The Plateau-Douglas problem for singular configurations and in general metric spaces
aa r X i v : . [ m a t h . DG ] A ug THE PLATEAU-DOUGLAS PROBLEM FOR SINGULARCONFIGURATIONS AND IN GENERAL METRIC SPACES
PAUL CREUTZ AND MARTIN FITZIA bstract . Assume you are given a finite configuration Γ of disjoint rectifiableJordan curves in R n . The Plateau-Douglas problem asks whether there exists aminimizer of area among all compact surfaces of genus at most p which span Γ .While the solution to this problem is well-known, the classical approaches breakdown if one allows for singular configurations Γ where the curves are potentiallynon-disjoint or self-intersecting. Our main result solves the Plateau-Douglasproblem for such potentially singular configurations. Moreover, our proof worksnot only in R n but in general proper metric spaces. Thus we are also able toextend previously known existence results of J¨urgen Jost as well as of the sec-ond author together with Stefan Wenger for regular configurations. In particular,existence is new for disjoint configurations of Jordan curves in general com-plete Riemannian manifolds. A minimal surface of fixed genus p bounding agiven configuration Γ need not always exist, even in the most regular settings.Concerning this problem, we also generalize the approach for singular configu-rations via minimal sequences satisfying conditions of cohesion and adhesion tothe setting of metric spaces.
1. I ntroduction and statement of main results
Introduction.
The classical Plateau problem asked whether any given rectifi-able Jordan curve Γ in R n bounds a Sobolev disc of least area. The positive answerwas obtained independently by Douglas and Rad´o in the early 1930’s, [Rad30,Dou31]. Over the years their result was generalized from R n to so-called homoge-neously regular Riemannian manifolds, metric spaces satisfying curvature boundsin the sense of Alexandrov and particular classes of homogeneously regular Finslermanifolds, [Mor48, Nik79, MZ10, OvdM14, PvdM17]. The solution of Plateau’sproblem in proper metric spaces given by Lytchak-Wenger in [LW17a] covers allthese settings. However, even in R n , the arguments break down if Γ is allowed toself-intersect. Still the generality of [LW17a] and a simple extension trick allowedthe first author to solve the Plateau problem for possibly self-intersecting curves inproper metric spaces which satisfy a local quadratic isoperimetric inequality, [Cre].In R n this improved a previous existence result due to Hass, [Has91].The Plateau-Douglas problem is a variation of the Plateau problem, where oneallows for various boundary components and surfaces of nontrivial topology. Oneway to state the solution obtained by Douglas in [Dou39] is the following: assumeyou are given a finite configuration of disjoint rectifiable Jordan curves Γ in R n anda natural number p ≥
0. Then there exists an area minimizer among all compact
Date : August 21, 2020.The first author was partially supported by the DFG grant SPP 2026. The second author waspartially supported by the Swiss National Science Foundation Grants 165848 and 182423. Thiswork was also partially supported by the grant 346300 for IMPAN from the Simons Foundation andthe matching 2015-2019 Polish MNiSW fund. urfaces which have genus at most p and span Γ . Douglas’ result has since been ex-tended by Jost to homogeneously regular Riemannian manifolds and recently evenfurther by the second author together with Stefan Wenger to proper metric spacesadmitting a local quadratic isoperimetric inequality, [Jos85, FWa]. Again, the ma-chinery fails if one allows for singular, possibly non-disjoint or self-intersectingconfigurations. Our main result, Theorem 1.2 below, solves the Plateau-Douglasproblem for such possibly singular configurations and in general proper metricspaces. The solution for singular configurations is new even in R n . Theorem 1.2also generalizes the main results of [FWa] and [Cre] as we are able to drop theassumption that X admits a local quadratic isoperimetric inequality. In particu-lar, existence is new for regular configurations in complete Riemannian manifoldswhich might not be homogeneously regular. It is not surprising that existence inthis case is harder to obtain, since already for such a setting discontinuous solutionscan only be excluded under additional geometric assumptions, cf. [Mor48].Note that the somewhat more modern approach to Plateau’s problem via currentsas in [FF60, AK00] does not allow for bounding the topology of solutions, andfor singular configurations currents would consider the boundary curves rather asunparametrized objects and could not keep track of the order in which they aretraversed, in contrast to our approach. Moreover, beyond the Riemannian setting,there is no appropriate regularity theory available.1.2. Main result.
Simple examples show that, without additional assumptions,one cannot hope for reasonably regular area minimizers of prescribed topologicaltype to bound a given contour Γ . For example, a Jordan curve in R n which isconvex and contained in a plane does not span a minimal surface of genus p > X be a smoothcomplete Riemannian manifold and M be a smooth, orientable, compact surface(which might be disconnected). Assume furthermore that all connected compo-nents of M have nonempty boundary. For a map u in the Sobolev space W , ( M , X )we denote by Area( u ) the parametrized Riemannian area of u .Assume now that M has k ≥ ∂ M i and Γ is a collection of k rectifiable closed curves Γ j in X . By a rectifiable closed curve we mean an equiv-alence class of parametrized rectifiable curves γ : S → X . We identify two suchparametrized curves if they are reparametrizations of each other, meaning moreprecisely that their constant speed parametrizations agree up to a homeomorphismof S . We say that a map u ∈ W , ( M , X ) spans Γ if for each curve Γ j there existsa boundary component ∂ M i such that the trace u | ∂ M i is a parametrization of Γ j . Let Λ ( M , Γ , X ) be the family of Sobolev maps u ∈ W , ( M , X ) which span Γ . We define a ( M , Γ , X ) : = inf { Area( u ) : u ∈ Λ ( M , Γ , X ) } and a p ( Γ , X ) : = a ( M , Γ , X ) if M is the (up to a di ff eomorphism) unique connectedsurface of genus p with k boundary components. We say that the Douglas condition olds for p , Γ and X if a p ( Γ , X ) is finite and(1.1) a p ( Γ , X ) < a ( M , Γ , X )for every M as in the previous paragraph and of one of the following types. Either M is connected and of genus strictly smaller than p , or M is disconnected andof total genus at most p . Note that in the case where Γ is a single curve and p =
0, which corresponds to the classical Plateau problem, the Douglas conditionis equivalent to the assumption that there is at least one Sobolev disc spanning Γ . Theorem 1.1.
Let X be a smooth complete Riemannian manifold and Γ a config-uration of k ≥ rectifiable closed curves. Let M be a compact, connected andorientable surface with k boundary components and of genus p ≥ . If the Dou-glas condition holds for p , Γ and X, then there exists u ∈ Λ ( M , Γ , X ) as well as aRiemannian metric g on M such that Area( u ) = a p ( Γ , X ) and u is weakly conformal with respect to g on M \ u − ( Γ ) . Furthermore, if...(i) ... X is homogeneously regular, then u may be chosen H ¨older continuouson M and smooth on M \ u − ( Γ ) .(ii) ... X is homogeneously regular and Γ is C , then u may be chosen locallyLipschitz on M \ ∂ M.(iii) ... Γ is a union of disjoint Jordan curves, then u and g may be chosen suchthat u is weakly conformal with respect to g on M. Here, by weakly conformal we mean that almost everywhere the weak di ff eren-tial of u either vanishes or is angle preserving. Already the most simple example ofa figure eight curve in R shows that self-intersecting curves need not always boundglobally weakly conformal area minimizing discs, cf. [Has91]. So the assumptionof (iii) seems quite sharp. Note that the existence of globally H ¨older continuousarea minimizers guaranteed by (i) is new already for topologically regular con-figurations in R n which potentially are of low analytic regularity. Compare therespective discussion for the Plateau problem in [Cre]. Without geometric assump-tions one cannot hope for the conclusion of (i) to be true. See [Mor48, p. 809] fora complete Riemannian manifold X and a Jordan curve Γ ⊂ X which only boundsdiscontinuous area minimizers. Parts (i) and (ii), respectively (ii) and (iii), arecompatible in the sense that when both respective assumptions are satisfied thenone can achieve the conclusion simultaneously for a single map u , compare Re-mark 4.4. However, if both the assumptions in (i) and (iii) hold, we can only cookup a single area minimizer which is simultaneously weakly conformal and globallyH ¨older continuous in the previously known case where all the curves of Γ satisfy achord-arc condition.We sketch the main ideas entering in the proof of Theorem 1.2. For (i), the pro-cedure is conceptually similar to the respective disc type result obtained in [Cre].Namely, we attach a cylinder to each of the curves in Γ . This way we obtain ametric space X Γ , which admits a local quadratic isoperimetric inequality and con-tains X isometrically, as well as a regular configuration ˜ Γ ⊂ X Γ . Now we apply[FWa] to solve the Plateau-Douglas problem for the new pair ( X Γ , ˜ Γ ) and projectthe obtained solution down to X . This gives the desired solution for ( X , Γ ). For (ii),the proof follows essentially the same lines. However, the construction is now per-formed in a way that is more sensitive to the concrete geometric situation. The onstruction scheme, which is a generalization of the funnel extensions introducedby Stadler in [Sta], allows us to obtain an extension space ˆ X Γ which admits a lo-cal quadratic isoperimetric inequality and is locally of curvature bounded abovein the sense of Alexandrov. This latter feature allows to apply the regularity the-ory for harmonic maps into spaces of curvature bounded above as developed e.g.in [KS93, Ser95, BFH + ε -thickenings as introduced in [Wen08] to approximate X by metric spaces ( X n ) n ∈ N which admit local quadratic isoperimetric inequalitiesand contain X isometrically. Then we apply again [FWa] to obtain solutions ( u n ) n ∈ N for the pairs ( X n , Γ ) respectively. A variant of the Rellich-Kondrachov compactnesstheorem allows us to pass to a limit surface in X which is our desired solution. Theproof of the remaining general case involves a mix of the arguments discussed for(i) and (iii).At this point, we would like to emphasize the following remarkable feature ofTheorem 1.1 and its proof: despite major additional complications that arise, theresults and methods developed in [FWa] for the Plateau-Douglas problem in metricspaces are in principle adaptations of respective ones developed for the classicalPlateau-Douglas problem in smooth ambient spaces. However, the flexibility ofthe metric setting therein allows us to draw new conclusions in the smooth settingthat seem out of reach within the classical methods.A theory of metric space valued Sobolev maps has been developed over thelast 30 years. With this language at hand, one can generalize all the introducedterminology to the setting where X is a complete metric space, see Sections 2and 3 below. Recall that a metric space X is called proper if all closed and boundedsubsets of X are compact. In fact, Theorem 1.1 is a special case of the followingvery general result. Theorem 1.2.
Let X be a proper metric space and Γ a configuration of k ≥ rectifiable closed curves. Let M be a compact, connected and orientable surfacewith k boundary components and of genus p ≥ . If the Douglas condition holdsfor p, Γ and X, then there exists u ∈ Λ ( M , Γ , X ) as well as a Riemannian metric gon M such that Area( u ) = a p ( Γ , X ) and u is infinitesimally isotropic with respect to g on M \ u − ( Γ ) . Furthermore, if...(i) ... X admits a local quadratic isoperimetric inequality, then u may bechosen H ¨older continuous on M and to satisfy Lusin’s property (N).(ii) ... X is geodesic, admits a local quadratic isoperimetric inequality and islocally of curvature bounded above, and Γ is of finite total curvature, thenu may be chosen locally Lipschitz on M \ ∂ M.(iii) ... Γ is a union of disjoint Jordan curves, then u and g may be chosen suchthat u is infinitesimally isotropic with respect to g on M. The respective assumptions and conclusions in Theorem 1.2 are natural met-ric generalizations of the respective smooth ones in Theorem 1.1. For examplehomogeneously regular Riemannian manifolds admit a local quadratic isoperimet-ric inequality. In fact, the huge class of metric spaces admitting a local quadraticisoperimetric inequality includes also homogeneously regular Finsler manifolds,CAT( κ ) spaces, compact Alexandrov spaces as well as more exotic examples such s higher dimensional Heisenberg groups, cf. [LW17a]. In particular, the assump-tion on X in Theorem 1.2.(ii) is satisfied if X is a CAT( κ ) space.We would also like to remark that, despite the fact that we exclusively restrict ourdiscussion to the parametrized Hausdor ff area (see Definition 2.3), an appropriatevariant of Theorem 1.2 holds for any area functional which induces quasi-convex2-volume densities in the sense of [LW17b, ´AT04] such as the Holmes-Thompsonarea functional. In order to obtain the respective results, only minor modificationsin the proof of the theorem are needed.1.3. Conditions of cohesion and adhesion.
As discussed above, in general onecannot hope for a given configuration Γ of disjoint Jordan curves to bound a min-imal surface of prescribed topological type if the Douglas condition for p , Γ and X fails. However, there are still situations where the Douglas condition fails butone can show the existence of such a desired surface. Namely, if the area infimummay be approximated by a sequence of surfaces which satisfies a geometric non-degeneracy condition, called condition of cohesion . In increasingly more generalsettings this has been shown to hold true in [Cou37, Shi39, TT88, FWa]. Addi-tional di ffi culties arise if one allows for singular configurations Γ . Imposing anadditional so-called condition of adhesion , Iseri was able to show a statement ofsimilar spirit for singular configurations in R n , [Ise96]. In Section 6 we general-ize the definition of adhesion and Iseri’s result to the setting of metric spaces. Forregular configurations in su ffi ciently nice ambient spaces, the Douglas conditionimplies the condition of cohesion for any sequence of surfaces approaching the en-ergy infimum. Note however that nothing similar is true for singular configurationsand the condition of adhesion. Hence these results can only be applied to obtainexistence for very particular configurations, cf. [Ise96].1.4. Organization.
After recalling some basic notions in section 2, we discussthe proof of Theorem 1.2.(i) in Section 3, where we first recall some terminologyand the main result of [FWa] in Subsection 3.1 before giving the actual proof of(i) in Subsection 3.2. Moving forward, we discuss a generalization of the Cartan-Hadamard theorem due to Bowditch and a gluing result due to Stadler in Sub-section 4.1, and the proof of Theorem 1.2.(ii) is performed in Subsection 4.2.Section 5 is then dedicated to the proofs of Theorems 1.2 and 1.1 in the generalcase. In Subsection 5.1, we first discuss how general proper metric spaces X canbe approximated by more regular spaces admitting local quadratic isoperimetricinequalities and when one can pass from a sequence of fillings within the approxi-mating spaces to a limit filling in X . Then in Subsection 5.2, we recall two devicesfrom [FWa] that allow, in spaces admitting a local quadratic isoperimetric inequal-ity, to lower the topological type of an area minimizing sequence whenever thissequence degenerates. These devices are combined in Section 5.3 with the approx-imating spaces discussed before. The proof of Theorem 1.2 is then completed inSection 5.4. In Section 5.5 we briefly discuss how Theorem 1.1 follows from The-orem 1.2. Finally in Section 6, we discuss the method using minimizing sequencessatisfying conditions of cohesion and adhesion. Acknowledgements.
We wish to thank our respective PhD advisors AlexanderLytchak and Stefan Wenger for their great support. Furthermore, advancements inthis project were made during a research visit to the Institute of Mathematics of the olish Academy of Sciences in 2019 and we would like to thank the IMPAN forits hospitality. 2. P reliminaries Basic notation.
We write | v | for the Euclidean norm of a vector v ∈ R , D : = { z ∈ R : | z | < } for the open unit disc in R and ¯ D for its closure. The di ff erential at z of a (weakly)di ff erentiable map ϕ between smooth manifolds is denoted D ϕ z .For a subset A ⊂ R , | A | denotes its Lebesgue measure. If ( X , d ) is a metric spacethen we use the notation H X ( A ) for the 2–dimensional Hausdor ff measure of asubset A ⊂ X . The normalizing constant is chosen such that H X coincides with the2–dimensional Lebesgue measure when X is Euclidean R . Thus, the Hausdor ff H g : = H M , g ) on a 2–dimensional Riemannian manifold ( M , g ) coincideswith the Riemannian area.2.2. Seminorms.
The (Reshetnyak) energy of a seminorm s on R is defined by I + ( s ) : = max { s ( v ) : v ∈ R , | v | = } . If s is a norm on R , then the Jacobian of s is defined as the unique number J ( s )satisfying H R , s ) ( A ) = J ( s ) · | A | for some and thus every subset A ⊂ R such that | A | >
0. For a degenerate semi-norm s we set J ( s ) : =
0. A seminorm s on R is isotropic if s = { v ∈ R : s ( v ) ≤ } is a Euclideanball. If s is a Euclidean seminorm, i.e. if s is induced by a (potentially degenerate)inner product, then s is isotropic precisely if it is a scalar multiple of the standardEuclidean norm | · | .If s is a seminorm on a 2-dimensional Euclidean vector space V then we definethe concepts of Jacobian, energy, and isotropy by identifying V with Euclidean( R , | · | ) via a linear isometry.2.3. Metric space valued Sobolev maps.
Let ( X , d ) be a proper metric space andlet M be a smooth, compact, orientable 2–dimensional manifold, possibly discon-nected and with non-empty boundary. We fix a Riemannian metric g on M and let Ω ⊂ M be an open set. Definition 2.1.
A measurable u : Ω → X belongs to the Sobolev space W , ( Ω , X ) if there exists h ∈ L ( Ω ) with the following property. For every real-valued –Lipschitz function f on X the composition f ◦ u belongs to the classical Sobolevspace H , ( Ω \ ∂ M ) and | D ( f ◦ u ) z | g ≤ h ( z ) for almost every z ∈ Ω . If u ∈ W , ( Ω , X ) then for almost every z ∈ Ω there exists a seminorm ap md u z on T z M , called approximate metric derivative , such thatap lim v → d ( u (exp z ( v )) , u ( z )) − ap md u z ( v ) | v | g = , here the approximate limit is taken within T z M and exp z denotes the exponentialmap of g at z . See [EG15] for the definition of approximate limits.Assume N = ( N , h ) is a smooth complete Riemannian manifolds. Then, byNash’s theorem, there is an isometric embedding ι : N → R m (in the Riemanniansense). Equivalently one may define W , ( Ω , N ) as the set of measurable mappings u : Ω → N such that ι ◦ u lies in the classical Sobolev space H , ( Ω \ ∂ M , R m );compare e.g. Lemma 9.3.3 and Exercise 2 in Section 9 of [Jos17]. In particular,for every Sobolev map u ∈ W , ( Ω , N ) there is a measurable weak di ff erential Du : T Ω → T N ⊂ N × R m . At almost every z ∈ Ω the approximate metric derivativeis given by(2.1) ap md u z ( v ) = | Du z ( v ) | h for all v ∈ T z Ω , compare Theorem 6.4 and the subsequent remark in [EG15].The approximate metric derivative allows one to define the Reshetnyak energyand the parametrized Hausdor ff area of a Sobolev map using the pointwise quanti-ties introduced in Section 2.2 above. Definition 2.2.
The (Reshetnyak) energy of u ∈ W , ( Ω , X ) with respect to g isdefined by E + ( u , g ) : = Z Ω I + (ap md u z ) d H g ( z ) . The energy E + is conformally invariant in the sense that E + ( u ◦ ϕ, g ′ ) = E + ( u , g )whenever ϕ : ( M ′ , g ′ ) → ( M , g ) is a conformal di ff eomorphism. Definition 2.3.
The parametrized (Hausdor ff ) area of u ∈ W , ( Ω , X ) is defined by Area( u ) : = Z Ω J (ap md u z ) d H g ( z ) . If A ⊂ Ω is measurable, then the area of the restriction u | A is defined analogously. It is easy to see that Area( u ◦ ϕ ) = Area( u )for any biLipschitz homeomorphism ϕ : Ω ′ → Ω . In particular, Area( u ) is inde-pendent of the choice of the Riemannian metric g . A measurable map u : Ω → X satisfies Lusin’s property (N) if H X ( u ( A )) = A ⊂ Ω . If u ∈ W , ( Ω , X ), then by the area formulaArea( u ) ≤ Z X { z ∈ Ω : u ( z ) = x } d H X ( x ) , with equality if u satisfies Lusin’s property (N); see [Kar07]. Definition 2.4.
A map u ∈ W , ( M , X ) is infinitesimally isotropic with respect to themetric g on a measurable subset A ⊂ M if for almost every z ∈ A the approximatemetric derivative ap md u z is isotropic with respect to g ( z ) . If no subset A ⊂ M isspecified, it is understood that u is infinitesimally isotropic with respect to g on M.
It is not hard to see that Area( u ) ≤ E + ( u , g ) , here equality holds precisely if u is infinitesimally isotropic and the approxi-mate metric derivative of u at almost every z ∈ M is a Euclidean seminorm, com-pare [LW17b].If Ω ⊂ M \ ∂ M is a Lipschitz domain, then for every u ∈ W , ( Ω , X ) thereis a well defined trace tr( u ) ∈ L ( ∂ Ω , X ). If u extends to a continuous map ¯ u on ¯ Ω , then the trace is simply given by ¯ u | ∂ Ω . Hence, in abuse of notation, we alsodenote the trace of u by u | ∂ Ω . If no continuous extension exists, define tr( u ) locallyaround p ∈ ∂ Ω in the following way. Choose an open neighborhood U of p and abiLipschitz map ψ : (0 , × [0 , → M such that ψ ((0 , × (0 , = U ∩ Ω and ψ ((0 , × { } ) = U ∩ ∂ Ω . Then for almost every s ∈ (0 ,
1) the trace at ψ ( s ,
0) isgiven by lim t ց ( u ◦ ψ )( s , t ), compare [KS93].3. P roof for regular metric spaces The Plateau-Douglas problem for regular configurations.
Let M ( k ) be thefamily of compact, orientable, smooth surfaces M with k boundary componentsand such that each connected component of M has non-empty boundary. Denoteby M k , p the, up to a di ff eomorphism, unique connected surface in M ( k ) of genus p .A reduction of M k , p is a surface M ∗ ∈ M ( k ) with one of the following properties.Either M ∗ is connected and has genus at most p − M ∗ has several connectedcomponents and the total genus of M ∗ is at most p . Since the Euler characteristicof M k , p is given by χ ( M k , p ) = − p − k , it follows that χ ( M ∗ ) > χ ( M k , p ) for any reduction M ∗ of M k , p , and hence χ ( M ∗ ) = k if and only if M ∗ is the union of k smooth discs. For M ∈ M ( k ) with n > M ∗ is a reduction of M if there exists a partition M ∗ = M ∗ ∪ ... ∪ M ∗ n such that each M ∗ l is the reduction of exactly one connectedcomponent of M . Notice that for any M ∈ M ( k ) there are only finitely manyreductions M ∗ up to di ff eomorphism, and that any reduction M ∗∗ of such M ∗ isalso a reduction of M .Let Γ = S Γ j be a configuration of k ≥ X and p ≥
0. By defining a ∗ p ( Γ , X ) : = min { a ( M ∗ , Γ , X ) : M ∗ is a reduction of M k , p } , the Douglas condition (1.1) can be rewritten as a p ( Γ , X ) < a ∗ p ( Γ , X ) . We would like to point out that the notion of reduction used here is broader than theone given in [FWa], where a reduction of the second type consists of exactly twoconnected components. Consequently, the Douglas condition used in [FWa] is `apriori a weaker assumption than the respective one in this article, which turns out tobe more convenient for us. However, the two conditions are in fact equivalent. Thisfollows since a p ( Γ , X ) < ∞ implies that all curves Γ j lie in the same componentof rectifiable connectedness of X , i.e. the curves can be joined pairwise by pathsof finite length, and using this fact one can show that a ( M ∗ , Γ , X ) ≤ a ( M ∗∗ , Γ , X )whenever M ∗∗ is a reduction of a reduction M ∗ of M k , p .The basis for our proof of Theorem 1.2 in the special cases (i) and (ii) will bethe existence results [FWa, Theorem 1.2] and [FWa, Theorem 1.4.(iii)] for Jordancurves, which we now state as a combined theorem for convenience of the reader. heorem 3.1. Let X be a proper metric space admitting a local quadratic isoperi-metric inequality, Γ ⊂ X the disjoint union of k ≥ rectifiable Jordan curves andp ≥ . If the Douglas condition (1.1) holds for p, Γ and X, then there exists acontinuous u ∈ Λ ( M k , p , Γ , X ) and a Riemannian metric g on M k , p such that Area( u ) = a p ( Γ , X ) and u is infinitesimally isotropic with respect to g. Furthermore, if every Jordancurve in Γ is chord-arc, then any such u is H ¨older continuous on M k , p and satisfiesLusin’s property (N). Here, a metric space X is said to admit a ( C , ℓ ) -quadratic isoperimetric inequal-ity if every closed Lipschitz curve c : S → X of length ℓ ( c ) ≤ ℓ is the trace of aSobolev disc u ∈ W , ( D , X ) satisfyingArea( u ) ≤ C · ℓ ( c ) . If there is no need to specify the constants C , ℓ >
0, we simply say that X admitsa local quadratic isoperimetric inequality . A Jordan curve Γ is called chord-arc ifit is biLipschitz equivalent to S .The following replacement lemma will be used in the proof of Lemma 3.4. Itfollows from the proof of [LW18, Lemma 4.8] and the gluing result [KS93, Theo-rem 1.12.3]. While [LW18, Lemma 4.8] is stated for disc-type surfaces, the argu-ments in the proof thereof are local around the boundary curve and can be appliedwithout changes to the present situation. Lemma 3.2.
Let X be a complete metric space admitting a local quadratic isoperi-metric inequality, Γ ⊂ X a configuration of k ≥ rectifiable closed curves andM ∈ M ( k ) . Then for every u ∈ Λ ( M , Γ , X ) and ε > there is v ∈ Λ ( M , Γ , X ) suchthat Area( v ) ≤ Area( u ) + ε and the continuous representative of tr( v ) | ∂ M i is a constant speed parametrizationfor each i ∈ { , . . . , k } . Lemma 3.2 is applied in the proofs of Propositions 5.1 and 6.1 in [FWa]. It isone of the implications in [FWa] making use of the assumption of a local quadraticisoperimetric inequality. In fact the only implications needing this assumption andused in the proof of the existence result therein may be phrased as Lemmas 5.3and 5.4 below. While these lemmas seem to heavily rely on the assumption, it is anopen question whether Lemma 3.2, which enters in their proofs, holds true withoutit or not.3.2.
Proof of Theorem 1.2.(i).
Let X be a complete metric space and Γ a config-uration of k ≥ Γ j in X . Since the Douglas conditionfails as soon as k > Γ j is constant, and since the minimiza-tion problem is trivial for a single constant curve Γ , we may assume without lossof generality that Γ , . . . , Γ k are all nonconstant. For each j , let S j be a geodesiccircle of circumference ℓ ( Γ j ), let γ j : S j → X be a unit speed parametrization of Γ j and Z j : = S j × [0 ,
1] be the cylinder equipped with the product metric. We definethe quotient space X Γ as the disjoint union X ⊔ Z ⊔ · · · ⊔ Z k under the identification γ j ( p ) ∼ ( p ,
0) for every p ∈ Z j , and we equip this space with the quotient met-ric, see for example [BH99]. Furthermore, let P Γ : X Γ → X be the projection given y P Γ ( x ) : = x x ∈ X ,γ j ( p ) x = ( p , t ) ∈ Z j . The proof of [Cre, Lemma 4.1] shows that X ⊂ X Γ isometrically and P Γ : X Γ → X is a 1-Lipschitz retraction. Lastly, we define ˜ Γ j as the (equivalence class of the)rectifiable curve p ( p , ∈ Z j , p ∈ S j , and ˜ Γ as the configuration consisting ofthe curves ˜ Γ , . . . , ˜ Γ k . Then ˜ Γ is a configuration of disjoint chord-arc curves and P Γ ◦ ˜ Γ j = Γ j for each j . Lemma 3.3.
Let X be a complete metric space, Γ ⊂ X a configuration of k ≥ rectifiable closed curves and M ∈ M ( k ) . Then for every u ∈ Λ ( M , ˜ Γ , X Γ ) one hasP Γ ◦ u ∈ Λ ( M , Γ , X ) and Area( u ) ≥ Area( P Γ ◦ u ) + k X j = H ( Z j ) . In particular, one has the inequalitya ( M , ˜ Γ , X Γ ) ≥ a ( M , Γ , X ) + k X j = H ( Z j ) . Proof.
Let u ∈ Λ ( M , ˜ Γ , X Γ ). Without loss of generality, we may assume that M isconnected. By the 1-Lipschitz continuity of P Γ , we have that P Γ ◦ u ∈ Λ ( M , Γ , X ).Since P Γ ( Z j ) is contained in the rectifiable curve Γ j , the area formula in Section 2.3implies that Area (cid:16) ( P Γ ◦ u ) | u − ( Z j ) (cid:17) = . Thus, since the restriction P Γ | X is an isometry, we obtainArea( u ) = Area( u | u − ( X ) ) + X j Area (cid:16) u | u − ( Z j ) (cid:17) = Area( P Γ ◦ u ) + X j Area (cid:16) u | u − ( Z j ) (cid:17) . To complete the proof, it therefore su ffi ces to show that(3.1) Area (cid:16) u | u − ( Z j ) (cid:17) ≥ H ( Z j )for each j . In order to see this, fix j and define Y j as the quotient space X Γ / A , where A : = X ∪ S i , j Z i . Then Y j is isometric to Z j / ( S j × { } ). Hence Y j is homeomorphicto ¯ D and, by [Cre20b, Theorem 3.2], admits a local quadratic isoperimetric inequal-ity. Furthermore, let Q j : X Γ → Y j be the 1-Lipschitz map given by Q j ( x ) : = [ x ].Then the composition Q j ◦ u is an element in Λ ( M , Q j ◦ ˜ Γ , Y j ) with(3.2) Area( Q j ◦ u ) = Area (cid:16) u | u − ( Z j ) (cid:17) . Let ∂ M i be the boundary component of M such that tr( u ) | ∂ M i is an element of Γ j ,and consider M embedded into a smooth compact surface ˜ M ∈ M (1) of samegenus as that of M such that each boundary component ∂ M l bounds a topologicaldisc in ˜ M except for ∂ M i , which agrees with the boundary component of ˜ M . Themap Q j ◦ u extends naturally onto ˜ M by setting its value on ˜ M \ M to be [ x ] for any x ∈ X , yielding a map v j ∈ Λ ( ˜ M , Q j ◦ ˜ Γ j , Y j ) satisfying(3.3) Area( v j ) = Area( Q j ◦ u ) . pparently, there exists a surface M ∗ , either being equal to ˜ M or else being areduction of it, such that a ( ˜ M , Q j ◦ ˜ Γ j , Y j ) = a ( M ∗ , Q j ◦ ˜ Γ j , Y j )and the Douglas condition holds for M ∗ , Q j ◦ ˜ Γ j and Y j . Hence by Theorem 3.1there exists a continuous map w j ∈ Λ ( M ∗ , Q j ◦ ˜ Γ j , Y j ) satisfying Lusin’s prop-erty (N) and(3.4) Area( w j ) ≤ Area( v j ) . Since Y j is homeomorphic to ¯ D with boundary curve Q j ◦ ˜ Γ j , it follows that w j issurjective. Otherwise assume p ∈ Y j \ w j ( M ∗ ). Then Q j ◦ ˜ Γ j , considered as a 1-cycle, would be a generator of H ( Y j \ { p } ) (cid:27) H ( ¯ D \ { } ) (cid:27) Z and at the same timewould bound the 2-chain defined in Y j \ { p } by w j , which is a clear contradiction.Hence, by the area formula, we have(3.5) Area( w j ) = Z Y j n w − j ( x ) o d H ( x ) ≥ H ( Y j ) = H ( Z j ) . Combining (3.2), (3.3), (3.4) and (3.5), we finally obtain (3.1). (cid:3)
While we did not need to assume a local quadratic isoperimetric inequality on X in the previous lemma, this assumption is required in the proof of the upcomingreverse inequality. Lemma 3.4.
Let X be a complete metric space admitting a local quadratic isoperi-metric inequality, Γ ⊂ X a configuration of k ≥ rectifiable closed curves andM ∈ M ( k ) . Then one hasa ( M , ˜ Γ , X Γ ) ≤ a ( M , Γ , X ) + k X i = H ( Z j ) . Proof.
Let ε >
0. By Lemma 3.2 there exists v ∈ Λ ( M , Γ , X ) such thatArea( v ) ≤ a ( M , Γ , X ) + ε and such that tr( v ) | ∂ M i is a constant speed parametrization for each i . We relabelthe boundary components of M such that tr( v ) | ∂ M j is an element of Γ j for each j .Embed M di ff eomorphically into a smooth compact surface ˜ M ∈ M ( k ) such that˜ M \ int( M ) is the disjoint union of k smooth cylinders Ω j with boundary, each Ω j having ∂ M j as one boundary component. Notice that ˜ M is di ff eomorphic to M .Now if ˜ γ j : S j → X Γ is a constant speed parametrization of ˜ Γ j , then the inclusion ι j : Z j → X Γ is a Lipschitz homotopy between ˜ γ j and γ j of area H ( Z j ). Thus, byidentifying Ω j with Z j via a biLipschitz homeomorphism, there exist maps w j ∈ W , ( Ω j , X Γ ) with trace ˜ γ j respectively γ j = tr( v ) | ∂ M j and of area H ( Z j ). Let w : ˜ M → X Γ be the mapping obtained by stitching v together with every w j along ∂ M j , which is a well-defined element in W , ( ˜ M , X Γ ) = W , ( M , X Γ ) by [KS93,Thm. 1.12.3]. Then w spans ˜ Γ and satisfies a ( M , ˜ Γ , X Γ ) ≤ Area( w ) = Area( v ) + k X j = Area( w j ) ≤ a ( M , Γ , X ) + k X j = H ( Z j ) + ε. Since ε > (cid:3) ith these preparations at hand, it is now not hard to give a proof of Theo-rem 1.2.(i). Proof of Theorem 1.2.(i).
Since X admits a local quadratic isoperimetric inequal-ity, it follows from the proof of [Cre20b, Theorem 3.2] that X Γ admits a localquadratic isoperimetric inequality as well. Lemma 3.3 together with Lemma 3.4imply that one has the equality(3.6) a ( ˜ M , ˜ Γ , X Γ ) = a ( ˜ M , Γ , X ) + k X j = H ( Z j )for every ˜ M ∈ M ( k ). Hence the Douglas condition a p ( ˜ Γ , X Γ ) < a ∗ p ( ˜ Γ , X Γ )holds for p , ˜ Γ and X Γ . Since ˜ Γ is a disjoint configuration of chord-arc curves, wehave by Theorem 3.1 that there is a H ¨older continuous v ∈ Λ ( M , ˜ Γ , X Γ ) satisfyingLusin’s property (N) and a Riemannian metric g on M such thatArea( v ) = a p ( ˜ Γ , X Γ )and v is infinitesimally isotropic with respect to g . By Lemma 3.3 and equa-tion (3.6) the projection u : = P Γ ◦ v ∈ Λ ( M , Γ , X ) then satisfiesArea( u ) = a p ( Γ , X ) . Moreover, since P Γ is isometric on X , the map u is infinitesimally isotropic withrespect to g on M \ u − ( Γ ) ⊂ M \ v − ( X Γ \ X ). Thus the proof of (i) is complete. (cid:3)
4. I nterior L ipschitz regularity Upper curvature bounds.
Let X be a metric space. Closed piecewise ge-odesic curves in X will be denoted x x . . . x m , where x i ∈ X indicate the end-points of the geodesic segments. For κ ∈ R , let D κ be the diameter of the modelspace M κ of constant curvature κ . That is, D κ = π/ √ κ for κ > D κ = ∞ for κ ≤
0. A geodesic triangle xyz will be called κ -admissible if ℓ ( xyz ) < D κ .For every κ -admissible triangle xyz , there is a (up to isometry) unique comparisontriangle x κ y κ z κ in M κ which has the same side lengths. A κ -admissible triangle xyz is called CAT( κ ) if there is a 1-Lipschitz map f : x κ y κ z κ → xyz such that f ( x κ ) = x , f ( y κ ) = y and f ( z κ ) = z . We say that X is a CAT( κ ) space if X is geodesic and every κ -admissible triangle in X is CAT( κ ), and call X locally CAT( κ ) if every point in X has a neighbourhood which is a CAT( κ ) space. Two standard facts are that CAT( κ )spaces are also CAT( κ ′ ) for any κ ′ ≥ κ , and that balls of radius at most D κ / κ ) spaces are themselves CAT( κ ) spaces. Finally, we say that X is locally ofcurvature bounded above if every point p ∈ X has a neighbourhood U p which is aCAT( κ p ) space for some κ p ∈ R . By the preceeding observations, we may alwaysassume that κ p > U p is a small ball.If X is geodesic and locally CAT(0), then the Cartan-Hadamard theorem statesthat X is a CAT(0) space if and only if X is simply connected. Aiming to handlealso spaces satisfying positive upper curvature bounds, we discuss a variant of thisresult due to Bowditch. For Lipschitz curves γ , γ : S → A ⊂ X , we say that γ is monotonically homotopic to γ in A if there exists a continuous homotopy h : [0 , × S → A such that h (0 , · ) = γ , h (1 , · ) = γ and ℓ ( h ( t , · )) ≤ ℓ ( γ ) for all t ∈ [0 , γ is monotonically nullhomotopic in A if γ is monotonically omotopic to a constant curve in A . If X is a CAT( κ ) space, then Reshetnyak’s ma-jorization theorem (see for example [AKP19]) implies that every closed Lipschitzcurve in X of length smaller than 2 D κ is monotonically nullhomotopic. Dually, thefollowing holds by Theorem 3 . . Theorem 4.1.
Let X be a proper geodesic metric space, κ ∈ R and A ⊂ X becompact such that the D κ -neighbourhood of A is locally CAT( κ ) . If a κ -admissibletriangle ∆ ⊂ A is monotonically nullhomotopic in A, then ∆ is CAT( κ ) . Theorem 3.1.2 in [Bow95] is stated under the assumption that the entire space X is locally CAT( κ ). However, as discussed in Section 3.6 of [Bow95], the ar-gument is local in the D κ -neighbourhood of any set in which ∆ is monotonicallynullhomotopic, and hence the proof readily gives Theorem 4.1. As a corollaryof Theorem 4.1, we obtain the following result allowing to derive quantitativelycontrolled ”local globalizations”. Corollary 4.2.
Let X be a proper geodesic metric space, κ ∈ R and B ( p , r ) ⊂ Xa ball which is locally
CAT( κ ) . If every triangle ∆ ⊂ ¯ B ( p , r / is monotonicallynullhomotopic in ¯ B ( p , r / , then ¯ B ( p , ¯ r ) is a CAT(¯ κ ) space, where ¯ κ = ¯ κ ( κ, r ) and ¯ r = ¯ r ( κ, r ) only depend on κ and r.Proof. Set ¯ κ : = max { κ, π r − } and ¯ r : = D ¯ κ /
4. Note that ¯ κ is chosen such that D ¯ κ ≤ r /
2. To see that ¯ B ( p , ¯ r ) is convex, let x , y ∈ ¯ B ( p , ¯ r ) and observe that any geo-desic triangle pxy is ¯ κ -admissible and contained in ¯ B ( p , r ) ⊂ ¯ B ( p , r / B ( p , r / pxy is CAT(¯ κ ). Since ¯ r < D ¯ κ /
2, it follows that pxy ⊂ ¯ B ( p , ¯ r ), and weconclude that ¯ B ( p , ¯ r ) is convex. Now let xyz ⊂ ¯ B ( p , ¯ r ). Then xyz is ¯ κ -admissibleand monotonically nullhomotopic in ¯ B ( p , r / xyz is CAT(¯ κ ). (cid:3) For α ≥ r >
0, we let S α, r be the ball of radius r around the vertex inthe cone over a compact interval of length α (see [BBI01] for the definition ofcones), and call S α, r the sector of radius r and angle α . On any sector, we fix anorientation so that the left leg and the right leg of S α, r are defined. The followinglemma generalizes [Sta, Lemma 21] to spaces satisfying positive upper curvaturebounds. Lemma 4.3.
Let κ ≥ , < r ≤ D κ / , X be a proper CAT( κ ) space, p ∈ X and η , . . . , η l , ν , . . . , ν l ⊂ X geodesic segments all of length r and starting at p. Fori = , . . . , l, let α i ∈ [0 , π ] be the angle at p between η i and ν i , and let S i be thesector of angle π − α i and radius r. Then the space Z, obtained by gluing eachsector S i to X via isometric identifications of its left leg with η i and its right legwith ν i , is a CAT( κ ) space. In the lemma, the isometric identifications are chosen such that p correspondsto the vertex point in S i . In the following, we assume without further mentioningthat the orientations of isometric identifcations are chosen in such a natural way. Proof.
By induction, it is su ffi cient to prove the statement for l =
1, and hencewe set η : = η , ν : = ν and α : = α . Reshetnyak’s gluing theorem (see forexample [BH99]) implies that the space Y , obtained by gluing S π − α, r to X via anisometric identification of the left leg of S π − α, r and η , is a CAT( κ ) space. Observethat the angle in Y between the right leg η ′ of S π − α, r and ν equals π and that the ength of the concatenation η ′ ∪ ν is at most D k . Hence the curve η ′ ∪ ν is a geodesicin Y and in particular a convex subset of Y , see [BH99, Proposition 1.7]. Thus theclaim follows from another application of Reshetnyak’s theorem upon noting that Z may be constructed alternatively by gluing the sector S π, r to Y via isometricidentifications of its left leg with η ′ and its right leg with ν . (cid:3) Proof of Theorem 1.2.(ii).
Let X be a metric space which is locally of cur-vature bounded above. The total curvature of a closed piecewise geodesic curve x x . . . x m in X is defined by σ ( x x . . . x m ) : = m X i = ( π − β i ) , where β i denotes the angle at x i between the geodesic segments x i x i − and x i x i + .Let L be a closed rectifiable curve. The curve x x . . . x m is called inscribed to L if the points x , x , . . . , x m lie on L and are traversed by L in cyclic order. The total curvature of L , denoted σ ( L ), may be defined as lim n →∞ σ ( L n ), where ( L n )is a sequence of closed piecewise geodesic curves which are inscribed to L andconverge uniformly to L , see [ML03, Proposition 2.4]. Proof of Theorem 1.2.(ii).
Let X be as in the statement of the theorem. Assumefirst L = x x . . . x m is a closed piecewise geodesic curve in X . For i = , . . . , m ,we set S i : = S π − β i , and Q i : = I i × [0 , I i ⊂ R is a compact interval oflength d ( x i , x i + ). We define a geodesic metric cylinder ˆ Z L by gluing the left endinterval of each Q i isometrically to the right leg of S i and the right end intervalof each Q i to the left leg of S i + . Then, by Reshetnyak’s gluing theorem, balls ofradius at most ℓ ( L ) / Z L are CAT(0) spaces. Denote the inner boundary curveof ˆ Z L by ¯ L and the outer boundary curve of ˆ Z L by ˆ L . There exist a 1-Lipschitzretraction ˆ P L : ˆ Z L → ¯ L such that ˆ P L ◦ ˆ L = ¯ L , as well as a ( ℓ ( L ) + σ ( L ))-Lipschitzhomotopy h L : S × [0 , → ˆ Z L between ¯ L and ˆ L such that Area( h ) = H ( ˆ Z L ).In particular, ¯ L is a geodesic circle of circumference ℓ ( L ) and there is a canonicalunit-speed parametrization c L : ¯ L → L . Now let L be any closed rectifiable curve offinite total curvature. All the properties discussed for piecewise geodesic curve arequantitative and hence stable under ultralimits; see e.g. [AKP19] for the definitionand properties of ultralimits. Thus we may approximate L by a sequence ( L n ) of L -inscribed piecewise geodesic curves, perform the construction for each L n , passto an ultralimit and obtain that there exist ˆ Z L , ˆ L , ¯ L , c L , h L , ˆ P L as above, all enjoyingthe very same properties.Let ˆ Z j : = ˆ Z Γ j for j = , . . . , k . We define the quotient space ˆ X Γ as the disjointunion X ⊔ ˆ Z ⊔ · · · ⊔ ˆ Z k under the identification c Γ j ( p ) ∼ p for p ∈ ¯ Γ j , and we equipthis space with the quotient metric. Also, we let ˆ P Γ : ˆ X Γ → X be the 1-Lipschitzretraction given by ˆ P Γ ( x ) : = x for x ∈ X and ˆ P Γ ( x ) = ˆ P Γ j ( x ) for x ∈ ˆ Z j . ByReshetnyak’s majorization theorem each ˆ Z j admits a local quadratic isoperimetricinequality. This, together with the facts that ˆ P Γ is 1-Lipschitz and X admits a localquadratic isoperimetric inequality, makes it straight forward to modify the proofof [Cre20b, Theorem 3.2] and derive that the space ˆ X Γ admits a local quadraticisoperimetric inequality. Let ˆ Γ be the configuration formed by ˆ Γ , ..., ˆ Γ k . The prop-erties discussed above allow us to imitate the proofs of Lemmas 3.3 and 3.4 for the onfiguration ˆ Γ ⊂ ˆ X Γ , and hence derive that(4.1) a ( ˜ M , ˆ Γ , ˆ X Γ ) = a ( ˜ M , Γ , X ) + k X i = H ( ˆ Z i )for every ˜ M ∈ M ( k ).So far we have not achieved any advantage from our more complicated con-struction over the one in Section 3.2. However, and this is the crucial di ff erence,now we claim that ˆ X Γ is locally of curvature bounded above. Since ˆ X Γ \ X is locallyCAT(0), it su ffi ces to show that every p ∈ X has a CAT neighbourhood within ˆ X Γ .So let p in X and choose κ > < r < D κ / B X ( p , r ) isa CAT( κ ) space. The proof that X is locally of curvature bounded above will becompleted by showing that ¯ B ˆ X Γ ( p , ¯ r ) is a CAT(¯ κ ) space, where ¯ κ and ¯ r are as in thestatement of Corollary 4.2. Since ¯ κ and ¯ r are independent of Γ and the CAT(¯ κ ) con-dition is stable under ultralimits, we lose no generality in assuming that Γ , . . . , Γ k are piecewise geodesic curves. Thus it remains to verify the assumptions of Corol-lary 4.2. Clearly, B ˆ X Γ ( p , r ) \ Γ is locally CAT( κ ). Since we assumed Γ consistsof piecewise geodesic curves, for q ∈ B ˆ X Γ ( p , r ) ∩ Γ and s > ffi ciently smallthe ball ¯ B ˆ X Γ ( q , s ) is obtained from ¯ B X ( q , s ) as the space Z is obtained from X inLemma 4.3. Thus the lemma states that ¯ B ˆ X Γ ( q , s ) is a CAT( κ ) space and hencewe conclude that B ˆ X Γ ( p , r ) is locally CAT( κ ). To verify the other assumption ofCorollary 4.2, let ∆ ⊂ ¯ B ˆ X Γ ( p , r /
2) be a geodesic triangle. Sliding ∆ down to X we see that ∆ is monotonically homotopic in ¯ B ˆ X Γ ( p , r /
2) to a curve η ⊂ X . Since¯ B X ( p , r /
2) is a CAT( κ ) space and ℓ ( η ) < D κ , Reshetnyak’s majorization theoremimplies in turn that η is monotonically nullhomotopic in ¯ B X ( p , r / X Γ admits a local quadratic isoperimet-ric inequality, we can proceed as we did when proving (i) in the last section. Theadvantage is now that by [Ser95], see also [BFH +
18, Theorem 1.3], the minimizer v ∈ Λ ( M , ˆ X Γ , ˆ Γ ) is locally Lipschitz on M \ ∂ M , and hence so is our final solution u = ˆ P Γ ◦ v . In order to apply these regularity results, note that v is a continuous har-monic map into a space which is locally of curvature bounded above. Harmonic-ity of v follows since v is infinitesimally isotropic and ˆ X Γ is locally of curvaturebounded from above and hence has property (ET), see [LW17a, Section 11]. (cid:3) Remark 4.4.
The map u we produce in the proof of Theorem 1.2.(ii) is also globallyH ¨older continuous on M. This follows as in the proof of Theorem 1.2.(i) uponnoting that the configuration ˆ Γ we construct consists of chord-arc curves.
5. G eneral C ase Throughout this section, we use the terminology introduced in the beginning ofSection 3.5.1.
Approximating sequences.
Let X be a complete metric space. We call ametric space Y an ε -thickening of X if Y contains X isometrically and X is ε -dense in Y . We will need the following variant of the thickening results obtainedin [Wen08] and [LWY20]. emma 5.1. There is a universal constant C ≥ such that for every proper metricspace X and ε > , there exists a ( C ε ) -thickening Y of X such that Y is proper andadmits a ( C , ε ) -quadratic isoperimetric inequality. If X is geodesic, then Lemma 5.1 follows readily from [LWY20, Lemma 3.3]and in this case, the space Y may also be chosen geodesic. This version su ffi ces toobtain Theorem 1.2 in the special case that X is geodesic, and hence in particularto obtain Theorem 1.1. Thus for the convenience of a reader who is only inter-ested in Theorem 1.2 for geodesic target spaces, the general proof of Lemma 5.1 ispostponed to the appendix.Let X be a proper metric space and ( Y n ) n ∈ N a sequence of proper ε n -thickeningsof X . We call ( Y n ) an X-approximating sequence if ε n →
0. The following conse-quence of the generalized Rellich-Kondrachov compactness theorem, [KS93, The-orem 1.13], allows to pass from a sequence of maps in approximating spaces to alimit map in X . Proposition 5.2.
Let X be a proper space and Γ be a configuration of k ≥ disjoint rectifiable Jordan curves in X. Let M ∈ M ( k ) be connected and endowedwith a Riemannian metric g. Assume that there exist an X-approximating sequence ( Y n ) n ∈ N and mappings u n ∈ Λ ( M , Γ , Y n ) of uniformly bounded energies E + ( u n , g ) and such that the traces tr( u n ) : ∂ M → Γ are equicontinuous with respect to g.Then there is u ∈ Λ ( M , Γ , X ) such that (5.1) Area( u ) ≤ lim sup n →∞ Area( u n ) & E + ( u , g ) ≤ lim sup n →∞ E + ( u n , g ) . The proof is the following standard argument, which is similar to respectivesteps e.g. in the proofs of [GW20, Theorem 1.5] and [LWY20, Theorem 5.1].
Proof.
Let Z be the proper metric space obtained by gluing all the spaces Y n along X . Note that Y n ⊂ Z isometrically and hence Λ ( M , Γ , Y n ) ⊂ Λ ( M , Γ , Z ) foreach n ∈ N . For fixed p ∈ Γ , [FWa, Lemma 2.4] implies that there is a constant C such that Z M d ( p , u n ( z )) d H g ( z ) ≤ C · (cid:16) diam( Γ ) + E + ( u n , g ) (cid:17) for all n ∈ N . In particular,sup n ∈ N "Z M d ( p , u n ( z )) d H g ( z ) + E + ( u n , g ) < ∞ . Thus by the metric space version of the Rellich-Kondrachov compactness theo-rem, [KS93, Theorem 1.13], there is v ∈ W , ( M , Z ) such that v j → v in L ( M , Z ).In fact, since ( Y n ) n ∈ N is an approximating sequence, we may assume that v takesvalues in X ⊂ Z and hence v ∈ W , ( M , X ). By lower semicontinuity of areaand energy, see e.g. [LW17a], the inequalities (5.1) are satisfied for u . Finally, theArzel`a-Ascoli theorem and [KS93, Theorem 1.12.2] imply that v ∈ Λ ( M , Γ , X ). (cid:3) Reductions of fillings.
Let X be a complete metric space, p ≥ Γ ⊂ X a configuration of k ≥ Γ j . The two followingresults are needed for the proof of Lemma 5.6 and can be extracted from the proofsof [FWa, Proposition 6.1] and [FWa, Proposition 5.1] respectively. For the firstlemma, we assume that k + p >
2, which is equivalent to the assumption that thesurface M k , p is neither of disc- nor of cylindrical type. In this case M k , p may beendowed with a hyperbolic metric, which we define to be a Riemannian metric g f constant sectional curvature − ∂ M k , p is geodesicwith respect to g . By a relative geodesic in ( M k , p , g ) we mean either a simpleclosed geodesic in M k , p or a geodesic arc with endpoints on ∂ M k , p that is non-contractible via a homotopy of curves of the same type. We define sys rel ( M k , p , g )as the infimal length of relative geodesics in ( M k , p , g ). Furthermore, we choose foreach ρ > ρ ′ Γ = ρ ′ Γ ( ρ ) as in the first paragraph in the proof of [FWa,Proposition 6.1]. That is, for each ρ > < ρ ′ Γ < ρ such that whenevertwo points x , x ′ ∈ Γ satisfy d X ( x , x ′ ) ≤ ρ ′ Γ , then they lie on the same Jordan curve Γ j and the shorter segment of Γ j between x and x ′ has length at most ρ . The notationemphasizes that ρ ′ Γ only depends on the induced metric on Γ ⊂ X . Lemma 5.3.
Let C , K , ρ > . Assume X admits a ( C , ρ ) -quadratic isoperimetricinequality and g is a hyperbolic metric on M k , p such that sys rel ( M k , p , g ) < min ρ ′ Γ ( ρ )4 K , arsinh ! . Then for every u ∈ Λ ( M k , p , Γ , X ) with E + ( u , g ) ≤ K, there exist a reduction M ∗ of M k , p and a map u ∗ ∈ Λ ( M ∗ , Γ , Y ) such that Area( u ∗ ) ≤ Area( u ) + C ρ . An analogue of the above lemma holds for cylindrical M k , p endowed with a flat metric, which we define as a Riemannian metric with vanishing sectional curvatureand such that the Riemannian area of ( M k , p , g ) is equal to 1 and the boundary ∂ M k , p geodesic. The analogue follows by using a basic flat collar (instead of a hyperbolicone) in the proof of [FWa, Proposition 6.1]. Compare also the respective remark inthe proof of [FWa, Theorem 1.2].For the second lemma, we assume that k + p ≥
2, hence we only exclude that M k , p is of disc-type. Let g be a Riemannian metric on M k , p and 0 < δ g < z ∈ ∂ M k , p has a neighbourhood in ( M k , p , g ) which is theimage of the set B : = { z ∈ C : | z | ≤ | z − | < p δ g } under a 2-biLipschitz di ff eomorphism ψ with z = ψ (1). Lemma 5.4.
Let C , K , ρ > . Assume that X admits a ( C , ρ ) -quadratic isoperi-metric inequality and < δ ≤ δ g is so small that π · K | log( δ ) | ! < ρ ′ Γ ( ρ ) . If there exist u ∈ Λ ( M k , p , Γ , Y ) with E + ( u , g ) ≤ K and a subarc γ − ⊂ ∂ M k , p satisfy-ing ℓ g ( γ − ) ≤ δ & ℓ X (tr( u ) ◦ γ − ) > ρ, then there exist a reduction M ∗ of M k , p and a map u ∗ ∈ Λ ( M ∗ , Γ , X ) such that Area( u ∗ ) ≤ Area( u ) + C ρ . .3. Reductions of approximating sequences.
Let X be a proper metric spaceand Γ be a configuration of k ≥ X and p ≥ Proposition 5.5.
Let ( Y n ) be an X-approximating sequence. If there exist mapsu n ∈ Λ ( M k , p , Γ , Y n ) satisfyinga : = lim sup n →∞ Area( u n ) < a ∗ p ( Γ , X ) , then there exists u ∈ Λ ( M k , p , Γ , X ) such that Area( u ) ≤ a. Moreover, for anysequence ( g n ) of Riemannian metrics on M k , p , there exists u as above and a Rie-mannian metric g on M k , p such thatE + ( u , g ) ≤ lim sup n →∞ E + ( u n , g n ) . The proposition follows by repeatedly applying the next lemma.
Lemma 5.6.
Let ( Y n ) be an X-approximating sequence, M ∈ M ( k ) , ( g n ) be asequence of Riemannian metrics on M and u n ∈ Λ ( M , Γ , Y n ) be fillings such that Area( u n ) is uniformly bounded. Then one of the following two options holds. Eitherthere is u ∈ Λ ( M , Γ , X ) and a Riemannian metric g on M such that Area( u ) ≤ lim sup n →∞ Area( u n ) & E + ( u , g ) ≤ lim sup n →∞ E + ( u n , g n ) , or there exist a reduction M ∗ of M, an X-approximating sequence ( Y ∗ n ) and mapsu ∗ n ∈ Λ ( M ∗ , Γ , Y ∗ n ) such that (5.2) lim sup n →∞ Area( u ∗ n ) ≤ lim sup n →∞ Area( u n ) . Proof of Proposition 5.5.
Let M , Y n , u n and g n be as in the proposition. If the firstpossibility in Lemma 5.6 when applied to these elements is true, i.e. if the existenceof u ∈ Λ ( M , Γ , X ) and a metric g on M as in this lemma is given, then the proposi-tion follows immediately. We claim that the second possibility in the lemma cannotoccur. Otherwise, we could iteratedly apply Lemma 5.6 to M ∗ , the sequences ( Y ∗ n )and ( u ∗ n ) given by the lemma and arbitrarily chosen metrics g ∗ n on M ∗ , as well astheir respective successors, until eventually the first possibility holds. This has tobe the case after finitely many iterations, since the Euler characteristic strictly in-creases when passing to a reduction, but is also bounded from above by k in oursetting. Thus we would obtain a reduction M ∗ of M and a map u ∈ Λ ( M ∗ , Γ , X )such that Area( u ) ≤ lim sup n →∞ Area( u n ) < a ∗ p ( Γ , X ) , which gives a contradiction. (cid:3) At the end of this section, we give a proof for Lemma 5.6. It is based on Propo-sition 5.2 as well as Lemmas 5.3 and 5.4.
Proof of Lemma 5.6.
Without loss of generality, we may assume that M is con-nected. Define a : = lim sup n →∞ Area( u n ) < ∞ & e : = lim sup n →∞ E + ( u n , g n ) . If e is infinite, we choose a sequence of auxiliary metrics g ′ n on M satisfying E + ( u n , g ′ n ) ≤ π Area( u n ) + , hich exist by [FWb, Theorem 1.2] and [FWb, Section 5]. Thus, after potentiallyredefining g n : = g ′ n , we may assume that e is finite.We first address the special setting where Γ is a single Jordan curve and M adisc-type surface. We may assume that M = ¯ D and, since all Riemannian metricson ¯ D are conformally equivalent, that each g n is equal to the standard Euclideanmetric g Eucl . Now precompose each u n with a conformal di ff eomorphism ϕ n of ¯ D such that tr( u n ◦ ϕ n ) satisfies for each n the same prefixed three-point conditionon ∂ D and Γ , see p. 1149 in [LW17a]. Note that the maps v n : = u n ◦ ϕ n satisfyArea( v n ) = Area( u n ) and E + ( v n , g Eucl ) = E + ( u n , g Eucl ). It then follows by [LW17a,Proposition 7.4] that the family { tr( v n ) : n ∈ N } is equicontinuous, and therefore byProposition 5.2 that there exists u ∈ Λ ( ¯ D , Γ , X ) withArea( u ) ≤ lim sup n →∞ Area( v n ) = a & E + ( u , g Eucl ) ≤ lim sup n →∞ E + ( v n , g Eucl ) = e as in the first option proposed by the lemma.From now on, we assume that M is a connected surface which is not of disc-type.Since every conformal class of Riemannian metrics on M has a hyperbolic repre-sentative (respectively a flat one if M is of cylindrical type), we lose no generalityin assuming that all the metrics g n are hyperbolic (respectively flat). In the rest ofthe proof, we discuss three di ff erent cases of outcomes in which ultimately eitherLemma 5.3, Lemma 5.4 or Proposition 5.2 is used to deduce one of the optionsstated in the lemma itself.First assume that(5.3) inf { sys rel ( M , g n ) : n ∈ N } > . Then by [FWa, Theorem 3.3] (respectively its analogue for flat metrics) there existdi ff eomorphisms ϕ n of M and a metric g on M such that the pullback-metrics ϕ ∗ n g n converge (up to a subsequence) smoothly to g . This convergence implies for themaps v n : = u n ◦ ϕ n ∈ Λ ( M , Γ , Y n ) that E + ( v n , g ) ≤ C n · E + ( u n , g n ) , where C n ≥ n → ∞ . In particular, the energies E + ( v n , g ) areuniformly bounded. Now assume furthermore that the family(5.4) { tr( v n ) : n ∈ N } is equicontinuouswith respect to the metric g . Then by Proposition 5.2 there exists u ∈ Λ ( M , Γ , X )with Area( u ) ≤ a & E + ( u , g ) ≤ e as in the first option of the lemma.In the remaining two cases, we discuss the outcomes if either the bound (5.3)does not hold; or if it does indeed, but property (5.4) fails for the traces of theconstructed maps v n ∈ Λ ( M , Γ , Y n ). Let ρ j : = √ C j + , where C ≥ ρ ′ j : = ρ ′ Γ ( ρ j ) foreach j ∈ N . We claim that in either of these subcases, there exist a sequence ofreductions M ∗ j of M , a subsequence ( u n j ) ⊂ Λ ( M , Γ , Y n j ), (2 C ρ j )-thickenings Y ∗ j of Y n j and fillings u ∗ j ∈ Λ ( M ∗ j , Γ , Y ∗ j ) uch that Area( u ∗ j ) ≤ Area( u n j ) + − j . The existence of a sequence as implied in the lemma is then true by the follow-ing two observations. Firstly, there are only finitely many reductions of M up todi ff eomorphism, hence we may assume that each M ∗ j is equal to the same reduc-tion M ∗ of M by passing to a subsequence of M ∗ j . Secondly, the spaces Y ∗ j are( ε n j + C ρ j )-thickenings of X , where ε n is the thickening parameter of Y n , and thus( Y ∗ j ) an X -approximating sequence.We continue by showing the claim and first suppose that (5.3) is violated. Weonly discuss the case for hyperbolic metrics, the situation for flat metrics beinganalogous. The assumption on the systoles of g n implies that there exists a subse-quence ( g n j ) such that sys rel ( M , g n j ) = : λ j → . Choosing this subsequence appropriately, we may assume that λ j < min ρ ′ j K , arsinh ! , where we define K : = sup n E + ( u n , g n ) < ∞ . By Lemma 5.1, for each j thereexists a (2 C ρ j )-thickening Y ∗ j of Y n j admitting a ( C , ρ j )-quadratic isoperimetricinequality. Since the spaces Y ∗ j contain X (and hence Γ ) isometrically and since themetrics g n are all hyperbolic, we have by Lemma 5.3 that there exist reductions M ∗ j of M and maps u ∗ j ∈ Λ ( M ∗ j , Γ , Y ∗ j ) withArea( u ∗ j ) ≤ Area( u n j ) + C ρ j ≤ Area( u n j ) + − j . This shows the claim in the first subcase.Lastly, we address the case where (5.3) is true, but (5.4) is violated for the ob-tained metric g . Choose for each j ∈ N a number 0 < δ j ≤ δ g such that π · K | log( δ j ) | ! ≤ ρ ′ j . From the assumption of nonequicontinuity of { tr( v n ) } , it follows that there exists ε > j there exists a map tr( v n j ) : M → Y n j and a segment γ − j ⊂ ∂ M satisfying ℓ g ( γ − j ) ≤ δ j & ℓ X (tr( v n j ) ◦ γ − j ) > ε. Notice that for all j big enough we have that ρ j ≤ ε , so in particular ℓ X (tr( v n j ) ◦ γ − j ) > ρ j . Let Y ∗ j be given analogously as in the previous subcase. Then by Lemma 5.4 thereexist reductions M ∗ j of M and mappings u ∗ j ∈ Λ ( M ∗ j , Γ , Y ∗ j ) satisfyingArea( u ∗ j ) ≤ Area( v n j ) + C ρ j ≤ Area( u n j ) + − j . This shows the claim in the second subcase and completes the proof of the lemma. (cid:3) .4. Proof of the main result.
Finally, we are able to complete the proof of The-orem 1.2.
Proof of Theorem 1.2.
The statements (i) and (ii) of the theorem have already beenproved in Sections 3.2 and 4.2. Thus it remains to show (iii) as well as existence inthe general case, where X might not admit a local quadratic isoperimetric inequalityand Γ might be a configuration of overlapping or self-intersecting curves.We begin with the proof of part (iii) and assume that Γ is a collection of dis-joint rectifiable Jordan curves. For n ∈ N we set Y n : = X and choose maps u n ∈ Λ ( M , Γ , X ) such that Area( u n ) ≤ a p ( Γ , X ) + − n . Since we assumed that the Douglas condition holds for p , Γ and X , we may applyProposition 5.5 to the sequences ( Y n ) and ( u n ). This shows that Λ min : = { u ∈ Λ ( M , Γ , X ) : Area( u ) = a p ( Γ , X ) } is nonempty. Choose sequences of maps u n ∈ Λ min and Riemannian metrics g n on M such thatlim n →∞ E + ( u n , g n ) = inf { E + ( w , h ) : w ∈ Λ min , h a Riemannian metric on M } = : e . Applying Proposition 5.5 to the sequences ( Y n ), ( g n ) and ( u n ), one sees that thereexist u ∈ Λ min and a Riemannian metric g on M such that E + ( u , g ) = e . Then by[FWb, Corollary 1.3] u is infinitesimally isotropic with respect to g . This completesthe proof in the special case that the configuration is assumed to consist of disjointJordan curves.We move on to the general case. Let ( X n ) be an X -approximating sequence,where every X n admits some local quadratic isoperimetric inequality: such an ap-proximating sequence exists by Proposition 5.1. Then ( Y n ) : = (( X n ) Γ ) defines an X Γ -approximating sequence, where the collar extensions are performed as definedin Section 3.2. By Lemma 3.4, there exist maps u n ∈ Λ ( M , ˜ Γ , Y n ) such thatArea( u n ) ≤ a p ( Γ , X n ) + k X j = H ( Z j ) + − n ≤ a p ( Γ , X ) + k X j = H ( Z j ) + − n . Then by Lemma 3.3, and since the Douglas condition holds for p , Γ and X , one haslim sup n →∞ Area( u n ) ≤ a p ( Γ , X ) + k X j = H ( Z j ) < a ∗ p ( Γ , X ) + k X j = H ( Z j ) ≤ a ∗ p ( ˜ Γ , X Γ ) . Thus applying Proposition 5.5 to the sequences ( Y n ) and ( u n ) shows that the Dou-glas condition holds for p , ˜ Γ and X Γ and that a p ( ˜ Γ , X Γ ) ≤ a p ( Γ , X ) + k X j = H ( Z j ) . Since ˜ Γ is a configuration of disjoint Jordan curves, the Douglas condition and thefirst part of the proof imply that there exist v ∈ Λ ( M , ˜ Γ , X Γ ) and a Riemannianmetric g on M such that Area( v ) = a p ( ˜ Γ , X Γ ) and v is infinitesimally isotropic with espect to g . For the projection u : = P Γ ◦ v Lemma 3.3 implies that u ∈ Λ ( M , Γ , X )with Area( u ) ≤ Area( v ) − k X j = H ( Z j ) ≤ a p ( Γ , X ) , and thus Area( u ) = a p ( Γ , X ). Furthermore, the composition P Γ ◦ v agrees with v on the complement of v − ( Z ) = u − ( Γ ), hence u is infinitesimally isotropic on M \ u − ( Γ ) with respect to g . This concludes the proof of the theorem in the generalcase. (cid:3) Translation to the smooth setting.
To obtain Theorem 1.1, we make the fol-lowing observations, where M ∈ M ( k ), ( X , h ) is a complete Riemannian manifoldand u ∈ W , ( M , X ). • By the Hopf-Rinow theorem, X defines a proper geodesic metric space. • Homogeneously regular Riemannian manifolds admit a local quadraticisoperimetric inequality. See [Jos85] for the definition and compare Sec-tion 4.3 in [Cre20a] for the simple argument. • Smooth Riemannian manifolds are locally of curvature bounded above,compare for example [BH99, Theorem II.1A.6]. • Compact C curves in smooth Riemannian manifolds have finite total cur-vature, see [CFM10]. • As a consequence of (2.1), for almost every z ∈ M the approximate metricderivative ap md u z defines a Euclidean seminorm on T z M , and hence u isinfinitesimally isotropic if and only if it is weakly conformal. • Weakly conformal area minimizers in X are minimizers of the Dirich-let energy, and thus weakly harmonic in the classical sense. Continu-ous weakly harmonic maps between Riemannian manifolds are howeversmooth by [Jos17, Theorem 9.4.1].With these observations at hand, Theorem 1.2 is easily seen to imply Theorem 1.1.6. M inimizers under the conditions of cohesion and adhesion Let X be a complete metric space, M a smooth compact and connected surfaceand η >
0. A mapping u : M → X is said to be η -cohesive if u is continuous and ℓ ( u ( c )) ≥ η for every non-contractible closed curve c in M . Definition 6.1.
A family F of maps from M to X is said to satisfy the condition ofcohesion if there exists η > such that every map in F is η -cohesive. Now let c ⊂ M be an embedded arc such that the endpoints of c lie on ∂ M and let u : M → X be continuous. If the endpoints of c lie on a single component ∂ M j , thenthey divide ∂ M j into two components c − and c + , where the notation is chosen suchthat ℓ ( u ( c − )) ≤ ℓ ( u ( c + )). Let ¯ ρ : (0 , ∞ ) → (0 , ∞ ) be a function such that ¯ ρ ( ρ ) ≤ ρ for every ρ ∈ (0 , ∞ ). We say that u : M → X is ¯ ρ -adhesive if u is continuous andfor every arc c with endpoints in ∂ M and of image-length ℓ ( u ( c )) ≤ ¯ ρ ( ρ ), one hasthat the endpoints lie in the same connected component of ∂ M and ℓ ( u ( c − )) < ρ. efinition 6.2. A family F of maps from M to X is said to satisfy the conditionof adhesion if there exists a function ¯ ρ : (0 , ∞ ) → (0 , ∞ ) as above such that everymap in F is ¯ ρ -adhesive. Let Γ be a configuration of k ≥ X and M ∈ M ( k ).Set e ( M , Γ , X ) : = inf { E + ( u , g ) : u ∈ Λ ( M , Γ , X ) , g a Riemannian metric on M } . An energy minimizing sequence in Λ ( M , Γ , X ) is a sequence of pairs ( u n , g n ) ofmappings u n ∈ Λ ( M , Γ , X ) and Riemannian metrics g n on M such that E + ( u n , g n ) → e ( M , Γ , X )as n tends to infinity. Theorem 6.3.
Let X be a proper metric space and Γ ⊂ X a configuration of k ≥ rectifiable closed curves. Let M ∈ M ( k ) be connected. If there exist an energy min-imizing sequence in Λ ( M , Γ , X ) satisfying the conditions of cohesion and adhesion,then there exist u ∈ Λ ( M , Γ , X ) and a Riemannian metric g on M such thatE + ( u , g ) = e ( M , Γ , X ) . For any such u and g the map u is infinitesimally isotropic with respect to g. If X is a complete Riemannian manifold, then energy minimizers are preciselyweakly conformal area minimizers. For more general spaces X however, the rela-tion is more complicated and energy minimizers need not be area minimizers, seefor example [LW17b, LW17a]. Nevertheless, one can obtain existence of area min-imizers for singular configurations in proper metric spaces if there exists an areaminimizing sequence satisfying the conditions of cohesion and adhesion by mod-ifying the proofs of [FWb, Theorem 1.6] and [FWb, Proposition 5.3] accordingly.However, as in [FWb, Theorem 1.6] and [FWb, Proposition 5.3], either the ob-tained area minimizers are potentially not infinitesimally isotropic, or one has tochoose a somewhat di ff erent interpretation of the term ’area’. Proof of Theorem 6.3.
It follows from [FWb, Corollary 1.3] that any energy mini-mizing pair ( u , g ) is infinitesimally isotropic. Thus it remains to show existence ofsuch a pair.First assume that M is not of disc-type. If Γ is a configuration of disjoint Jordancurves, then any continuous u ∈ Λ ( M , Γ , X ) satisfies a ρ ′ Γ -condition of adhesion,where ρ ′ Γ is as in Section 5.2. In fact, under this observation, the proof of Theo-rem 6.3 for such M is a straightforward generalization of the proof of [FWa, The-orem 8.2]. Namely, if one replaces in the statements of Propositions 8.3 and 8.4 in[FWa] the assumption that Γ consists of disjoint Jordan curves by the assumptionthat u is ¯ ρ -adhesive, the proofs become virtually identical upon replacing ρ ′ = ρ ′ Γ by ¯ ρ . With these modified propositions at hand, the proof of Theorem 6.3 is com-pleted as is that of [FWa, Theorem 8.2].Finally assume that Γ is a single curve and that M = ¯ D . If Γ is constant, the resultis trivial. Otherwise we may represent Γ as a composition of 3 curves Γ , Γ , Γ ofequal length. We also decompose S into three consecutive arcs ¯ Γ , ¯ Γ , ¯ Γ of equallength. We say that a continuous map u ∈ Λ ( M , Γ , X ) satisfies the 3 -arc condition if u | ¯ Γ i is a parametrization of Γ i for every i = , ,
3. Fix K ≥ ρ : (0 , ∞ ) → (0 , ∞ ). Let F be the family of maps u ∈ Λ ( M , Γ , X ) which re ¯ ρ -adhesive, satisfy the 3-arc condition and have energy E + ( u , g Eucl ) ≤ K . Weclaim that the trace family { u | S : u ∈ F } is equicontinuous. To prove this claim,we fix 0 < ε < ℓ ( Γ ) / p ∈ S and u ∈ F . Let 0 < δ < π K | log δ | ! < ¯ ρ ( ε ) . For 0 < r <
1, denote by c r the arc { z ∈ ¯ D : | z − p | = r } . By the Courant-Lebesguelemma, [LW17a, Lemma 7.3], there is r ∈ ( δ, √ δ ) such that ℓ ( u ◦ c r ) ≤ ¯ ρ ( ε ). The¯ ρ -adhesiveness then implies that ℓ ( u ◦ c − r ) ≤ ε , and hence it follows from the 3-arc condition together with the choice of ε that c − r = B ( p , r ) ∩ S . Thus, for any x ∈ B ( p , δ ) ∩ S , one has d ( u ( x ) , u ( p )) ≤ ε . Since the choice of δ was independentof u and p , the claimed equicontinuity follows.Now let ( u n , g n ) be an energy minimizing sequence which is ¯ ρ -adhesive. Sinceall metrics on the disc are conformally equivalent, we may assume that g n = g Eucl for each n ∈ N . Furthermore, after precomposing with Moebius transforms, onehas that all u n satisfy the 3-arc condition. Thus by the claim the sequence ( u n | S )is equicontinuous and hence Proposition 5.2 implies the existence of the desiredenergy minimizer. (cid:3)
7. A ppendix
In this section we discuss the proof of Lemma 5.1. A metric space X will becalled δ -geodesic , where δ >
0, if for all x , y ∈ X satisfying d ( x , y ) < δ there isa curve γ in X joining x to y such that ℓ ( γ ) = d ( x , y ). Lemma 5.1 is only a slightstrengthening of the following consequence of [LWY20, Lemma 3.3]. Lemma 7.1.
There is a universal constant C ≥ such that for every proper, δ -geodesic metric space X and < ε ≤ δ , there exists an ε -thickening Y of X suchthat Y is proper and satisfies a ( C , ε/ C ) -quadratic isoperimetric inequality. [LWY20, Lemma 3.3] is stated for spaces which are globally geodesic, thoughthe proof readily gives the claimed result for δ -geodesic spaces. Namely, in theproof the assumption only comes into play when estimating the diameter of thesmall ball B z with respect to its induced intrinsic metric by twice the radius. Thisestimate holds in a δ -geodesic space as soon as the radius of the ball is boundedfrom above by δ . More precisely, this estimate is used twice: on p. 241 of [Wen08]to estimate the diameter of X z and on p. 242 to find the curves ¯ γ j .For the proof of Lemma 5.1, recall that the injective hull E ( X ) of a compactmetric space X is a compact geodesic metric space. Furthermore, X ⊂ E ( X ) iso-metrically and diam( E ( X )) = diam( X ), see for example [Lan13]. Proof of Lemma 5.1.
We claim that for any δ >
0, there is an (8 δ )-thickening Z of X such that Z is proper and δ -geodesic. Lemma 5.1 then follows by first ap-plying the claim to X , yielding a (8 C ε )-thickening Z of X which is proper and( C ε )-geodesic, where C is as in Lemma 7.1; and then applying Lemma 7.1 to Z to obtain a ( C ε )-thickening Y of Z which is proper and admits a ( C , ε )-quadraticisoperimetric inequality. It remains to note that Y is a (9 C ε )-thickening of X andredefine C .In order to prove the claim, we perform a variation of the construction discussedin [Wen08] and [LWY20]. Let S be a maximal δ -separated subset in X . For z ∈ S et B z : = B ( z , δ ) and X z : = E ( B z ). Then diam( B z ) ≤ δ and hence diam( X z ) ≤ δ .We set Z : = (cid:16) G z ∈ S X z (cid:17)(cid:14) ∼ , where x ∼ y if x ∈ B z ⊂ X z , y ∈ B w ⊂ X w and x = y . The space Z is endowedwith the quotient metric. It follows from the construction that Z is proper and a(4 δ )-thickening of X , compare also [LWY20].It remains to show that Z is δ -geodesic. To this end, let x , y ∈ Z such that d ( x , y ) < δ . Then either x and y lie in a common X z and d ( x , y ) = d X z ( x , y ) or thereare z , w ∈ S , u ∈ X z ∩ X and v ∈ X w ∩ X such that d ( x , y ) = d X z ( x , u ) + d X ( u , v ) + d X w ( v , y ) . In the former case, the distance is realized by a curve because X z is geodesic. By thesame reasoning, it su ffi ces to show that d ( u , v ) is realized by the length of a curvein Z in the latter case. By maximality of S there exists s ∈ S such that d X ( s , u ) ≤ δ and hence u , v ∈ X s . As X s ⊂ Z is a geodesic subset, the claim follows. (cid:3) R eferences [AK00] Luigi Ambrosio and Bernd Kirchheim. Currents in metric spaces. Acta Math. ,185(1):180, 2000.[AKP19] Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin.
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