on the data of X, though the converse need not be true unless \(p=\infty \); see [11, 12]. For \(p<q\), the assumption of pthick geodecity is strictly stronger than that of qthick geodecity, see Example 3.3 below. However, under the assumption of a suitable Poincaré inequality the particular value of q is immaterial. Indeed, for Poincaré spaces the essential metrics of all indices coincide with the essential metric \(d_{ess}\) introduced in [1]. The essential metric \(d_{ess}\) is given by
Proposition 1.5
Let X be a doubling metric measure space satisfying a pPoincaré inequality. Then \(d_q=d_{ess}\) for every \(q\in [p,\infty ]\).
In the proof of Theorem 1.2 we will set \(\widehat{Z}:=(Z,d_2)\). A posteriori, the chosen metric agrees with the essential metric (1.3). However from this characterization it is unclear that the disc obtained in Theorem 1.2 is unique. This follows from the use of 2modulus in the construction of the metric, and its quasiinvariance under quasisymmetric maps.
1.3 The SobolevtoLipschitz property, thick geodecity, and volume rigidity
The SobolevtoLipschitz property and thick geodecity are defined and equivalent in a much broader framework.
Definition 1.6
A metric measure space X is said to have the pSobolevtoLipschitz property, if every \(f\in N^{1,p}(X)\) for which the minimal pweak upper gradient \(g_f\) satisfies \(g_f\le 1\) almost everywhere has a 1Lipschitz representative.
To illustrate this equivalence, and contrast the difference of the SobolevtoLipschitz property with merely being geodesic, recall that a proper metric space is geodesic if and only if, whenever a function f has (genuine) upper gradient 1, it is 1Lipschitz. The pSobolevtoLipschitz property requires we can reach the same conclusion from the weaker assumption that f has pweak upper gradient 1. The next result translates this condition to having an abundance of nearly geodesic curves in the space.
Theorem 1.7
Let X be an infinitesimally doubling metric measure space and \(p\ge 1\). Then X has the pSobolevtoLipschitz property if and only if it is pthick geodesic.
These properties are also related to volume rigidity, which asks whether any volume preserving 1Lipschitz map of a given class is an isometry. Recall that a Borel map \(f:(X,\mu )\rightarrow (Y,\nu )\) between metric measure spaces is volume preserving if \(f_*\mu =\nu \). When X and Y are nrectifiable, we equip them with the Hausdorff measure \(\mathcal {H}^n\), unless otherwise explicitly stated. Volume preserving 1Lipschitz maps between rectifiable spaces are essentially length preserving, see Proposition 4.1, but need not be homeomorphisms. For volume preserving 1Lipschitz homeomorphisms with quasiconformal inverse, the SobolevtoLipschitz property guarantees isometry.
Proposition 1.8
Let \(f:X\rightarrow Y\) be a volume preserving 1Lipschitz homeomorphism between nrectifiable metric spaces. If \(f^{1}\) is quasiconformal and Y has the nSobolevtoLipschitz property, then f is an isometry.
Note that the inverse of a quasiconformal map is not quasiconformal in general; cf. [44, Remark 4.2]. This is guaranteed under the assumption that the spaces are Ahlfors nregular and X supports an nPoincaré inequality. The assumption that Y has the SobolevtoLipschitz property is essential for the validity of Proposition 1.8; cf. Example 4.4. Thus, Proposition 1.8 connects volume rigidity, the SobolevtoLipschitz property and thick geodecity under these fairly restrictive assumptions, and will be used in the proof of Theorem 1.2.
1.4 Application: essential minimal surfaces and essential pullback metrics
For the definitions and terminology in the forthcoming discussion, we refer the reader to Sect. 2.
A smooth map \(u:M\rightarrow X\) between Riemannian manifolds gives rise to a pullback metric on M. In the case that M and X are merely metric spaces and u a continuous map, various nonequivalent definitions of pullback metric are discussed in [37, 38]. The metric
where \(\gamma \) varies over all continuous curves in M joining p to q, was used in [33, 38, 42] to study solutions \(u:{\overline{\mathbb {D}}}\rightarrow X\) of the Plateau problem in X, also referred to as minimal discs. If X satisfies a quadratic isoperimetric inequality and u bounds a Jordan curve of finite length, then \(d_u\) gives rise to a factorization
where:

(1)
\(Z_u\) is a geodesic disc which satisfies the same quadratic isoperimetric inequality as X.

(2)
\(P_u\) is a minimal disc bounding \(\partial Z_u\) and a uniform limit of homeomorphisms.

(3)
\(\bar{u}\) is 1Lipschitz and \(\ell (P_u\circ \gamma )=\ell (u\circ \gamma )\) for every curve \(\gamma \) in \(\overline{\mathbb {D}}\).
The fact that \(Z_u\) satisfies the same quadratic isoperimetric inequality as X reflects the fact that the intrinsic curvature of a classical minimal surface is bounded above by that of the ambient manifold; cf. [34].
Our definition of essential metrics suggests the following variant of pullback metrics in the minimal disc setting. Namely for a family of curves \(\Gamma \) in \(\overline{\mathbb {D}}\) let \(ess\ell _{u}(\Gamma )\) be the essential infimum of \(\ell (u\circ \gamma )\) with respect to 2modulus; cf. Definition 5.1. Then for \(p,q\in \overline{\mathbb {D}}\) we define the essential pullback metric by
This construction gives rise to the following factorization result.
Theorem 1.9
Let X be a proper metric space satisfying a \((C,l_0)\)quadratic isoperimetric inequality. Let \(u:\overline{\mathbb {D}}\rightarrow X\) be a minimal disc bounding a Jordan curve \(\Gamma \) which satisfies a chordarc condition. Then there exists a geodesic disc \(\widehat{Z}_u\) and a factorization
such that

(1)
\({\widehat{Z}}_u\) has the SobolevtoLipschitz property and satisfies a \((C,l_0)\)quadratic isoperimetric inequality,

(2)
\(\widehat{P}_u\) a minimal disc bounding \(\partial \widehat{Z}_u\) and a uniform limit of homeomorphisms, and

(3)
\(\widehat{u}:\widehat{Z}_u\rightarrow X\) is 1Lipschitz and for 2almost every curve \(\gamma \) in \(\overline{\mathbb {D}}\) one has \(\ell (\widehat{P}_u \circ \gamma )=\ell (u \circ \gamma )\).
Furthermore the factorization is maximal in the following sense: If \({\widetilde{Z}}\) is a metric disc and \(u={{\tilde{u}}}\circ {\widetilde{P}}\), where \({\widetilde{P}}:{\overline{\mathbb {D}}}\rightarrow {\widetilde{Z}}\) and \(\tilde{u}:{\widetilde{Z}}\rightarrow X\) satisfy (2) and (3), then there exists a surjective 1Lipschitz map \(f:\widehat{Z}_u\rightarrow \widetilde{Z}\) such that \({\widetilde{P}}= f\circ \widehat{P}_u\).
A Jordan curve is said to satisfy a chordarc condition if it is biLipschitz equivalent to \(S^1\). The fact that the map \(\widehat{P}_u\) is a minimal disc implies additional regularity. Indeed, \({\widehat{P}}_u\) is infinitesimally quasiconformal, Hölder continuous and its minimal weak upper gradient has global higher integrability; cf. [31,32,33, 35].
Compared to the construction in [33], we trade off some regularity of \(\overline{u}\) in exchange for the SobolevtoLipschitz property on \(\widehat{Z}_u\). Indeed, \(\widehat{u}\) only preserves the length of almost every curve, while \(\widehat{Z}_u\) is nicer in analytic and geometric terms. Note however that some geometric properties of \(\widehat{Z}_u\), established for \(Z_u\) in [33, 42], remain open for \({\widehat{Z}}_u\). In particular we do not know whether \(\mathcal {H}^1(\partial {\widehat{Z}}_u)<\infty \).
Note that, in contrast to Theorem 1.2, no characterization in terms of the SobolevtoLipschitz property in Theorem 1.9 is possible. In fact it may happen that \(Z_u\) already has the SobolevtoLipschitz property but still \(\widehat{Z}_u\) and \(Z_u\) are not isometric, see Example 6.4.
More generally, when M supports a QPoincaré inequality where \(Q\ge 1\) satisfies (2.2), maps \(u\in N^{1,p}_{loc}(M;X)\), with a priori higher exponent \(p>Q\), give rise to a pessential pullback metric on M and the resulting metric measure space has the QSobolevtoLipschitz property; cf. Definition 1.6 and Theorem 5.16. Higher integrability of minimal weak upper gradients for quasiconformal maps is known to hold when M and X both have Qbounded geometry, and also in the situation of [31, Theorem 1.4], but not in general; cf. [20, 21, 26]. Together with the Poincaré inequality and Morrey’s embedding, higher integrability of minimal weak upper gradients implies finiteness of the essential pullback metric. The assumption in Theorem 1.9 that \(\Gamma \) is not only of finite length but satisfies a chordarc condition is needed to guarantee this additional regularity.
1.5 Organization
The paper is organized as follows. After the preliminaries in Sect. 2, we treat the equivalence of thick geodecity and the SobolevtoLipschitz property in Sect. 3, where we prove Theorem 1.7. Volume rigidity is discussed in Sect. 4, which includes the proof of Proposition 1.8.
The construction of essential metrics is presented in Sect. 5. We develop some general tools in Sect. 5.1, and prove Theorem 1.4 and Proposition 1.5 in Sect. 5.2. In Sect. 5.3 we apply our constuction to Sobolev maps from a Poincaré space, and derive some basic properties of the resulting metric measure space; cf. Theorem 5.15.
Finally, in Sect. 6, we apply the results from Sects. 5.2 and 5.3 to obtain Theorems 1.2 and 1.9. First in Sect. 6.1 we prove Theorem 1.2. Then in Sect. 6.2 we prove a weaker but more general result for discs, cf. Theorem 6.3, and finally in Sect. 6.3 we discuss Theorem 1.9.
2 Preliminaries
We refer to the monographs [2, 22] for the material discussed below. Given a metric space (X, d), balls in X are denoted B(x, r) and, if \(B=B(x,r)\subset X\) is an open ball and \(\sigma >0\), \(\sigma B\) denotes the ball with the same center as B and radius \(\sigma r\). The Hausdorff measure on X is denoted \(\mathcal {H}^n\) or, if we want to stress the space or metric, by \(\mathcal {H}^n_X\) or \(\mathcal {H}^n_d\). The normalizing constant is chosen so that \(\mathcal {H}^n_{\mathbb {R}^n}\) agrees with the Lebesgue measure.
By a measure \(\mu \) on a metric space X we mean an outer measure which is Borel regular. A triple \((X,d,\mu )\) where (X, d) is a proper metric space and \(\mu \) is a measure on X which is nontrivial on balls, is called a metric measure space. Note that all metric measure spaces in our convention are complete, separable and every closed bounded subset is compact. Furthermore all balls have positive finite measure and in particular the measures are Radon, cf. [22, Corollary 3.3.47]. We often abbreviate \(X=(X,d,\mu )\) when the metric and measure are clear from the context.
Given a curve \(\gamma :[a,b]\rightarrow X\), we denote by \(\ell (\gamma )\) its length. When \(\gamma \) is absolutely continuous we write \(\gamma _t'\) for the metric speed (at \(t\in [a,b]\)) of \(\gamma \). Moreover, when \(\gamma \) is rectifiable (i.e. \(\ell (\gamma )<\infty \)) we denote by \({\bar{\gamma }}:[a,b]\rightarrow X\) the constant speed parametrization of \(\gamma \). This is the (Lipschitz) curve satisfying
The arc length parametrization \(\gamma _s\) of \(\gamma \) is the reparametrization of \({\bar{\gamma }}\) to the interval \([0,\ell (\gamma )]\). With the exception of Proposition 5.7, we assume curves are defined on the interval [0, 1].
Given a metric measure space X, a Banach space V, and \(p\ge 1\), we denote by \(L^p(X;V)\) and \(L^p_{\mathrm {loc}}(X;V)\) the a.e.equivalence classes of pintegrable and locally pintegrable \(\mu \)measurable maps \(u:X\rightarrow V\). We also denote \(L^p(X):=L^p(X;\mathbb {R})\) and \(L^p_{\mathrm {loc}}(X):=L^p_{\mathrm {loc}}(X;\mathbb {R})\) and abuse notation by writing \(f\in L^p(X;V)\) or \(f\in L_{\mathrm {loc}}^p(X;V)\) for maps f (instead of equivalence classes).
2.1 Properties of measures
A measure \(\mu \) on X is called doubling if there exists \(C\ge 1\) such that
for every \(x\in X\) and \(0<r<{\text {diam}}X\). The least such constant is denoted \(C_\mu \) and called the doubling constant of \(\mu \). Doubling measures satisfy a relative volume lower bound
for some constants \(C>0\) and \(Q\le \log _2C_\mu \) depending only on \(C_\mu \). The opposite inequality with the same exponent Q need not hold. If there are constants \(C,Q >0\) such that
we say that \(\mu \) is Ahlfors Qregular.
If \(\mu \) is doubling and
denotes the (restricted) centered maximal function of f at \(x\in X\) the sublinear operator \(\mathcal {M}_r\) satisfies the usual boundedness estimates
for some constant C depending only on \(C_\mu \) and p. Consequently almost every point of a \(\mu \)measurable set \(E\subset X\) is a Lebesgue density point.
If the measure \(\mu \) satisfies the infinitesimal doubling condition
for \(\mu \)almost every \(x\in X\), the claim about density points still remains true.
Proposition 2.1
[22, Theorem 3.4.3] If \((X,d,\mu )\) is infinitesimally doubling metric measure space and \(f\in L^1_{\mathrm {loc}}(X)\) then
for \(\mu \)almost every \(x\in X\). In particular, \(\mu \)almost every point of a Borel set \(E\subset X\) is a Lebesgue density point.
2.1.1 Sobolev spaces
Let \((X,d,\mu )\) be a metric measure space, Y a metric space, and \(p\ge 1\). If \(u:X\rightarrow Y\) is a map and \(g:X\rightarrow [0,\infty ]\) is Borel, we say that g is an upper gradient of u if
for every curve \(\gamma :[a,b]\rightarrow X\). Recall that the line integral of g over \(\gamma \) is defined as
if \(\gamma \) is rectifiable and \(\infty \) otherwise. Suppose \(Y=V\) is a separable Banach space. A \(\mu \)measurable map \(u\in L^p(X;V)\) is called pNewtonian if it has an upper gradient \(g\in L^p(X)\). The Newtonian seminorm is
where the infimum is taken over all upper gradients g of u. The Newtonian space \(N^{1,p}(X;V)\) is the vector space of equivalence classes of pNewtonian maps, where two maps \(u,v:X\rightarrow V\) are equivalent if \(\Vert uv\Vert _{1,p}=0\). The quantity (2.5) defines a norm on \(N^{1,p}(X;V)\) and \((N^{1,p}(X;V),\Vert \cdot \Vert _{1,p})\) is a Banach space.
We say that a \(\mu \)measurable map \(u:X\rightarrow V\) is locally pNewtonian if every point \(x\in X\) has a neighbourhood U such that
and denote the vector space of locally pNewtonian maps by \(N^{1,p}_{loc}(X;V)\).
2.1.2 Minimal upper gradients
Let \(\Gamma \) be a family of curves in X. We define the pmodulus of \(\Gamma \) as
We say that a Borel function \(g:X\rightarrow [0,\infty ]\) is a pweak upper gradient of u if there is a path family \(\Gamma _0\) with \({\text {Mod}}_p(\Gamma _0)=0\) so that (2.4) holds for all curves \(\gamma \notin \Gamma _0\). The infimum in (2.5) need not be attained by upper gradients of u but there is a minimal pweak upper gradient \(g_u\) of u so that
The minimal pweak upper gradient is unique up to sets of measure zero.
If Y is a complete separable metric space then there is an isometric embedding \(\iota :Y\rightarrow V\) into a separable Banach space. We may define
We remark that maps \(u\in N^{1,p}_{loc}(X;Y)\) have minimal pweak upper gradients that are unique up to equality almost everywhere and do not depend on the embedding.
2.1.3 Poincare inequalities
A metric measure space \(X=(X,d,\mu )\) supports a pPoincaré inequality if there are constants \(C,\sigma \ge 1\) so that
whenever \(u\in L^1_{\mathrm {loc}}(X)\), g is an upper gradient of u, and B is a ball in X. Doubling metric measure spaces supporting a Poincaré inequality enjoy a rich theory.
Theorem 2.2
(Morrey embedding) Let \((X,d,\mu )\) be a complete doubling metric measure space, where the measure satisfies (2.2) for \(Q\ge 1\), and suppose X supports a QPoincaré inequality. If \(p>Q\), there is a constant \(C\ge 1\) depending only on the data of X so that for any \(u\in N^{1,p}_{loc}(X;Y)\) and any ball \(B\subset X\) we have
Here the data of X refers to p, the doubling constant of the measure and the constants in the Poincaré inequality.
2.1.4 Rectifiability and metric differentials
A separable metric space X is said to be nrectifiable if there exist Borel sets \(E_i\subset \mathbb {R}^n\) and Lipschitz maps \(g_i:E_i\rightarrow X\) for each \(i\in \mathbb {N}\), satisfying
Unless otherwise stated we always equip nrectifiable spaces with the Hausdorff nmeasure \(\mathcal {H}^n\).
Metric differentials play an important role in the study of rectifiable metric spaces. Let \(E\subset \mathbb {R}^n\) be measurable and \(f:E\rightarrow X\) be a Lipschitz map into a metric space X. By a fundamental result of Kirchheim [25], f admits a metric differential \({\text {md}}_xf\) at almost every point \(x\in E\). The metric differential is a seminorm on \(\mathbb {R}^n\) and such that
If \(\Omega \subset \mathbb {R}^n\) is a domain and \(p\ge 1\), a Sobolev map \(u\in N^{1,p}_{loc}(\Omega ;X)\) admits an approximate metric differential \({\text {ap md}}_x u\) for almost every \(x\in \Omega \); cf. [24] and [31]. If \(\Omega \subset M\) is a domain in a Riemannian nmanifold, the approximate metric differential of \(u\in N^{1,p}(\Omega ;X)\) may be defined, for almost every \(x\in \Omega \), as a seminorm on \(T_xM\) by
for any chart \(\psi \) around x; see the discussion after [14, Definition 2.1] and the references therein for the details.
We formulate the next results for maps defined on domains of manifolds; that is, we assume that \(\Omega \subset M\) for a Riemannian nmanifold M. Note that, although in the references the statements are proved in the Euclidean setting, the definition (2.7) easily implies the corresponding statements for general Riemannian manifolds. For the following area formula for Sobolev maps with Lusin’s property (N), recall that a Borel map \(f:(X,\mu )\rightarrow (Y,\nu )\) is said to have Lusin’s property (N) if \(\nu (f(E))=0\) whenever \(\mu (E)=0\).
Theorem 2.3
[24, Theorems 2.4 and 3.2] Let \(u\in N^{1,p}_{loc}(\Omega ;X)\) for some \(p>n\). Then u has Lusin’s property (N), and
for any Borel function \(\varphi :\Omega \rightarrow [0,\infty ]\).
Here the Jacobian of a seminorm s on \(\mathbb {R}^n\) is defined by
Following [31] we also define the maximal stretch of a seminorm s by
Note that the inequality \(J(s)\le I_+^n(s)\) always holds, and \(I_+({\text {ap md}}_xu)=g_u\) almost everywhere for \(u\in N^{1,n}_{loc}(\Omega ;X)\), see [31, Section 4]. We say that s is Kquasiconformal if \(I_+^n(s)\le KJ(s)\). A Sobolev map \(u\in N^{1,n}_{loc}(\Omega ;X)\) is called infinitesimally Kquasiconformal, if \({\text {ap md}}_x u\) is Kquasiconformal for almost every \(x\in \Omega \).
The (Reshetnyak) energy and (parametrized Hausdorff) area of \(u\in N^{1,n}_{loc}(\Omega ;X)\) are defined as
2.1.5 Quasiconformality
We refer to [17,18,19,20] for various definitions of quasiconformality and their relationship in metric spaces, and only mention what is sometimes known as geometric quasiconformality. A homeomorphism \(u:(X,\mu )\rightarrow (Y,\nu )\) between metric measure spaces is said to be Kquasiconformal with “index” \(Q\ge 1\), if
for any curve family \(\Gamma \) in X. Without further assumptions on the geometry of the spaces, the modulus condition (2.10) is fundamentally onesided; cf. the discussion in the introduction. We refer to [44] for an equivalent characterization in terms of analytic quasiconformality and note that there is a corresponding result for monotone maps. Recall that a map is monotone if the preimage of every point is a connected set.
Proposition 2.4
[35, Proposition 3.4] Let M be \({\overline{\mathbb {D}}}\) or \(S^2\). If \(u\in N^{1,2}(M;X)\) is a continuous, surjective, monotone and infinitesimally Kquasiconformal map into a complete metric space X, then
for any curve family \(\Gamma \) in \({\overline{\mathbb {D}}}\).
We remark that in [35] the result is proven for \(M={\overline{\mathbb {D}}}\). The same proof however carries over to \(S^2\) with minor modifications.
2.1.6 Quadratic isoperimetric inequality and minimal discs
Given a proper metric space X and a closed Jordan curve \(\Gamma \) in X, we denote
Here, a function \(u\in N^{1,2}({\overline{\mathbb {D}}};X)\) is said to span \(\Gamma \) if \(u_{S^1}\) agrees a.e. with a monotone parametrization of \(\Gamma \). We call \(u\in \Lambda (\Gamma ,X)\) a minimal disc spanning \(\Gamma \) if u has least area among all maps in \(\Lambda (\Gamma ,X)\) and is furthermore of minimal Reshetnyak energy \(E^2_+\) among all area minimizers in \(\Lambda (\Gamma ,X)\). In [31] it has been shown that, as soon as \(\Lambda (\Gamma ,X)\ne \varnothing \), there exists a minimal disc u spanning \(\Gamma \) and every such u is infinitesimally \(\sqrt{2}\)quasiconformal.
Definition 2.5
A metric space X is said to satisfy a \((C,l_0)\)quadratic isoperimetric inequality if, for any Lipschitz curve \(\gamma :S^1\rightarrow X\) with \(\ell (\gamma )<l_0\), there exists \(u\in N^{1,2}(\overline{\mathbb {D}};X)\) with \(u_{S^1}=\gamma \) and
If X satisfies a quadratic isoperimetric inequality then minimal discs \(u:\overline{\mathbb {D}}\rightarrow X\) lie in \(N^{1,p}_{loc}(\mathbb {D},X)\) for some \(p>2\) and, if furthermore \(\Gamma \) satisfies a chordarc condition, then \(u\in N^{1,p}(\overline{\mathbb {D}},X)\).
If a proper geodesic metric space Z is homeomorphic to a 2manifold, then it satisfies a \((C,l_0)\)quadratic isoperimetric inequality if and only if every Jordan curve \(\Gamma \) in Z such that \(\ell (\Gamma )<l_0\) bounds a Jordan domain \(U\subset Z\) which satisfies
This follows from the proof of [35, Theorem 1.4] together with the observation leading to [6, Corollary 1.5]. We will use this fact in Sect. 6 to establish the quadratic isoperimetric inequality for \({\widehat{Z}}_u\) in Theorem 1.9.
3 A geometric characterization of the SobolevtoLipschitz property
In this section we prove Theorem 1.7. We assume throughout this section that X is a metric measure space and \(p\ge 1\). We will need the following consequence of the SobolevtoLipschitz property for measurable maps with 1 as pweak upper gradients.
Lemma 3.1
Suppose X has the pSobolevtoLipschitz property, and let V be a separable Banach space. Then any measurable function \(f:X\rightarrow V\) with 1 as pweak upper gradient has a 1Lipschitz representative.
Proof
We prove the claim first when \(V=\mathbb {R}\). Fix \(x_0\in X\) and consider the function
for each \(k\in \mathbb {N}\). Note that \(w_k\) is 1Lipschitz and \(w_k\in N^{1,p}(X)\). The functions
have 1 as pweak upper gradient (see [22, Proposition 7.1.8]) and \(f_k\in N^{1,p}(X)\). By the SobolevtoLipschitz property there is a set \(N\subset X\) with \(\mu (N)=0\) and 1Lipschitz functions \({\bar{f}}_k\) such that \(f_k(x)={\bar{f}}_k(x)\) if \(x\notin N\), for each \(k\in \mathbb {N}\). Since \(f_k\rightarrow f\) pointwise everywhere we have, for \(x,y\notin N\), that
Thus f has a 1Lipschitz representative.
Next, let V be a separable Banach space and let \(\{x_n\}\subset V\) be a countable dense set. Given f as in the claim, the functions \(f_n:={\text {dist}}(x_n,f)\) have 1 as pweak upper gradient, and thus there is a null set \(E\subset X\) and 1Lipschitz functions \({\bar{f}}_n\) so that \(f_n(x)={\bar{f}}_n(x)\) whenever \(n\in \mathbb {N}\) and \(x\notin E\). For \(x,y\notin N\), we have
We again conclude that f has a 1Lipschitz representative. \(\square \)
Proposition 3.2
Let X be infinitesimally doubling and pthick quasiconvex with constant C. Then every \(u\in N^{1,p}(X)\) satisfying \(g_u\le 1\), has a CLipschitz representative.
Proof
Let \(\Gamma _0\) a path family of zero pmodulus such that
whenever \(\gamma \notin \Gamma _0\). By Lusin’s theorem there is a decreasing sequence of open sets \(U_m\subset X\) such that, for each \(m\in \mathbb {N}\), \(\mu (U_m)<2^{m}\) and \(u_{X\setminus U_m}\) is continuous. Fix \(m\in \mathbb {N}\), and let x and y be distinct density points of \(X\setminus U_m\). Then for every \(\delta >0\) we have that
For \(\gamma ^\delta \in \Gamma (B(x,\delta )\setminus U_m,B(y,\delta )\setminus U_m;C)\setminus \Gamma _0\) one has
Letting \(\delta \rightarrow 0\) and using the continuity of \(u_{X\setminus U_m}\) we obtain
Letting \(m \rightarrow \infty \) one sees that (3.1) holds for \(\mu \)almost every \(x,y \in X\) and hence u has a CLipschitz representative. \(\square \)
Proof of Theorem 1.7
One implication is directly implied by the preceeding proposition. Indeed, assume X is pthick geodesic and let \(f\in N^{1,p}(X)\) have 1 as a pweak upper gradient. By Proposition 3.2, f has a CLipschitz representative for every \(C>1\). So the continuous representative of f is 1Lipschitz.
Now assume X has the pSobolevtoLipschitz property but is not pthick geodesic. Let \(E,F\subset X\) measurable and \(C>1\) be such that \(\mu (E)\mu (F)>0\) and \({\text {Mod}}_p\Gamma (E,F;C)=0\). By looking at density points of E and F respectively we may assume without loss of generality that
There exists a nonnegative Borel function \(g\in L^p(X)\) for which \(\displaystyle \int _\gamma g=\infty \) for every \(\gamma \in \Gamma (E,F;C)\). Denote
whence \(\Gamma (E,F;C)\subset \Gamma _0\), \({\text {Mod}}_p(\Gamma _0)=0\) and for \(\gamma _1,\gamma _2\notin \Gamma _0\) one has \(\gamma _1 \gamma _2 \notin \Gamma _0\) whenever the concatenation \(\gamma _1\gamma _2\) is defined. Define the function \(v:X\rightarrow [0,\infty ]\) by
The function v is measurable by [22, Theorem 9.3.1] and satisfies \(v_{E}\equiv 0\). We also have
whenever \(y\in F\) and \(x\in E\).
For any \(\gamma \notin \Gamma _0\) by closedness under composition one has \(v(\gamma (0))=\infty \) iff \(v(\gamma (1))=\infty \). Furthermore if \(v(\gamma (0))\ne \infty \), then
So \(g \equiv 1\) is a pweak upper gradient of v.
If we had that \(v \in N^{1,p}(X)\), the SobolevtoLipschitz property and (3.2) would lead to a contradiction. A simple cut off argument remains to complete the proof.
Define \(w:X\rightarrow [0,\infty ]\) by
Then w has 1 as a upper gradient, \(w\in N^{1,p}(X)\) is compactly supported and for \(x\in E\), \(y \in F\) one has
Set \(u:=\min \{v,w\}\). Then \(u\in N^{1,p}(X)\) and \(g_u\le 1\). By the SobolevtoLipschitz property u has a 1Lipschitz representative. But for every \(x\in E\), \(y\in F\)
As \(\mu (E)\mu (F)>0\) these two observations lead to a contradiction. \(\square \)
Example 3.3
For \(p\in [1,\infty )\) let \(Z_p:=\{(x,y)\in \mathbb {R}^2:\ y\le x^p\}\) be endowed with the intrinsic length metric and Lebesgue measure. Then Z is qthick geodesic if and only if \(q>p+1\). This follows by Example 1 in [12]. Since \(Z_p\) is doubling, it also has the qSobolevtoLipschitz property if and only if \(q>p+1\).
4 Volume rigidity
We prove Proposition 1.8 using the fact that volume preserving 1Lipschitz maps between rectifiable spaces are essentially length preserving. We say that a map \(f:X\rightarrow Y\) between nrectifiable spaces is volume preserving, if \(f_*\mathcal {H}^n_X=\mathcal {H}^n_Y\).
Proposition 4.1
Let \(f:X\rightarrow Y\) volume preserving 1Lipschitz map between nrectifiable metric spaces. There exists a Borel set \(N\subset X\) with \(\mathcal {H}^n(N)=0\) so that for every absolutely continuous curve \(\gamma \) in X with \(\gamma ^{1}N=0\) we have
Proof
Let \(E_i\subset \mathbb {R}^n\) and \(g_i:E_i\rightarrow X\) be Borel sets and Lipschitz maps, respectively, satisfying \(\mathcal {H}^n(E)=0\), where
We may assume that the sets \(g_i(E_i)\) are pairwise disjoint. For each i, fix Lipschitz extensions \({\bar{g}}_i:\mathbb {R}^n\rightarrow l^\infty \) and \({\bar{f}}_i:\mathbb {R}^n\rightarrow l^\infty \) of \(g_i\) and \(f\circ g_i\), respectively (here we embed X and Y isometrically into \(l^\infty \)).
Lemma 4.2
Let \(i\in \mathbb {N}\). For almost every \(x\in E_i\), we have
Proof of Lemma 4.2
Since f is 1Lipschitz we have, at almost every point \(x\in E_i\) where both \({\text {md}}_x {\bar{g}}_i\) and \({\text {md}}_x{\bar{f}}_i\) exist, the inequality \({\text {md}}_x {\bar{f}}_i\le {\text {md}}_x {\bar{g}}_i\), which in particular implies \(J({\text {md}}_x{\bar{f}}_i)\le J({\text {md}}_x{\bar{g}}_i)\) for \(\mathcal {L}^n\)almost every \(x\in E_i\). By the area formula [25, Corollary 8] and the fact that f is volume preserving, it follows that
cf. Theorem 2.3. Consequently \(J({\text {md}}_x{\bar{g}}_i)= J({\text {md}}_x{\bar{f}}_i)\) for \(\mathcal {L}^n\)almost every \(x\in E_i\). Thus \({\text {md}}_x{\bar{g}}_i={\text {md}}_x{\bar{f}}_i\) for \(\mathcal {L}^n\)almost every \(x\in E_i\). \(\square \)
For each \(i\in \mathbb {N}\), let \(U_i\) denote the set of those density points x of \(E_i\) for which \({\text {md}}_x {\bar{g}}_i={\text {md}}_x{\bar{f}}_i\). Lemma 4.2 implies that \(\mathcal {L}^n(E_i\setminus U_i)=0\). Set
whence \(\mathcal {H}^n(N)=0\). Let \(\gamma :[0,1]\rightarrow X\) be a Lipschitz path. For each \(i\in \mathbb {N}\), denote \(K_i=\gamma ^{1}(g_i(U_i))\subset [0,1]\), and suppose \(\gamma _i:[0,1]\rightarrow \mathbb {R}^n\) is a Lipschitz extension of the “curve fragment” \(g_i^{1}\circ \gamma :K_i\rightarrow \mathbb {R}^n\).
Let \(t\in K_i\) be a density point of \(K_i\), for which \({{\dot{\gamma }}}_t\), \((f\circ \gamma )_t'\) and \(\gamma _i'(t)\) exist, and \(\gamma (t)\notin N\). Then \(\gamma _i(t)\in U_i\) and we have
The same argument with \({\bar{f}}_i\) in place of \({\bar{g}}_i\) yields
Since \(\gamma _i(t)\in U_i\), we have
Suppose now that \(\gamma :[0,1]\rightarrow X\) is a absolutely continuous path with \(\gamma ^{1}(N)=0\). We may assume that \(\gamma \) is constant speed parametrized. Since \(\gamma ^{1}(N)=0\) it follows that the union of the sets \(K_i\) over \(i\in \mathbb {N}\) has full measure in [0, 1]. Since, for each \(i\in \mathbb {N}\), a.e. \(K_i\) satisfies the conditions listed above, we may compute
\(\square \)
The proof yields the following corollary.
Corollary 4.3
Let \(f:X\rightarrow Y\) be a surjective 1Lipschitz map between rectifiable spaces with \(\mathcal {H}^n(X)=\mathcal {H}^n(Y)<\infty \). Then f is volume preserving and, in particular, \(\ell (f\circ \gamma )=\ell (\gamma )\) for \(\infty \)almost every \(\gamma \) in X.
We close this section with the proof of Proposition 1.8, which connects volume rigidity and the SobolevtoLipschitz property.
Proof of Proposition 1.8
Since \(f^{1}\) is quasiconformal, \(f^{1}\circ \gamma \) is rectifiable for nalmost every curve \(\gamma \) in Y. Proposition 4.1 implies that
Since Y has the nSobolevtoLipschitz property, it follows that \(f^{1}\) has a 1Lipschitz representative, cf. Lemma 3.1. By continuity of \(f^{1}\) it coincides with this representative. \(\square \)
The SobolevtoLipschitz property is crucial for the conclusion of Proposition 1.8, as the next example shows.
Example 4.4
Let \(Y=({\overline{\mathbb {D}}},d_w)\), where
and the metric \(d_w\) is given by,
The map \(\mathrm {id}:{\overline{\mathbb {D}}}\rightarrow Y\) is a volume preserving 1Lipschitz homeomorphism and the inverse is quasiconformal, but not Lipschitz continuous.
Indeed, since \([0,1]\times \{0\}\) has zero measure, we have \(J({\text {ap md}}\mathrm {id})=1\) almost everywhere, whence the area formula implies that \(\mathcal {H}^2({\overline{\mathbb {D}}})=\mathcal {H}^2_{d_w}({\overline{\mathbb {D}}})\).
5 Construction of essential pullback metrics
Let X be a metric measure space and let \({\overline{\Gamma }}(X)\) denote the set of Lipschitz curves \([0,1]\rightarrow X\). Recall that palmost every curve in X admits a Lipschitz reparametrization. In this section we construct essential pullback distances by Sobolev maps. The key notion here is the essential infimum of a functional over a path family.
Definition 5.1
Let \(p\ge 1\), \(F:{\overline{\Gamma }}(X)\rightarrow [\infty ,\infty ]\) be a function and \(\Gamma \subset {\overline{\Gamma }}(X)\) a path family. Define the pessential infimum of F over \(\Gamma \) by
with the usual convention \(\inf \varnothing =\infty \).
It is clear that the supremum may be taken over curve families \(\Gamma _\rho \) for nonnegative Borel functions \(\rho \in L^p(X)\), where
Remark 5.2
We have the following alternative expression for \({\text {essinf}}_p\):
Here
Let Y be a complete metric space, \(p\ge 1\), and \(u\in N^{1,p}_{loc}(X;Y)\). Given a path family \(\Gamma \) in X we set
Definition 5.3
Let \(u\in N^{1,p}_{loc}(X;Y)\). Define the essential pullback distance
\(\displaystyle d'_{u,p}:X\times X\rightarrow [0,\infty ]\) by
In general, the essential pullback distance may assume both values 0 and \(\infty \) for distinct points, and it need not satisfy the triangle inequality. We denote by \(d_{u,p}\) the maximal pseudometric not greater than \(d_{u,p}'\), given by
The maximal pseudometric below \(d_{u,p}'\) may also fail to be a finite valued. We give two situations that guarantee finiteness and nice properties of the arising metric space. Firstly, in Sect. 5.2 we consider the simple case \(u=\mathrm {id}:X\rightarrow X\) and use it to prove Theorem 1.4. Secondly, in Proposition 5.12 we prove that, when X supports a Poincaré inequality and \(u\in N^{1,p}_{loc}(X;Y)\) for large enough p, the distance in Definition 5.3 is a finite valued pseudometric.
For both we need the notion of regular curves, whose properties we study next.
5.1 Regular curves
Throughout this subsection \((X,d,\mu )\) is a metric measure space. We define a metric on \({\overline{\Gamma }}(X)\) by setting
for any two Lipschitz curves \(\alpha ,\beta \). By a simple application of the ArzelaAscoli theorem it follows that \(({\overline{\Gamma }}(X),d_\infty )\) is separable.
Definition 5.4
A curve \(\gamma \in {\overline{\Gamma }} (X)\) is called (u, p)regular if \(u\circ \gamma \) is absolutely continuous and
for every \(\delta >0\).
Proposition 5.5
palmost every Lipschitz curve is (u, p)regular.
For the proof we will denote \(\Gamma _{u,p}(\lambda )=\{\gamma : \ell (u\circ \gamma )<\lambda \}\).
Proof of Proposition 5.5
Denote by \(\Gamma _0\) the set of curves in \(\overline{\Gamma }(X)\) which are not (u, p)regular. We may write \(\Gamma _0=\Gamma _1\cup \Gamma _2\), where
By the fact that \(u\in N^{1,p}_{loc}(X;Y)\) we have \({\text {Mod}}_p(\Gamma _1)=0\). It remains to show that \({\text {Mod}}_p(\Gamma _2)=0\). For any \(\delta >0\) and \(\gamma \in {\overline{\Gamma }}(X)\), set
Note that \(\delta (\gamma )>0\) if and only if \(\gamma \in \Gamma _2\). Using Remark 5.2 we make the following observation which holds for \(\delta ,\varepsilon >0\):
Moreover, for any \(\gamma \in \Gamma _2\) and \(\delta >0\), we have
Let \(\{\gamma _i \}_{i\in \mathbb {N}}\subset \Gamma _2\) be a countable dense set. For each \(i,k\in \mathbb {N}\) and rational \(r>0\) let \(\gamma _{i,k,r}\in B(\gamma _i,r)\cap \Gamma _2\) satisfy
By (5.2) it suffices to prove that
For any \(\gamma \in \Gamma _2\) let \(i\in \mathbb {N}\) be such that \(d_{\infty }(\gamma ,\gamma _i)<\delta (\gamma )/8\). Choose rational numbers \(r,\delta \in \mathbb {Q}_+\) such that \(d_\infty (\gamma _i,\gamma )<r<\delta (\gamma )/8\) and \(\delta (\gamma )/4<\delta <\delta (\gamma )/2\), and a natural number \(k\in \mathbb {N}\) so that \(1/k<\varepsilon (2\delta ,\gamma )\).
We will show that
Indeed, since \(\gamma _{i,k,r}\in B(\gamma _i,r)\cap \Gamma _2\), the triangle inequality yields
In particular
and also
To bound the length of \(u\circ \gamma \), observe that
Setting \(\varepsilon :=\ell (u\circ \gamma )+1/k\ell (u\circ \gamma _{i,k,r})>0\) it follows that
Combining (5.5) and (5.6) with (5.2) we obtain
which, by (5.1) yields \(\varepsilon \le \varepsilon (\delta ,\gamma _{i,k,r})\). Thus
which, together with (5.4), implies (5.3). This completes the proof. \(\square \)
For the next two results we assume that \(u\in N^{1,p}_{loc}(X;Y)\) is continuous. We record the following straightforward consequence of the definition of \(d_{u,p}'\) and the continuity of u as a lemma.
Lemma 5.6
For each \(x,y\in X\) we have
For the next proposition, we denote by \(\ell _{u,p}(\gamma )\) the length of a curve \(\gamma \) with respect to the pseudometric \(d_{u,p}\).
Proposition 5.7
If \(\gamma :[a,b]\rightarrow X\) is (u, p)regular and \(a\le t\le s\le b\) then \(\gamma _{[t,s]}\) is (u, p)regular and
Proof
Denote \(\eta :=\gamma _{[s,t]}\). Let
Since \(\gamma \) is (u, p)regular we have
We claim that for every \(\varepsilon >0\) there exists \(n_0\in \mathbb {N}\) so that
for all \(n\ge n_0\). Indeed, otherwise there exists \(\varepsilon _0>0\) and a sequence
so that
Thus
yielding
By taking \(\liminf _{k\rightarrow \infty }\) in (5.8) we obtain
which is a contradiction.
Thus (5.7) holds true. If \(\varepsilon ,\delta >0\), let \(n\in \mathbb {N}\) be such that (5.7) holds and \(\delta >1/n\). We have
implying \(ess\ell _{u,p}B(\eta ,\delta )\le \ell (u\circ \eta )+\varepsilon \). Since \(\varepsilon >0\) is arbitrary, \(\eta \) is (u, p)regular.
To prove the equality in the claim note that, since \(\gamma \) is (u, p)regular we have
for any \(s\le t\). It follows that
On the other hand Lemma 5.6 implies that
whenever \(s\le t\), from which the opposite inequality readily follows. \(\square \)
5.2 The SobolevtoLipschitz property in thick quasiconvex spaces
In this subsection let \(p\in [1,\infty ]\) and \(X=(X,d,\mu )\) be pthick quasiconvex with constant \(C\ge 1\). Consider the map \(u=\mathrm {id}\in N^{1,p}_{loc}(X;X)\). We denote by \(d_p\) the pseudometric \(d_{u,p}\) associated to u.
Lemma 5.8
\(d_p\) is a metric on X, and satisfies \(d\le d_p\le Cd\).
Proof
Let \(x,y\in X\) be distinct and \(\delta >0\). Since
it follows that
implying \(d_{\mathrm {id},p}'(x,y)\le Cd(x,y)\). Thus \(d_p\le Cd\) and in particular \(d_p\) is finitevalued. Lemma 5.6 implies that \(d(x,y)\le d_p(x,y)\) for every \(x,y\in X\). These estimates together prove the claim. \(\square \)
We are now ready to prove Theorem 1.4. For the proof, we denote by \(X_p\) the space \((X,d_p,\mu )\) and by \(B_p(x,r)\) balls in X with respect to the metric \(d_p\); \(\ell _p\) and \({\text {Mod}}_{X_p,p}\) refer to the length of curves and pmodulus taken with respect to \(X_p\).
Proof of Theorem 1.4
By Lemma 5.8 it follows that \(X_p\) is an infinitesimally doubling metric measure space and \({\text {Mod}}_{X_p,p}\Gamma =0\) if and only if \({\text {Mod}}_p\Gamma =0\). In particular if \(\Gamma _*\) denotes the set of curves \(\gamma \) in X such that \(\ell (\gamma )\ne \ell _p(\gamma )\) then by Propositions 5.5 and 5.7
From (5.9) and the definition of modulus it follows that \({\text {Mod}}_{X_p,p}\Gamma = {\text {Mod}}_p\Gamma \) for every family of curves \(\Gamma \) in X.
By (5.9) and Proposition 3.2 every \(f\in N^{1,p}(X_p)\) with \(g_f\le 1\) has a CLipschitz representative. If \(x,y\in X\) are distinct and \(\delta , \varepsilon >0\) note that the curve family
satisfies
by Remark 5.2 and the fact that \(\varepsilon >0\). Note also that
for \(\gamma \in \Gamma _1\). Let \(f\in N^{1,p}(X_p)\) satisfy \(g_f\le 1\) almost everywhere. Let \({\bar{f}}\) be the Lipschitz representative of f, and \(\Gamma _0\) a curve family with \({\text {Mod}}_{p}\Gamma _0=0\) and
whenever \(\gamma \notin \Gamma _0\). We have
so that there exists \(\gamma _\delta \in \Gamma _1\setminus \Gamma _0\). We obtain
Letting \(\delta \rightarrow 0\) yields \({\bar{f}}(x){\bar{f}}(y)\le d_{\mathrm {id},p}'(x,y)+\varepsilon \). Since \(x,y\in X\) and \(\varepsilon >0\) are arbitrary it follows that \({\bar{f}}\) is 1Lipschitz with respect to \(d_p\). In particular \(X_p\) has the pSobolevtoLipschitz property and hence Theorem 1.7 implies it is pthick geodesic. We prove the minimality in Proposition 5.9. \(\square \)
The metric \(d_p\) is the minimal metric above d which has the pSobolevtoLipschitz property. Proposition 5.9 provides a more general statement, from which the minimality discussed in the introduction immediately follows.
Proposition 5.9
Let Y be a pthick geodesic metric measure space and \(f:Y\rightarrow X\) be a volume preserving 1Lipschitz map. Then the map
is volume preserving and 1Lipschitz.
Proof
It suffices to show that
For any \(A>1\), the curve family
has positive pmodulus in Y. Since f is volume preserving and 1Lipschitz we have
If \(\gamma \in \Gamma \) then \(f\circ \gamma \in \Gamma (B(f(x),\delta ),B(f(y),\delta ))\) and, moreover
It follows that \(ess\ell _{\mathrm {id},p}\Gamma (B(f(x),\delta ),B(f(y),\delta ))\le Ad(x,y)+2A\delta \), and thus
Since \(A>1\) and \(\delta >0\) are arbitrary, the claim follows. \(\square \)
We have the following immediate corollary.
Corollary 5.10
Assume X is infinitesimally doubling and pthick geodesic. Then \(d=d_p\).
Before considering essential pullback metrics by nontrivial maps, we prove Proposition 1.5. The proof is based on the fact that spaces with Poincaré inequality are thick quasiconvex, and on the independence of the minimal weak upper gradient on the exponent.
Proposition 5.11
([12]) Let \(p\in [1,\infty ]\) and X be a doubling metric measure space satisfying a pPoincaré inequality. There is a constant \(C\ge 1\) depending only on the data of X so that X is pthick quasiconvex with constant C.
The inverse implication in Proposition 5.11 only holds if \(p=\infty \), see [11, 12].
Proof of Proposition 1.5
Assume X is a doubling metric measure space supporting a pPoincaré inequality, and \(q\ge p\ge 1\). Since \({\text {Mod}}_q\Gamma =0\) implies \({\text {Mod}}_p\Gamma =0\), see [2, Proposition 2.45], we have
for some constant C depending only on p and the data of X.
Fix \(x_0\in X\) and consider the function
Then f is Lipschitz and, by Propositions 5.5 and 5.7, it has 1 as a pweak upper gradient. Since X is doubling and supports a pPoincaré inequality, [2, Corollary A.8] implies that the minimal qweak and pweak upper gradients of f agree almost everywhere, and thus 1 is a qweak upper gradient of f, i.e.
for qalmost every curve. The space \(X_q\) has the qSobolevtoLipschitz property, see Theorem 1.7, and the 1Lipschitz representative of f agrees with f everywhere, since f is continuous. By this and the definition of \(d_q\) we obtain
Since \(x_0\in X\) is arbitrary the equality \(d_p=d_q\) follows.
For the remaining equality, note that \(d\le d_\infty \le d_{ess}\le Cd\). Indeed, X supports an \(\infty \)Poincaré inequality, whence [11, Theorem 3.1] implies the rightmost estimate with a constant C depending only on the data of the \(\infty \)Poincaré inequality. As above, the function
satisfies
for \(\infty \)almost every \(\gamma \), from which the inequality \(d_{ess}\le d_\infty \) follows. \(\square \)
Note that in the proof of Proposition 5.11 we use that in pPoincaré spaces the qweak upper gradient does not depend on \(q\ge p\). In general such an equality is not true, see [9], and we do not know whether we can weaken the assumptions in Proposition 1.5 from pPoincaré inequality to pthick quasiconvexity.
5.3 Essential pullback metrics by Sobolev maps
Throughout this subsection \((X,d,\mu )\) will be a doubling metric measure space satisfying (2.2) with \(Q\ge 1\) and supporting a weak (1, Q)Poincaré inequality, and \(Y=(Y,d)\) a proper metric space.
We will use the following observation without further mention. If \(p>Q\), and \(u:X\rightarrow Y\) has a Qweak upper gradient in \(L^p_\mathrm {loc}(X)\), then u has a representative \({\bar{u}}\in N^{1,p}_{loc}(X;Y)\) and the minimal pweak upper gradient of \({\bar{u}}\) coincides with the minimal Qweak upper gradient of u almost everywhere. See [2, Chapter 2.9 and Appendix A] and [22, Chapter 13.5] for more details.
The next proposition states that higher regularity of a map is enough to guarantee that the essential pullback distance in Definition 5.3 is a finitevalued pseudometric.
Proposition 5.12
Let \(p>Q\), and suppose \(u\in N^{1,p}_{loc}(X;Y)\). The pullback distance \(d_u:=d_{u,Q}'\) in Definition 5.3 is a pseudometric satisfying
whenever \(B\subset X\) is a ball and \(x,y\in B\), where the constant C depends only on p and the data of X.
For the proof of Proposition 5.12 we define the following auxiliary functions. Let \(x\in X\), \(\delta >0\) and \(\Gamma _0\) a curve family with \({\text {Mod}}_Q\Gamma _0=0\). Set \(f:=f_{x,\delta ,\Gamma _0}:X\rightarrow \mathbb {R}\) by
When \(\rho \in L^Q(X)\) is a nonnegative Borel function, whence \({\text {Mod}}_Q(\Gamma _\rho )=0\), we denote \(f_{x,\delta ,\rho }:=f_{x,\delta ,\Gamma _\rho }\).
Lemma 5.13
Let \(x,\delta \) and \(\rho \) be as above. The function \(f=f_{x,\delta ,\rho }\) is finite \(\mu \)almost everywhere and has a representative in \(N^{1,p}_{loc}(X)\) with pweak upper gradient \(g_u\). The continuous representative \({\bar{f}}\) of f satisfies
Proof
Let \(g\in L^p_{\mathrm {loc}}(X)\) be a genuine upper gradient of u and let \(\varepsilon >0\) be arbitrary. We fix a large ball \(B\subset X\) containing \({\bar{B}}(x,\delta )\) and note that there exists \(x_0\in {\bar{B}}(x,\delta )\) for which \(\mathcal {M}_B(g+\rho )^Q(x_0)<\infty \), since \((g+\rho )_B\in L^Q(B)\); cf. (2.3). Arguing as in [41, Lemma 4.6] we have that
for almost every \(y\in B\).
Let \(\gamma \notin \Gamma _\rho \) be a curve such that \(f(\gamma (1)),f(\gamma (0))<\infty \) and \(\int _\gamma g<\infty \). We may assume that \(f(\gamma (1))f(\gamma (0))=f(\gamma (1))f(\gamma (0))\ge 0\). If \(\beta \) is an element of \({\bar{\Gamma }}(B(x,\delta ),\gamma (0))\setminus \Gamma _\rho \) such that \(\ell (u\circ \beta )<f(\gamma (0))+\varepsilon \) then the concatenation \(\gamma \beta \) satisfies \(\gamma \beta \in \Gamma ({\bar{B}}(x,\delta ),\gamma (1))\setminus \Gamma _\rho \). Thus
It follows that \(g\in L^p_\mathrm {loc}(X)\) is a Qweak upper gradient for f. By [2, Corollary 1.70] we have that \(f(y)<\infty \) for Qquasievery \(y\in B\), and \(f\in N^{1,Q}_{loc}(X)\); cf. [22, Theorem 9.3.4].
Moreover, since \(g\in L^p_\mathrm {loc}(X)\), f has a continuous representative \({\bar{f}}\in N^{1,p}_{loc}(X)\) which satisfies (5.11), cf. [22, Theorem 9.2.14]. \(\square \)
Proof of Proposition 5.12
To prove the triangle inequality, let \(x,y,z\in X\) be distinct. Take \(\delta >0\) small, \(\rho \in L^Q(X)\) nonnegative, and let \(E\subset X\) be a set of Qcapacity zero such that \(f_{x,\delta ,\Gamma _\rho }\) and \(f_{y,\delta ,\Gamma _\rho }\) agree with their continuous representatives outside E. Remember that \(\alpha \beta \notin \Gamma _\rho \) whenever \(\alpha ,\beta \notin \Gamma _\rho \). We have that
Together with the estimate (5.11) this yields
where C depends on u, x and y as well as the data. Since \(\delta >0\) and \(\rho \) are arbitrary it follows that
Moreover, for any \(y'\in {\bar{B}}(y,\delta )\setminus E\),
Taking supremum over \(\rho \), and letting \(\delta \) tend to zero, we obtain (5.10). \(\square \)
Remark 5.14
A slight variation of the proof of Proposition 5.12 shows that also in the setting of Theorem 1.4 the essential pullback distance defines a metric. In particular it satisfies the triangle inequality and one does not have to pass to the maximal semimetric below. In this case in the argument instead of the Morrey embedding one applies Proposition 3.2.
Fix a continuous map \(u\in N^{1,p}_{loc}(X;Y)\), where \(p>Q\), and denote by \(Y_u\) the set of equivalence classes [x] of points \(x\in X\), where x and y are set to be equivalent if \(d_{u,Q}'(x,y)=0\). The pseudometric \(d_{u,Q}'\) defines a metric \(d_u\) on \(Y_u\) by
The natural projection map
is continuous by (5.10). The map u factors as \(u={\widehat{u}}\circ {\widehat{P}}_u\), where
is welldefined and 1Lipschitz; cf. Lemma 5.6.
Lemma 5.15
Under the given assumptions we obtain the following properties.

(1)
\({\widehat{P}}_u\in N^{1,p}_{loc}(X;Y_u)\) and \(g_{{\widehat{P}}_u}=g_u\) \(\mu \)almost everywhere.

(2)
If u is proper then \(Y_u\) is proper and the projection \({\widehat{P}}_u:X\rightarrow Y_u\) is proper and monotone.
Recall here that a map is called proper if the preimage of every compact set is compact or equivalently if the preimage of every singleton is compact.
Proof
(1) Suppose \(\gamma \) is a (u, Q)regular curve such that \(g_u\) is an upper gradient of u along \(\gamma \). Then by Proposition 5.7
By Proposition 5.5 this implies that \({\widehat{P}}_u\in N^{1,Q}_{loc}(X;Y_u)\) and \(g_{{\widehat{P}}_u}\le g_u\). The opposite inequality follows because \(\widehat{u}\) is 1Lipschitz.
(2) The factorization implies that \(K\subset {\widehat{P}}_u( u^{1}(\overline{{\widehat{u}}(K)}))\), for \(K\subset Y_u\) and hence \(Y_u\) is proper. Similarly \({\widehat{P}}_u^{1}(K)\subset u^{1}(\widehat{u}(K))\) and hence \(\widehat{P}_u\) is proper. The proof of monotonicity is an adaptation of the proof of [33, Lemma 6.3].
Assume \({\widehat{P}}_u^{1}(y)\) is not connected for some \(y\in Y_u\). Then there are compact sets \(K_1,K_2\) for which \({\text {dist}}(K_1,K_2)>0\) and \({\widehat{P}}_u^{1}(y)=K_1\cup K_2\). Let S denote the closed and nonempty set of points X whose distance to \(K_1\) and \(K_2\) agree. Let a be the minimum of \(x\mapsto {\text {dist}}_u(y,\widehat{P}_u(x))\) on S. Note here that \(\widehat{P}_u(S)\) is closed as \(Y_u\) and \(\widehat{P}_u\) are proper and hence the infimum is attained and positive. Let \(k_i\in K_i\) for \(i=1,2\). Since \(d_{u,Q}'(k_1,k_2)=0\), for every \(\varepsilon >0\) and small enough \(\delta >0\), Proposition 5.5 implies the existence of a (u, Q)regular curve \(\gamma \in \Gamma (B(k_1,\delta ),B(k_2,\delta ))\) with \(\ell (u\circ \gamma )<\varepsilon \). If \(\delta \) is chosen small enough the curve must intersect S at some point \(s:=\gamma (t)\). Since \(\gamma \) is (u, Q)regular, \(\gamma _{[0,t]}\) is (u, Q)regular and it follows that
cf. Proposition 5.7 and (5.10). Choosing \(\varepsilon >0\) and \(\delta >0\) small enough this yields a contradiction. Thus \({\widehat{P}}_u^{1}(y)\) is connected for every \(y\in Y_u\). \(\square \)
Assume additionally that Y is endowed with a measure \(\nu \) such that \(Y=(Y,d,\nu )\) is a metric measure space. We equip \((Y_u,d_u)\) with the measure \(\nu _u:={\widehat{u}}^*\nu \), which is characterized by the property
cf. [13, Theorem 2.10.10]. Note that in general \(\nu _u\) is not a \(\sigma \)finite measure. For the next theorem, we say that a Borel map \(u:X\rightarrow Y\) has Jacobian Ju, if there exists a Borel function \(Ju:X\rightarrow [0,\infty ]\) for which
holds for every Borel set \(E\subset X\). The Jacobian Ju, if it exists, is unique up to sets of \(\mu \)measure zero.
Theorem 5.16
Assume u is proper, nonconstant and has a locally integrable Jacobian Ju which satisfies
\(\mu \)almost everywhere for some \(K\ge 1\). Then the following properties hold.

(1)
\(Y_u\) is a metric measure space and has the QSobolevtoLipschitz property.

(2)
Ju is the locally integrable Jacobian of \(\widehat{P}_u\) and \(\#{\widehat{P}}_u^{1}(y)=1\) for \(\nu _u\)almost every \(y\in Y_u\).
Proof
Note that
for every Borel set \(E\subset Y_u\). Thus \(\nu _u\) is a locally finite measure, and the estimate above implies that \({\widehat{P}}_u\) has a locally integrable Jacobian \(J{\widehat{P}}_u\le Ju\). For any Borel set \(E\subset X\) we have
which yields \(J\widehat{P}_u=J u\) almost everywhere. Since \(\widehat{P}_u\) is monotone and satisfies (5.13) we have that \(\#P_u^{1}(y)=1\) for \(\nu _u\)almost every \(y\in Y_u\).
By Lemma 5.15 to see that \(Y_u\) is a metric measure space it remains to show that balls in \(Y_u\) have positive measure. The idea of of proof is borrowed from [33, Lemma 6.11]. Assume \(B=B(z,r)\) is a ball in \(Y_u\) such that \(\nu _u(B)=0\). Then we would have \(J {\widehat{P}}_u=J_u=0\) almost everywhere on \(U:=\widehat{P}_u^{1}(B)\). By (5.14) and Lemma 5.15\(g_{\widehat{P}_u}=g_u=0\) almost everywhere on U. Thus \(\widehat{P}_u\) is locally constant on U. But U is connected and hence \(\widehat{P}_u\) is constant.
To see that U is connected let \(x\in U\) satisfy \(\widehat{P}_u(x)=z\). Then, for \(y\in U\) and \(\delta >0\) arbitrary, there is a (u, Q)regular curve \(\gamma \in \Gamma (B(x,\delta ),B(y,\delta ))\) such that \(\ell (u\circ \gamma )<r\). For \(\delta \) small enough by Proposition 5.7 the image of \(\gamma \) is contained in U. As this holds for all small enough \(\delta \), the points x and y must lie in the same connected component of the open set U. Since y was arbitrary, U must be connected.
We have deduced that \(B=\widehat{P}_u(U)\) consists only of the single point z. So either \(Y_u\) is disconnected or consists of a single point. The former is impossible because X is connected and the later by the assumption that u is nonconstant.
Before showing the SobolevtoLipschitz property we first claim
for every curve family \(\Gamma \) in X. Indeed, suppose \(\rho \in L^Q(Y_u)\) is admissible for \({\widehat{P}}_u(\Gamma )\), and set \(\rho _1=\rho \circ {\widehat{P}}_u\). Let \(\Gamma _0\) a curve family with \({\text {Mod}}_Q\Gamma _0=0\) for which \(\gamma \) is \(({\widehat{P}}_u,Q)\) and (u, Q)regular, and \(g_u\) is an upper gradient of \({\widehat{P}}_u\) and u along \(\gamma \), whenever \(\gamma \notin \Gamma _0\). For any \(\gamma \in \Gamma \setminus \Gamma _0\) we have
i.e. \(\rho _1g_u\) is admissible for \(\Gamma \setminus \Gamma _0\). We obtain
The last equality follows, since Lemma 5.15 implies \(\widehat{P}_{u*}(Ju\ \mu )=\nu _u\). Taking infimum over admissible \(\rho \) yields (5.15).
We prove that \(Y_u\) has the QSobolevtoLipschitz property. Suppose \(f\in N^{1,Q}(Y_u)\) satisfies \(g_f\le 1\). There is a curve family \(\Gamma _0\) in \(Y_u\) with \({\text {Mod}}_{Q}(\Gamma _0)=0\) such that
whenever \(\gamma \notin \Gamma _0\). By (5.15) we have
Here \({\widehat{P}}_u^{1}\Gamma \) denotes the family of curves \(\gamma \) in X such that \({\widehat{P}}_u\circ \gamma \in \Gamma _0\). Together with Propositions 5.5 and 5.7 this implies that, for \({\text {Mod}}_{Q}\)almost every curve \(\gamma \) in X, we have
Since \(g_u\in L^p_{\mathrm {loc}}(X)\) for \(p>Q\), it follows that \(f\circ {\widehat{P}}_u\) has a continuous representative \({\bar{f}}\). Let \(x,y\in X\) be distinct, \(\varepsilon >0\) arbitrary, and \(\delta >0\) such that
whenever \(z\in {\bar{B}}(x,\delta )\) and \(w\in {\bar{B}}(y,\delta )\). Denote by \(\Gamma _1\) the curve family with \({\text {Mod}}_{X,Q}\Gamma _1=0\) so that
Then
Since \(x,y\in X\) and \(\varepsilon >0\) are arbitrary it follows that \({\bar{f}}(x)={\bar{f}}(y)\) whenever \(d_{u,Q}'(x,y)=0\) and that the map \([x]\mapsto {\bar{f}}(x)\) is 1Lipschitz with respect to the metric \(d_u\), and is a \(\nu _u\)representative of f. \(\square \)
6 Essential metrics on surfaces
In this section we apply the constructions from Sect. 5 to prove Theorems 1.2 and 1.9.
6.1 Ahlfors regular spheres
In what follows, Z is an Ahlfors 2regular metric sphere which is linearly locally connected. We consider Z as endowed with the Hausdorff 2measure \(\mathcal {H}^2_Z\). Denote by \(\Lambda (Z)\) the set of celllike maps \(u\in N^{1,2}(S^2,Z)\). Recall that a continuous map is called celllike if the preimage of every point is celllike. The definition of general celllike sets may be found in [33] but for our purposes it suffices to know that a compact subset K of \(S^2\) is celllike if and only if K and \(S^2\setminus K\) are connected. Let us recall the following result from [35].
Theorem 6.1
( [35]) The set \(\Lambda (Z)\) is nonempty and contains an element \(u\in \Lambda (Z)\) of least Reshetnyak energy \(E^2_+(u)\). Such u is a quasisymmetric homeomorphism and unique up to a conformal diffeomorphism of \(S^2\).
In the following we fix such energy minimzing quasisymmetric homeomorphism \(u\in N^{1,2}(S^2,Z)\). Then Z supports a 2Poincaré inequality, and u and \(u^{1}\) are quasiconformal; cf. [21, Corollary 8.15 and Theorem 9.8]. Moreover, by [26, Theorem 2.9 and Corollary 4.20], we have that \(u\in N^{1,p}(S^2;Z)\) for some \(p>2\).
We denote by \(\widehat{Z}=(Z,d_2)\) the metric space as constructed in Sect. 5.2 and by \(\widehat{Z}_u=(S^2,d_{u,2})\) the metric space as constructed in Sect. 5.3. It turns out that in fact the two constructions coincide.
Lemma 6.2
The map \(\widehat{Z}_u\rightarrow \widehat{Z}\) induced by u defines an isometry.
Proof
Since u is a homeomorphism, the claim follows immediately from the construction and the fact that
for every family of curves \(\Gamma \) in \(S^2\). \(\square \)
In the following we identify \(\widehat{Z}\) and \(\widehat{Z}_u\) under this isometry. Thus we have a canonical factorization \(u=\widehat{u}\circ \widehat{P}_u\) where \(\widehat{P}_u:S^2\rightarrow \widehat{Z}\) is a homeomorphism and \(\widehat{u}:\widehat{Z}\rightarrow Z\) a 1Lipschitz homeomorphism.
Note that Z as supports a 2Poincaré in particular it is 2thick quasiconvex. By Theorem 1.4 the space \(\widehat{Z}\) is biLipschitz equivalent to Z and hence itself an Ahlfors 2regular metric sphere which is linearly locally connected.
Another crucial observation is the following. If Y is another complete metric space and \(v\in N^{1,2}(S^2,Y)\), then
holds almost everywhere on \(S^2\) if and only if
holds for 2almost every curve \(\gamma \) in \(S^2\). In particular \(Z_u=Y_v\) if either of the two conditions holds, see [33, Lemma 3.1 and Corollary 3.2].
Proof of Theorem 1.2
By Theorem 5.15 we have that \(\widehat{P}_u\in N^{1,p}(S^2,\widehat{Z})\). By Proposition 5.5 and Proposition 5.7, 2almost every curve \(\gamma \) in \(S^2\) satisfies \(\ell (u\circ \gamma )=\ell (\widehat{P}_u\circ \gamma )\). In particular
holds almost everywhere on \(S^2\). Assume \(v\in \Lambda (\widehat{Z})\) is such that \(E^2_+(v)<E^2_+(\widehat{P}_u)\). Then \(\widehat{u}\circ v \in \Lambda (Z)\) and
This contradicts the assumption that u is of minimal energy and hence \(\widehat{P}_u\) is energy minimizing in \(\Lambda (\widehat{Z})\). By (6.1) we obtain that \(\widehat{Z}\) is analytically equivalent to Z.
By Theorem 1.4 we know that \(\widehat{Z}\) is thick geodesic with respect to \(\mathcal {H}^2_Z\). We want thick geodecity however to hold with respect to \(\mathcal {H}^2_{\widehat{Z}}\). In particular to see property (2) it suffices to show that
Equality (6.3) follows however immediately from (6.1), Theorem 2.3 and the fact that \(\widehat{P}_u\) and \(\widehat{u}\) are homeomorphisms which satisfy higher integrability.
Since \(\widehat{Z}_u\) is Ahlfors 2regular, Theorem 1.7 implies that \(\widehat{Z}\) has the 2SobolevtoLipschitz property. (We could derive this also directly from Theorem 5.16 by showing \(\nu _u=\mathcal {H}^2_{\widehat{Z}}\) where \(\nu =\mathcal {H}^2_Z\).)
Let Y be analytically equivalent to \(\widehat{Z}\) and hence also to Z. Let \(v\in \Lambda (Y)\) be the energy minimizing homeomorphism. Then \(Y_v=\widehat{Z}_u=\widehat{Z}\) and hence the desired surjective 1Lipschitz homeomorphism \(\widehat{Z}\rightarrow Y\) is given by \(f=\widehat{v}\). This proves (3).
Next we prove (4), i.e. volume rigidity. Let Y be an Ahlfors 2regular metric sphere which is linearly locally connected. Suppose \(f:Y\rightarrow \widehat{Z}\) is 1Lipschitz, celllike and volume preserving with respect to \(\mathcal {H}^2_Y\) and \(\mathcal {H}^2_Z\). Let \(v\in \Lambda (Y)\) be an energy minimizer. Then v is a quasisymmetric homeomorphism, and
Thus \(f\circ v\) is an area minimizer in \(\Lambda ({\widehat{Z}})\) and \({\text {ap md}}f\circ v ={\text {ap md}}v\) almost everywhere. Since v is infinitesimally quasiconformal, see [35, Theorem 6.6], the same holds for \(f\circ v\). By the proof of [35, Theorem 6.3] this shows that \(f\circ v\) is quasisymmetric. It follows that f is quasisymmetric and thus \(f^{1}\) is quasiconformal. Proposition 1.8 implies that f is an isometry.
It remains to check that \(\widehat{Z}\) is characterized uniquely by any of the listed properties. For the forthcoming discussion, we fix a metric sphere Y which is analytically and biLipschitz equivalent to Z. We moreover fix an energy minimal homeomorphism \(v:S^2 \rightarrow Y\) such that \({\text {ap md}}u={\text {ap md}}v\) almost everywhere.
If Y satisfies (1) (resp. (2)), then by Theorem 1.7 it satisfies (2) (resp. (1)) and, by Theorem 1.4 (see Corollary 5.10), we have that \({\widehat{Y}}=Y\). But under the usual identifications \(\widehat{Y}=\widehat{Y}_v=\widehat{Z}_u=\widehat{Z}\) and hence Y is isometric to \(\widehat{Z}\).
If Y satisfies (3), then there are surjective 1Lipschitz maps \(f:Y\rightarrow \widehat{Z}\) and \(g:\widehat{Z}\rightarrow Y\). Since Y and \(\widehat{Z}\) are compact, the composition \(f\circ g\) is an isometry, see [4, Theorem 1.6.15]. By the fact that f is 1Lipschitz it follows that g is an isometry.
Assume Y satisfies (4). The canonical map \({\widehat{Y}}\rightarrow Y\) is 1Lipschitz and volume preserving. In particular by volume rigidity \(\widehat{Y}\) is isometric to Y. However the former is isometric to \(\widehat{Z}\). This concludes the proof. \(\square \)
6.2 Metric discs
In this subsection let Z be a geodesic metric disc such that \((Z,\mathcal {H}^2_Z)\) is a metric measure space. First we discuss the following weaker variant of Theorem 1.2 in the disc setting.
Theorem 6.3
Assume \(u\in \Lambda (\partial Z,Z)\) is a minimizer of \(E^2_+\) which is moreover monotone and lies in \(N^{1,p}(\overline{\mathbb {D}},Z)\) for some \(p>2\). Then there is a geodesic metric disc \(\widehat{Z}_u\) and a factorization \(u=\widehat{u}\circ \widehat{P}_u\) such that:

(1)
\(\widehat{Z}_u\) satisfies the SobolevtoLipschitz property.

(2)
\(\widehat{P}_u\in \Lambda (\partial {\widehat{Z}}_u,{\widehat{Z}}_u)\) is a minimizer of \(E^2_+\), contained in \(N^{1,p}({\overline{\mathbb {D}}},\widehat{Z}_u)\), and a uniform limit of homeomorphisms.

(3)
For 2almost every curve \(\gamma \) in \({\overline{\mathbb {D}}}\) one has \(\ell (\widehat{P}_u\circ \gamma )=\ell (u\circ \gamma )\).

(4)
If Y is a disc, and \(v\in \Lambda (\partial Y,Y)\) satisfies \({\text {ap md}}u={\text {ap md}}v\) almost everywhere, then there is a surjective 1Lipschitz map \(j:\widehat{Z}\rightarrow Y\) such that \(v=j\circ \widehat{P}_u\).

(5)
If Z satisfies a \((C,l_0)\)quadratic isoperimetric inequality then \(\widehat{Z}_u\) satisfies a \((C,l_0)\)quadratic isoperimetric inequality.
Note that the existence of such u is guaranteed if Z satisfies a quadratic isoperimetric inequality and \(\partial Z\) satisfies a chordarc condition, see [31, 35].
Proof
Let \(\widehat{Z}_u=(\overline{\mathbb {D}},d_u)\) be the space constructed as in Sect. 5.3 and \(u=\widehat{u}\circ \widehat{P}_u\) be the associated factorization as discussed therein. Then as in the previous subsection we see that \({\text {ap md}}u={\text {ap md}}\widehat{P}_u\) holds almost everywhere and hence in particular \(\ell (u\circ \gamma )=\ell (\widehat{P}_u\circ \gamma )\) for 2almost every curve \(\gamma \) in \(\overline{\mathbb {D}}\). As in the previous subsection this implies that u and \(\widehat{P}_u\) are infinitesimally quasiconformal.
Next we show that \({\widehat{Z}}_u\) is a metric disc, and that \({\widehat{P}}_u\in \Lambda (\partial {\widehat{Z}}_u,{\widehat{Z}}_u)\) is a uniform limit of homeomorphisms. By [33, Corollary 7.12], to see this it suffices to prove that \(\widehat{P}_u\) is celllike and its restriction \(S^1\rightarrow \widehat{P}_u(S^1)\) is monotone. Since by Lemma 5.15\(\widehat{P}_u\) is monotone, the proof that \(\widehat{P}_u\) is celllike follows by the same argument as in the proof of [33, Theorem 8.1]. Thus it suffices to show that \({\widehat{P}}_u_{S^1}:S^1\rightarrow {\widehat{P}}_u(S^1)\) is monotone, i.e that \(\widehat{P}_u^{1}(y) \cap S^1\) is connected for every \(y\in \widehat{Z}_u\).
We identify \(\overline{\mathbb {D}}\) with the lower hemisphere of \(S^2\). Since \(\widehat{P}_u\) is celllike, the set \(K:=\widehat{P}_u^{1}(y)\) is a celllike subset of \(S^2\) and in particular K and \(S^2\setminus K\) are connected. Assume \(K\cap S^1\) is not connected. Then there exist \(z,w \in S^1\) such that neither of the of the arcs connecting z and w is contained in K. However \(u(z)=u(w)\in \partial Z\). Then since \(u\in \Lambda (\partial Z,Z)\), one of the arcs \(A \subset S^1\) connecting z and w gets mapped constantly to u(w). Set \(M:=K\cup A\). Clearly M is connected. We claim that \(S^2\setminus M\) is also connected. Assume it was not, and let O be a connected component of \(S^2 \setminus M\) contained in \(\overline{\mathbb {D}}\). Then as in the proof of [33, Theorem 8.1], the restriction of u to \(\partial O\) is identically constant u(w), and hence \(u_{O}\) is identically constant u(w). By the definition of \({\widehat{P}}_u\) it follows that \(\widehat{P}_u\) is constant on O. As \(\partial O \cap K \ne \emptyset \) we obtain \(O\subset K\), which is a contradiction.
Thus M is a celllike subset of \(S^2\). By Moore’s quotient theorem, see e.g. [33, Theorem 7.11], \(S^2/M\) is homeomorphic to \(S^2\). Since \(S^2/M\) is obtained from \(S^2/K \cong S^2\) by quotienting out a closed curve that only selfintersects at one point, the arising space would have a topological cutpoint which is not the case for \(S^2\). This contradiction shows that the restriction of \(\widehat{P}_u\) to \(S^1\) must be monotone and hence \(\widehat{Z}_u\) is a metric disc, and \(\widehat{P}_u\in \Lambda (\partial \widehat{Z}_u,\widehat{Z}_u)\) a uniform limit of homeomorphisms.
If \(\widehat{P}_u\) is not an energy minimizer, then there exists \(v\in \Lambda (\partial \widehat{Z}_u,\widehat{Z}_u)\) such that \(E^2_+(v)<E^2_+(\widehat{P}_u)\). This implies that \(\widehat{u}\circ v\in \Lambda (\partial Z,Z)\), and that
This contradiction shows that \(\widehat{P}_u\) is an energy minimizer.
Set \(\nu :=\mathcal {H}^2_Z\). By Theorem 5.16 we know that \(\widehat{Z}_u\) has the SobolevtoLipschitz property with respect to \(\nu _u\). Recall that u is infinitesimally quasiconformal and that, by Theorem 2.3 and monotonicity of u, the equality \(J({\text {ap md}}u)=J u\) holds almost everywhere. We want however the SobolevtoLipschitz property to hold with respect to \(\mathcal {H}^2_{\widehat{Z}_u}\). To this end we show that \(\nu _u=\mathcal {H}^2_{\widehat{Z}_u}\).
Since u and \({\widehat{P}}_u\) are monotone and surjective, it follows that the map \({\widehat{u}}:{\widehat{Z}}_u\rightarrow Z\) is monotone. This and (5.12) imply that \(\#{\widehat{u}}^{1}(z)=1\) for \(\mathcal {H}^2_Z\)almost every \(z\in Z\). Using the equality of the approximate metric differentials, we have
for any Borel set \(A\subset {\widehat{Z}}_u\). On the other hand
These two inequalities imply that \(\nu _u=\mathcal {H}^2_{\widehat{Z}_u}\). In particular we have shown that \(\widehat{Z}_u\) has the SobolevtoLipschitz property with respect to \(\mathcal {H}^2_{\widehat{Z}_u}\).
The proof of the maximality statement (4) is identical to the proof of maximality in Theorem 1.2.
Finally assume that Z satisfies a \((C,l_0)\)quadratic isoperimetric inequality. Note first that, by [33, Proposition 5.1] and Theorem 2.3, for every Jordan domain \(V\subset \overline{\mathbb {D}}\) with \(\ell (u_{\partial V})<l_0\), one has
The remaining proof is a variation of the proof of [33, Theorem 8.2]. We say that points \(p,q\in \overline{\mathbb {D}}\) are sufficiently connected if for every \(\epsilon >0\) and there exists a curve \(\eta \) joining p to q such that
If we could show that all points \(p,q \in \overline{\mathbb {D}}\) are sufficiently connected then the proof of [33, Theorem 8.2] would go through without changes. Unfortunately we do not know whether this is true.
Let A be the set of those \(p\in \overline{\mathbb {D}}\) such that p either lies in the interior of \(\overline{\mathbb {D}}\) or p lies in an open interval \(I\subset S^1\) satisfying \(\ell (\widehat{P}_u \circ I)<\infty \). We claim that if \(p,q\in A\) then p and q are sufficiently connected.
So let \(p,q\in A\). The proof of [33, Corollary 5.4] shows that for every \(\epsilon >0\) there exists \(\delta >0\) such that, for every \(y\in B_\delta (p)\) and \(z\in B_\delta (q)\), there are curves \(\gamma _p,\gamma _q\) connecting y to p and z to q, respectively, such that
By construction of the metric \(\widehat{d}_u\), there exists a curve \(\gamma \) connecting \(B_\delta (p)\) to \(B_\delta (q)\) such that
The concatenation of \(\gamma \) with correspondingly chosen \(\gamma _p\) and \(\gamma _q\) yields a curve \(\eta \) connecting p to q which satisfies
Since \(\epsilon >0\) was arbitrary the claim follows.
Now let \(U\subset \widehat{Z}_u\) a Jordan domain satisfying \(\ell (\partial U)<l_0\). Then the set of points in \(\partial U\) which have a preimage in A is dense in \(\partial U\). In particular we may choose the points \(t_i\) and \(x_i\) in the proof of [33, Theorem 8.2] such that all the points \(x_i\) lie in A and are hence pairwise sufficiently connected. Under this modification the argument in the proof of [33, Theorem 8.2] shows that
which completes the proof. \(\square \)
6.3 Essential minimal surfaces
Proof of Theorem 1.9
Let X be a proper metric space which satisfies a \((C,l_0)\)quadratic isoperimetric inequality, and \(u\in \Lambda (\Gamma ,X)\) a minimal disc spanning a Jordan curve \(\Gamma \) which satisfies a chordarc condition. By [30, Theorem 3.1] one has that \(u\in N^{1,p}(\overline{\mathbb {D}};X)\) for some \(p>2\). Let \(\widehat{Z}_u\) be the metric space discussed in Sect. 5.3 and \(u=\widehat{u}\circ \widehat{P}_u\) be the corresponding factorization. Let \(Z_u\) be the metric disc discussed in Sect. 1.4 and \(u=\bar{u}\circ P_u\) the corresponding factorization. Then, by [33] and the proof of [5, Theorem 2.7], the pair \((Z_u,P_u)\) satisfies the assumptions of Theorem 6.3. On the other hand, since \(\ell (P_u \circ \gamma )=\ell (u\circ \gamma )\) for every curve \(\gamma \) in \(\overline{\mathbb {D}}\), it follows that \(\widehat{Z}_u=\widehat{Z}_{\widehat{P}_u}\) and \(\widehat{P}_u=\widehat{P}_{P_u}\). Theorem 6.3 now implies the claim. \(\square \)
The following example from [35, Example 5.9], demonstrates that a unique characterization statement in Theorem 1.9 in terms of the SobolevtoLipschitz property does not hold.
Example 6.4
Let Z be the metric space obtained from the standard Euclidean disc by collapsing a segment I in its interior to a point. Then Z is a geodesic metric disc satisfying a quadratic isoperimetric inequality with constant \(\frac{1}{2\pi }\), compare the proof of [5, Theorem 3.2]. The canonical quotient map \(u:\overline{\mathbb {D}} \rightarrow Z\) is an energy minimizer in \(\Lambda (\partial Z,Z)\) and satisfies \({\text {ap md}}u={\text {ap md}}\mathrm {id}_{\overline{\mathbb {D}}}\) almost everywhere. In particular \(\widehat{Z}_u\) is isometric to \(\overline{\mathbb {D}}\). However \(Z_u\) is isometric to Z, see [5, Theorem 1.2]. It is also straightforward to see that \(Z_u\) has the SobolevtoLipschitz property.
To see that Z is thick geodesic let \(E\subset Z\) be a measurable subset of positive measure and \(C>1\). Furthermore let \(p\in \overline{\mathbb {D}}\) be such that \(u(p)\ne u(I)\) is a density point of E and \(q\in I\) the point which is closest to p. Then for \(\delta >0\) sufficiently small
see [12, Remark 3.4] for the first inequality. Now having Eq. (6.10) it is not hard to deduce that Z is thick geodesic. The example also shows that being thick quasiconvex with constant 1 is a strictly stronger condition than being thick geodesic.
The metric disc Z is not Ahlfors regular, since \(\mathcal {H}^2(B_Z(p,r))\) grows linearly in r, and thus Example 6.4 does not contradict Theorem 1.2. Note that, for Z in the example, the construction in [33] yields the original space Z, while the space \({\widehat{Z}}\) constructed in the proof of Theorem 6.3 coincides with \({\overline{\mathbb {D}}}\). This need not always be the case when collapsing a celllike subset in the interior of \(\overline{\mathbb {D}}\); the Euclidean disc with a small ball (in the interior) collapsed is a metric disc satisfying the assumptions of Theorem 6.3, where both constructions yield the original space; cf. [33, Example 11.3].
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Acknowledgements
The authors would like to thank Alexander Lytchak and Stefan Wenger for helpful discussions.
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The first author was partially supported by the DFG Grant SPP 2026. The second author was partially supported by the Vilho, Yrjö ja Kalle Väisalä Foundation (postdoc pool) and by the Swiss National Science Foundation Grant 182423.
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Creutz, P., Soultanis, E. Maximal metric surfaces and the SobolevtoLipschitz property. Calc. Var. 59, 177 (2020). https://doi.org/10.1007/s00526020018430
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DOI: https://doi.org/10.1007/s00526020018430
Mathematics Subject Classification
 30L10
 49Q05
 53B40