Document Zbl 07932383 - zbMATH Open

[1] Allard, W., On the first variation of a varifold, Ann. Math., 95, 417-491, 1972 · Zbl 0252.49028 [2] Almgren, FJ Jr, Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. Math. (2), 84, 277-292, 1966 · Zbl 0146.11905 [3] Almgren, FJ Jr, Almgren’s big regularity paper, World Scientific Monograph Series in Mathematics, 2000, River Edge: World Scientific Publishing Co. Inc., River Edge · Zbl 0985.49001 [4] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, 2000, Oxford: Clarendon Press, Oxford · Zbl 0957.49001 [5] Assimos, R., Jost, J.: The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2, Preprint [6] Barbosa, JLM, An extrinsic rigidity theorem for minimal immersion from \(S^2\) into \(S^n\), J. Differ. Geom., 14, 3, 355-368, 1980 · Zbl 0427.53028 [7] Bombieri, E.; De Giorgi, E.; Giusti, E., Minimal cones and the Bernstein problem, Invent. Math., 7, 243-268, 1969 · Zbl 0183.25901 [8] Bombieri, E.; De Giorgi, E.; Miranda, M., Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Rational Mech. Anal., 32, 255-267, 1969 · Zbl 0184.32803 [9] Bombieri, E.; Giusti, E., Harnack’s inequality for elliptic differential equations on minimal surfaces, Invent. Math., 15, 24-46, 1972 · Zbl 0227.35021 [10] Brendle, S., The isoperimetric inequality for a minimal submanifold in Euclidean space, J. Am. Math. Soc., 34, 595-603, 2021 · Zbl 1482.53012 [11] Cheeger, J.; Naber, A., Quantitative stratification and the regularity of harmonic maps and minimal currents, Commun. Pure Appl. Math., 66, 965-990, 2013 · Zbl 1269.53063 [12] Cheng, SY; Li, P.; Yau, ST, Heat equations on minimal submanifolds and their applications, Am. J. Math., 106, 1033-1065, 1984 · Zbl 0575.53038 [13] Chern, SS; Osserman, R., Complete minimal surfaces in Euclidean \(n-\) space, J. d’Anal. Math., 19, 15-34, 1967 · Zbl 0172.22802 [14] Colding, TH; Minicozzi, WP II, Liouville theorems for harmonic sections and applications, Commun. Pure Appl. Math., 51, 2, 113-138, 1998 · Zbl 0928.58022 [15] Colding, TH; Minicozzi, WP II, A course in minimal surfaces, Graduate Studies in Mathematics, 2011, Providence: American Mathematical Society, Providence · Zbl 1242.53007 [16] De Giorgi, E.: Sulla differentiabilitá e l’analiticitá delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino, s. III, parte I, pp 25-43 (1957) · Zbl 0084.31901 [17] De Giorgi, E.: Frontiere orientate di misura minima. Sem. Mat. Scuola Norm. Sup. Pisa, pp. 1-56 (1961) [18] De Lellis, C.; Spadaro, E., Regularity of area minimizing currents I: gradient \(L^p\) estimates, Geom. Funct. Anal., 24, 6, 1831-1884, 2014 · Zbl 1307.49043 [19] De Lellis, C.; Spadaro, E., Regularity of area minimizing currents II: center manifold, Ann. Math. (2), 183, 2, 499-575, 2016 · Zbl 1345.49052 [20] De Lellis, C.; Spadaro, E., Regularity of area minimizing currents III: blow-up, Ann. Math. (2), 183, 2, 577-617, 2016 · Zbl 1345.49053 [21] Ding, Q., Liouville type theorems and Hessian estimates for special Lagrangian equations, Math. Ann., 386, 1163-1200, 2023 · Zbl 1526.35160 [22] Ding, Q.; Jost, J.; Xin, YL, Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature, Am. J. Math., 138, 2, 287-327, 2016 · Zbl 1341.53089 [23] Ding, Q.; Jost, J.; Xin, YL, Existence and non-existence of minimal graphs, J. Math. Pure Appl., 179, 391-424, 2023 · Zbl 1532.53013 [24] Ding, Q.; Xin, YL; Yang, L., The rigidity theorems of self shrinkers via Gauss maps, Adv. Math., 303, 5, 151-174, 2016 · Zbl 1356.53065 [25] Ding, W.; Yuan, Yu, Resolving the singularities of the minimal Hopf cones (English summary), J. Partial Differ. Equ., 19, 3, 218-231, 2006 · Zbl 1101.49030 [26] Federer, H.; Fleming, W., Normal and integral currents, Ann. Math., 72, 458-520, 1960 · Zbl 0187.31301 [27] Fischer-Colbrie, D., Some rigidity theorems for minimal submanifolds of the sphere, Acta Math., 145, 29-46, 1980 · Zbl 0464.53047 [28] Fleming, WH, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2), 11, 69-90, 1962 · Zbl 0107.31304 [29] Giaquinta, M., Martinazzi, L.: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Scu. Norm. Sup. Pisa (2012) · Zbl 1262.35001 [30] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order, 1983, Berlin: Springer, Berlin · Zbl 0562.35001 [31] Giusti, E., Minimal Surfaces and Functions of Bounded Variation, 1984, Boston: Birkhäuser Boston Inc., Boston · Zbl 0545.49018 [32] Hasanis, T.; Savas-Halilaj, A.; Vlachos, T., Minimal graphs in \(\mathbb{R}^4\) with bounded Jacobians, Proc. Am. Math. Soc., 137, 3463-3471, 2009 · Zbl 1178.53061 [33] Hildebrandt, S.; Jost, J.; Widman, K., Harmonic mappings and minimal submanifolds, Invent. Math., 62, 269-298, 1980 · Zbl 0446.58006 [34] Jost, J.; Xin, YL, Bernstein type theorems for higher codimension, Calc. Var., 9, 277-296, 1999 · Zbl 0960.53035 [35] Jost, J.; Xin, YL; Yang, L., Curvature estimates for minimal submanifolds of higher codimension and small G-rank, Trans. AMS, 367, 12, 8301-8323, 2015 · Zbl 1327.53010 [36] Jost, J.; Xin, YL; Yang, L., The geometry of Grassmannian manifolds and Bernstein-type theorems for higher codimension, Ann della Scuola Normale Superiore di Pisa Serie V, XVI, 1-39, 2016 · Zbl 1339.58005 [37] Lawson, HB; Osserman, R., Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math., 139, 1-17, 1977 · Zbl 0376.49016 [38] Li, P.: Lecture notes on geometric analysis (1996) [39] Lin, FH; Yang, XP, Geometric Measure Theory: An Introduction, 2002, Beijing/Boston: Science Press/International Press, Beijing/Boston · Zbl 1074.49011 [40] Liu, L.; Wang, G.; Weng, L., The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space, J. Funct. Anal., 285, 2023 · Zbl 1526.53012 [41] Michael, J.; Simon, LM, Sobolev and mean-value inequalities on generalized submanifolds of \(\mathbb{R}^n\), Commun. Pure Appl. Math., 26, 361-379, 1973 · Zbl 0252.53006 [42] Moser, J., A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Commun. Pure Appl. Math., 13, 457-468, 1960 · Zbl 0111.09301 [43] Moser, J., On Harnack’s theorem for elliptic differential equations, Commun. Pure Appl. Math., 14, 577-591, 1961 · Zbl 0111.09302 [44] Osserman, R., Minimal varieties, Bull. Am. Math. Soc., 75, 1092-1120, 1969 · Zbl 0188.53801 [45] Ruh, EA; Vilms, J., The tension field of Gauss maps, Trans. AMS, 149, 569-573, 1970 · Zbl 0199.56102 [46] Schoen, R.; Simon, L., Regularity of stable minimal hypersurfaces, Commun. Pure Appl. Math., 34, 742-797, 1981 [47] Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Center for Mathematical Analysis, vol. 3. Australian National University (1983) · Zbl 0546.49019 [48] Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math. (2), 88, 62-105, 1968 · Zbl 0181.49702 [49] Wang, M-T, On graphic Bernstein type results in higher codimension, Trans. AMS, 355, 1, 265-271, 2003 · Zbl 1021.53005 [50] Wang, M-T, Interior gradient bounds for solutions to the minimal surface system, Am. J. Math., 126, 4, 921-934, 2004 · Zbl 1073.35091 [51] Wickramasekera, N., A general regularity theory for stable codimension 1 integral varifolds, Ann. Math., 179, 3, 843-1007, 2014 · Zbl 1307.58005 [52] Wickramasekera, N., A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces, Calc. Var. Partial Differ. Equ., 51, 3-4, 799-812, 2014 · Zbl 1320.35109 [53] Wong, Y-C, Differential geometry of Grassmann manifolds, Proc. N.A.S., 57, 589-594, 1967 · Zbl 0154.21404 [54] Xiaowei, X.; Yang, L.; Zhang, Y., Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system, J. Math. Pures Appl., 129, 266-300, 2019 · Zbl 1425.53011 [55] Xin, Y., Minimal Submanifolds and Related Topics, 2018, Singapore: World Scientific Publications, Singapore [56] Xin, YL; Yang, L., Convex functions on Grassmannian manifolds and Lawson-Osserman Problem, Adv. Math., 219, 4, 1298-1326, 2008 · Zbl 1147.49034