Geometry & Topology Volume 22, issue 1 (2018)

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Intrinsic structure of minimal discs in metric spaces

Alexander Lytchak and Stefan Wenger

Geometry & Topology 22 (2018) 591–644

DOI: 10.2140/gt.2018.22.591

Bibliography

1	J C Álvarez Paiva, A C Thompson, Volumes on normed and Finsler spaces, from: "A sampler of Riemann–Finsler geometry" (editors D Bao, R L Bryant, S S Chern, Z Shen), Math. Sci. Res. Inst. Publ. 50, Cambridge Univ. Press (2004) 1 MR2132656
2	L Ambrosio, B Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000) 527 MR1800768
3	A Bernig, Centroid bodies and the convexity of area functionals, J. Differential Geom. 98 (2014) 357 MR3238312
4	M Bonk, B Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002) 127 MR1930885
5	D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
6	D Chiron, On the definitions of Sobolev and BV spaces into singular spaces and the trace problem, Commun. Contemp. Math. 9 (2007) 473 MR2348841
7	U Dierkes, S Hildebrandt, F Sauvigny, Minimal surfaces, 339, Springer (2010) MR2566897
8	J Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263 MR1501590
9	R D Edwards, The topology of manifolds and cell-like maps, from: "Proceedings of the International Congress of Mathematicians" (editor O Lehto), Acad. Sci. Fennica (1980) 111 MR562601
10	H Federer, Geometric measure theory, 153, Springer (1969) MR0257325
11	D H Fremlin, Spaces of finite length, Proc. London Math. Soc. 64 (1992) 449 MR1152994
12	K P Hart, J i Nagata, J E Vaughan, editors, Encyclopedia of general topology, Elsevier (2003) MR2049453
13	J Heinonen, P Koskela, N Shanmugalingam, J T Tyson, Sobolev spaces on metric measure spaces: an approach based on upper gradients, 27, Cambridge Univ. Press (2015) MR3363168
14	J R Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964) 65 MR0182949
15	S V Ivanov, Volumes and areas of Lipschitz metrics, Algebra i Analiz 20 (2008) 74 MR2454453
16	S V Ivanov, Filling minimality of Finslerian 2–discs, Tr. Mat. Inst. Steklova 273 (2011) 192 MR2893545
17	M B Karmanova, Area and coarea formulas for mappings of the Sobolev classes with values in a metric space, Sibirsk. Mat. Zh. 48 (2007) 778 MR2355373
18	B Kirchheim, Rectifiable metric spaces : local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994) 113 MR1189747
19	N J Korevaar, R M Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993) 561 MR1266480
20	U Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013) 297 MR3096307
21	G Leoni, A first course in Sobolev spaces, 105, Amer. Math. Soc. (2009) MR2527916
22	A Lytchak, S Wenger, Isoperimetric characterization of upper curvature bounds, preprint (2016) arXiv:1611.05261
23	A Lytchak, S Wenger, Regularity of harmonic discs in spaces with quadratic isoperimetric inequality, Calc. Var. Partial Differential Equations 55 (2016) MR3528439
24	A Lytchak, S Wenger, Area minimizing discs in metric spaces, Arch. Ration. Mech. Anal. 223 (2017) 1123 MR3594354
25	A Lytchak, S Wenger, Canonical parametrizations of metric discs, preprint (2017) arXiv:1701.06346
26	A Lytchak, S Wenger, Energy and area minimizers in metric spaces, Adv. Calc. Var. 10 (2017) 407 MR3707085
27	A Lytchak, S Wenger, R Young, Dehn functions and Hölder extensions in asymptotic cones, preprint (2016) arXiv:1608.00082
28	F Morgan, M Ritoré, Isoperimetric regions in cones, Trans. Amer. Math. Soc. 354 (2002) 2327 MR1885654
29	C B Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. 49 (1948) 807 MR0027137
30	P Papasoglu, On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality, J. Differential Geom. 44 (1996) 789 MR1438192
31	A Petrunin, Metric minimizing surfaces, Electron. Res. Announc. Amer. Math. Soc. 5 (1999) 47 MR1679453
32	A Petrunin, On intrinsic isometries in Euclidean space, Algebra i Analiz 22 (2010) 140 MR2828830
33	A Petrunin, S Stadler, Metric minimizing surfaces revisited, preprint (2017) arXiv:1707.09635
34	C Pommerenke, Boundary behaviour of conformal maps, 299, Springer (1992) MR1217706
35	T Radó, On Plateau’s problem, Ann. of Math. 31 (1930) 457 MR1502955
36	K Rajala, Uniformization of two-dimensional metric surfaces, Invent. Math. 207 (2017) 1301 MR3608292
37	Y G Reshetnyak, Sobolev-type classes of functions with values in a metric space, II, Sibirsk. Mat. Zh. 45 (2004) 855 MR2091651
38	P Tukia, The planar Schönflies theorem for Lipschitz maps, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 49 MR595177
39	R L Wilder, Topology of manifolds, 32, Amer. Math. Soc. (1949) MR0029491
40	K Wildrick, Quasisymmetric parametrizations of two-dimensional metric planes, Proc. Lond. Math. Soc. 97 (2008) 783 MR2448247