Volume 24, issue 1 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 30
Issue 3, 835–1201
Issue 2, 389–833
Issue 1, 1–388

Volume 29, 9 issues

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Min-max minimal disks with free boundary in Riemannian manifolds

Longzhi Lin, Ao Sun and Xin Zhou

Geometry & Topology 24 (2020) 471–532
Bibliography
1 D R Adams, A note on Riesz potentials, Duke Math. J. 42 (1975) 765 MR458158
2 F J Almgren Jr., The homotopy groups of the integral cycle groups, Topology 1 (1962) 257 MR146835
3 F J Almgren Jr., The theory of varifolds, mimeographed notes (1965)
4 J Chen, G Tian, Compactification of moduli space of harmonic mappings, Comment. Math. Helv. 74 (1999) 201 MR1691947
5 R Coifman, P L Lions, Y Meyer, S Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993) 247 MR1225511
6 T H Colding, W P Minicozzi II, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008) 2537 MR2460871
7 T H Colding, W P Minicozzi II, Width and mean curvature flow, Geom. Topol. 12 (2008) 2517 MR2460870
8 R Courant, Dirichlet’s principle, conformal mapping, and minimal surfaces, Springer (1977) MR0454858
9 R Courant, N Davids, Minimal surfaces spanning closed manifolds, Proc. Nat. Acad. Sci. U.S.A. 26 (1940) 194 MR1472
10 C De Lellis, J Ramic, Min-max theory for minimal hypersurfaces with boundary, Ann. Inst. Fourier (Grenoble) 68 (2018) 1909 MR3893761
11 W Ding, J Li, Q Liu, Evolution of minimal torus in Riemannian manifolds, Invent. Math. 165 (2006) 225 MR2231956
12 W Ding, G Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom. 3 (1995) 543 MR1371209
13 L C Evans, Partial differential equations, 19, Amer. Math. Soc. (2010) MR2597943
14 A M Fraser, On the free boundary variational problem for minimal disks, PhD thesis, Stanford University (1998) MR2698288
15 A M Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000) 931 MR1755947
16 A Fraser, R Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math. 226 (2011) 4011 MR2770439
17 A Fraser, R Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2016) 823 MR3461367
18 L Grafakos, Modern Fourier analysis, 250, Springer (2009) MR2463316
19 M Grüter, J Jost, On embedded minimal disks in convex bodies, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986) 345 MR868522
20 F Hélein, Harmonic maps, conservation laws and moving frames, 150, Cambridge Univ. Press (2002) MR1913803
21 J Hohrein, Existence of unstable minimal surfaces of higher genus in manifolds of nonpositive curvature, PhD thesis, Universität Heidelberg (1994)
22 R A Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966) 249 MR223874
23 T Iwaniec, G Martin, Geometric function theory and non-linear analysis, Oxford Univ. Press (2001) MR1859913
24 J Jost, Existence results for embedded minimal surfaces of controlled topological type, II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13 (1986) 401 MR881099
25 J Jost, Two-dimensional geometric variational problems, Wiley (1991) MR1100926
26 J Jost, L Liu, M Zhu, The qualitative behavior at the free boundary for approximate harmonic maps from surfaces, Math. Ann. 374 (2019) 133 MR3961307
27 T Lamm, L Lin, Estimates for the energy density of critical points of a class of conformally invariant variational problems, Adv. Calc. Var. 6 (2013) 391 MR3199733
28 P Laurain, Analyse des problèmes conformément invariants, from: "Séminaire Laurent Schwartz–Équations aux dérivées partielles et applications, 2016–2017" (editors F Golse, F Merle), Ed. Éc. Polytech. (2017) MR3790939
29 P Laurain, L Lin, Energy convexity of intrinsic bi-harmonic maps and applications, I: Spherical target, preprint (2018) arXiv:1805.09428
30 P Laurain, R Petrides, Regularity and quantification for harmonic maps with free boundary, Adv. Calc. Var. 10 (2017) 69 MR3592578
31 M Li, X Zhou, Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory, preprint (2016) arXiv:1611.02612
32 F C Marques, A Neves, Min-max theory of minimal surfaces and applications, from: "Mathematical Congress of the Americas" (editors J A de la Peña, J A López-Mimbela, M Nakamura, J Petean), Contemp. Math. 656, Amer. Math. Soc. (2016) 13 MR3457593
33 R Montezuma, A mountain pass theorem for minimal hypersurfaces with fixed boundary, preprint (2018) arXiv:1802.04757
34 M Morse, C Tompkins, The existence of minimal surfaces of general critical types, Ann. of Math. 40 (1939) 443 MR1503471
35 F Müller, A Schikorra, Boundary regularity via Uhlenbeck–Rivière decomposition, Analysis (Munich) 29 (2009) 199 MR2554638
36 T H Parker, Bubble tree convergence for harmonic maps, J. Differential Geom. 44 (1996) 595 MR1431008
37 T H Parker, J G Wolfson, Pseudo-holomorphic maps and bubble trees, J. Geom. Anal. 3 (1993) 63 MR1197017
38 S Poornima, An embedding theorem for the Sobolev space W1,1, Bull. Sci. Math. 107 (1983) 253 MR719267
39 J Qing, Boundary regularity of weakly harmonic maps from surfaces, J. Funct. Anal. 114 (1993) 458 MR1223710
40 J Qing, G Tian, Bubbling of the heat flows for harmonic maps from surfaces, Comm. Pure Appl. Math. 50 (1997) 295 MR1438148
41 T Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007) 1 MR2285745
42 T Rivière, Conformally invariant variational problems, preprint (2012) arXiv:1206.2116
43 T Rivière, A viscosity method in the min-max theory of minimal surfaces, Publ. Math. Inst. Hautes Études Sci. 126 (2017) 177 MR3735867
44 T Rivière, M Struwe, Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008) 451 MR2383929
45 J Sacks, K Uhlenbeck, The existence of minimal immersions of 2–spheres, Ann. of Math. 113 (1981) 1 MR604040
46 C Scheven, Partial regularity for stationary harmonic maps at a free boundary, Math. Z. 253 (2006) 135 MR2206640
47 B Sharp, Higher integrability for solutions to a system of critical elliptic PDE, Methods Appl. Anal. 21 (2014) 221 MR3251921
48 B Sharp, M Zhu, Regularity at the free boundary for Dirac-harmonic maps from surfaces, Calc. Var. Partial Differential Equations 55 (2016) MR3465443
49 M Shiffman, The Plateau problem for non-relative minima, Ann. of Math. 40 (1939) 834 MR467
50 L Simon, Lectures on geometric measure theory, 3, Aust. National Univ. Centre for Mathematical Analysis (1983) MR756417
51 B Smyth, Stationary minimal surfaces with boundary on a simplex, Invent. Math. 76 (1984) 411 MR746536
52 M Struwe, On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984) 547 MR735340
53 M Struwe, Plateau’s problem and the calculus of variations, 35, Princeton Univ. Press (1988) MR992402
54 M Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, 34, Springer (2008) MR2431434
55 L Tartar, Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 1 (1998) 479 MR1662313
56 H C Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969) 318 MR243467
57 R Ye, On the existence of area-minimizing surfaces with free boundary, Math. Z. 206 (1991) 321 MR1095757
58 X Zhou, On the existence of min-max minimal torus, J. Geom. Anal. 20 (2010) 1026 MR2683775
59 X Zhou, On the free boundary min-max geodesics, Int. Math. Res. Not. 2016 (2016) 1447 MR3509932
60 X Zhou, On the existence of min-max minimal surface of genus g 2, Commun. Contemp. Math. 19 (2017) MR3665358