The Analysis of Willmore Surfaces and Its Generalizations in Higher Dimensions

Abstract

We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^{d}$ depending on the first and second fundamental forms and their covariant derivatives.

Keywords

Conformally invariant variational problems / higher order elliptic systems / Sobolev immersions / Willmore surfaces / minimal surfaces / integrability by compensation / concentration compactness

Acknowledgment

This project is financed by Swiss National Science Foundation, project (Grant No. SNF 200020 219429).

References

[1] Adams D R. A note on Riesz potentials. Duke Math. J., 1975, 42(4): 765-778.

[2] Adams D R. Morrey spaces. Lecture Notes in Applied and Birkhäuser/Springer,Numerical Harmonic Analysis. Cham, 2015.

[3] Adams D R, Hedberg L I.Function Spaces and Potential Theory. Grundlehren der Mathe-matischen Wissenschaften, Vol. 314. Springer-Verlag, Berlin, 1996.

[4] Ahlfors L V.Lectures on Quasiconformal Mappings, 2nd edition, with supplemental chap-ters by Earle C J, Kra I, Shishikura M and Hubbard J H. University Lecture Series, Vol. 38. American Mathematical Society, Providence, RI, 2006.

[5] Allard W K. On the first variation of a varifold: boundary behavior. Ann. Math., 1975, 101(2): 418-446.

[6] Allen B F, Gover A R. Higher Willmore energies from tractor coupled GJMS operators. Preprint,

[7] Anastasiou G, Araya I J, Bueno P, et al. Higher-dimensional Willmore energy as holo-graphic entanglement entropy. J. High Energy Phys., 2025, 1: 81.

[8] Aubin J-P, Frankowska H. Set-Valued Analysis. Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009.

[9] Aubin T. Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[10] Babich M, Bobenko A. Willmore tori with umbilic lines and minimal surfaces in hyperbolic space. Duke Math. J., 1993, 72(1): 151-185.

[11] Baraket S. Estimations of the best constant involving the L∞ norm in Wente’s inequality. Ann. Fac. Sci. Toulouse Math., 1996, 6(5): 373-385.

[12] Bauer M, Kuwert E. Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not., 2003, 10: 553-576.

[13] Bennett C, Sharpley R. Pure and Applied Mathematics,Interpolation of Operators. Vol. 129. Academic Press, Inc., Boston, MA, 1988.

[14] Bernard Y. Noether’s theorem and the Willmore functional. Adv. Calc. Var., 2016, 9(3): 217-234.

[15] Bernard Y.Structural equations for critical points of conformally invariant curvature ener-gies in 4d.

[16] Bernard Y, Lan T, Martino D, et al. The regularity of critical points to the Dirichlet energy of the mean curvature in dimension 4.

[17] Bernard Y, Laurain P, Marque N. Energy estimates for the tracefree curvature of Willmore surfaces and applications. Arch. Ration. Mech. Anal., 2023, 247(1): 8.

[18] Bernard Y, Rivière T.Local Palais-Smale sequences for the Willmore functional. Comm. Anal. Geom., 2011, 19(3): 563-599.

[19] Bernard Y, Rivière T. Energy quantization for Willmore surfaces and applications. Ann. Math., 2014, 180(2): 87-136.

[20] Bethuel F, Ghidaglia J-M. Improved regularity of solutions to elliptic equations involving Jacobians and applications. J. Math. Pures Appl., 1993, 72(9): 441-474.

[21] Blaschke W. Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III: Differentialgeometrie der Kreise und Kugeln. Grundlehren der mathematischen Wissenschaften, Vol. 29. Springer, Berlin, 1929.

[22] Blatt S. A singular example for the Willmore flow. Analysis (Munich), 2009, 29(4): 407-430.

[23] Blitz S, Gove A R, Waldron A. Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers. Commun. Contemp. Math., 2024, 26(5): 2350014.

[24] Bourgain J, Brezis H. On the equation div Y = f and application to control of phases. J. Amer. Math. Soc., 2003, 16(2): 393-426.

[25] Brazda K, Kruˇz´ık M, Stefanelli U. Generalized minimizing movements for the varifold Canham-Helfrich flow. Adv. Calc. Var., 2024, 17(3): 727-751.

[26] Brendle S. The isoperimetric inequality for a minimal submanifold in Euclidean space. J. Amer. Math. Soc., 2021, 34(2): 595-603.

[27] Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Univer-sitext, Springer, New York, 2011.

[28] Bryant R L. A duality theorem for Willmore surfaces. J. Differential Geom., 1984, 20(1): 23-53.

[29] Bryant R L. Surfaces in conformal geometry. In: The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 227-240; Proc. Sympos. Pure Math., Vol. 48. American Mathe-matical Society, Providence, RI, 1988.

[30] Bryant R L.On the conformal volume of 2-tori.

[31] Byun S-S. Elliptic equations with BMO coefficients in Lipschitz domains. Trans. Amer. Math. Soc., 2005, 357(3): 1025-1046.

[32] Chanillo S, Li Y. Continuity of solutions of uniformly elliptic equations in R2. Manuscripta Math., 1992, 77(4): 415-433.

[33] Chavel I. Eigenvalues in Riemannian geometry, including a chapter by Burton Randol, with an appendix by Jozef Dodziuk. Pure and Applied Mathematics,Vol. 115. Academic Press, Inc., Orlando, FL, 1984.

[34] Chen B-Y. Some conformal invariants of submanifolds and their applications. Boll. Un. Mat. Ital., 1974, 10(4): 380-385.

[35] Chen J, Lamm T. A Bernstein type theorem for entire Willmore graphs. J. Geom. Anal., 2013, 23(1): 456-469.

[36] Chen J, Li Y. Radially symmetric solutions to the graphic Willmore surface equation. J. Geom. Anal., 2017, 27(1): 671-688.

[37] Chern S-S, Lashof R K. On the total curvature of immersed manifolds. Amer. J. Math., 1957, 79: 306-318.

[38] Chiarenza F, Frasca M, Longo P. W2,p-solvability of the Dirichlet problem for nondiver-gence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc., 1993, 336(2): 841-853.

[39] Chodosh O, Maximo D. On the topology and index of minimal surfaces. J. Differential Geom., 2016, 104(3): 399-418.

[40] Chodosh O, Maximo D. On the topology and index of minimal surfaces II. J. Differential Geom., 2023, 123(3): 431-459.

[41] Cianchi A, Pick L. Sobolev embeddings into BMO, VMO, and L∞. Ark. Mat., 1998, 36(2): 317-340.

[42] Clop A, Faraco D, Mateu J, et al. Beltrami equations with coefficient in the Sobolev space W1,p. Publ. Mat., 2009, 53(1): 197-230.

[43] Coifman R, Lions P-L, Meyer Y, et al. Compensated compactness and Hardy spaces. J. Math. Pures. Appl., 1993, 72(9): 247-286.

[44] Costabel M, McIntosh A. On Bogovskiiˇ and regularized Poincare´ integral operators for de Rham complexes on Lipschitz domains. Math. Z., 2010, 265(2): 297-320.

[45] Curry S N, Gover A R. An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity. In: Asymptotic analysis in general relativity,86-170. London Mathematical Society Lecture Note Series, Vol. 443. Cambridge University Press, Cambridge, 2018.

[46] Da Lio F, Gianocca M, Rivière T. Morse index stability for critical points to conformally invariant Lagrangians. arXiv preprint arXiv: 2212. 03124, 2022.

[47] Da Lio F, Palmurella F, Rivière T. A resolution of the Poisson problem for elastic plates. Arch. Ration. Mech. Anal., 2020, 236(3): 1593-1676.

[48] Da Lio F, Rivière T. Critical chirality in elliptic systems. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2021, 38(5): 1373-1405.

[49] Da Lio F, Rivière T. Conservation laws for p-Harmonic systems with antisymmetric poten-tials and applications. Arch. Ration. Mech. Anal., 2025, 249(2): 21.

[50] Dall’Acqua A, M üller M, Schätzle R M, et al. The Willmore flow of tori of revolution. Anal. PDE, 2024, 17(9): 3079-3124.

[51] Dall’Acqua A, Schätzle R M. Rotational symmetric Willmore surfaces with umbilic lines. Preprint,

[52] Daners D.Domain perturbation for linear and semi-linear boundary value problems. Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 6, 1-81. Elsevier/North-Holland, Amsterdam, 2008.

[53] Demailly J P.Complex analytic and differential geometry. Université de Grenoble I, Greno-ble, 2012.

[54] Dorfmeister J F, Wang P. Willmore surfaces in spheres: the DPW approach via the confor-mal Gauss map. Abh. Math. Semin. Univ. Hambg., 2019, 89(1): 77-103.

[55] Durrett R. Probability: theory and examples, 5th edition. Cambridge Series in Statistical and Probabilistic Mathematics, Vol. 49. Cambridge University Press, Cambridge, 2019.

[56] Ejiri N.Willmore surfaces with a duality in SN(1). Proc. London Math. Soc. (3), 1988, 57(2): 383-416.

[57] Escher J, Mayer U F, Simonett G. The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal., 1998, 29(6): 1419-1433.

[58] Evans L C.Partial Differential Equations, 2nd edition. Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, RI, 2010.

[59] Federer H.Geometric measure theory, reprint of the 1969 edition. Classics in Mathematics, Vol. 153. Springer-Verlag, Berlin, 1996.

[60] Ferus D, Pedit F. S1-equivariant minimal tori in S4 and S1-equivariant Willmore tori in S3. Math. Z., 1990, 204(2): 269-282.

[61] Fischer-Colbrie D. On complete minimal surfaces with finite Morse index in three-manifolds. Invent. Math., 1985, 82(1): 121-132.

[62] Gallot S. Isoperimetric inequalities based on integral norms of Ricci curvature. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque No. 157-158, 1988, 191-216.

[63] Ge Y. Estimations of the best constant involving the L2 norm in Wente’s inequality and compact H-surfaces in Euclidean space. ESAIM Control Optim. Calc. Var., 1998, 3: 263-300.

[64] Germain S. Recherches sur la Théorie des Surfaces élastiques. Mme Ve Courcier, Libraire pour les sciences, Paris, 1821.

[65] Gianocca M. Optimal weighted Wente’s inequality. Comm. Partial Differential Equations, 2025, 50(1-2): 245-257.

[66] Gilbarg D, Trudinger N S.Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition. Classics in Mathematics, Vol. 224. Springer-Verlag, Berlin, 2001.

[67] Glaros M, Gover A R, Halbasch M, et al. Variational calculus for hypersurface functionals: singular Yamabe problem Willmore energies. J. Geom. Phys., 2019, 138: 168-193.

[68] Gover A R, Waldron A. Renormalized volume. Comm. Math. Phys., 2017, 354(3): 1205-1244.

[69] Gover A R, Waldron A. A calculus for conformal hypersurfaces and new higher Willmore energy functionals. Adv. Geom., 2020, 20(1): 29-60.

[70] Gover A R, Waldron A. Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem. Comm. Anal. Geom., 2021, 29(4): 779-836.

[71] Grafakos L.Modern Fourier Analysis, 3rd edition. Graduate Texts in Mathematics, Vol. 250. Springer, New York, 2014.

[72] Graham C R. Volume renormalization for singular Yamabe metrics. Proc. Amer. Math. Soc., 2017, 145: 1781-1792.

[73] Graham C R, Jenne R, Mason L J, et al. Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc., 1992, 46(3): 557-565.

[74] Graham C R, Karch A. Minimal area submanifolds in AdS × compact. J. High Energy Phys., 2014, 4: 168.

[75] Graham C R, Reichert N. Higher-dimensional Willmore energies via minimal submanifold asymptotics. Asian J. Math., 2020, 24(4): 571-610.

[76] Graham C R, Witten E. Conformal anomaly of submanifold observables in AdS/CFT cor-respondence. Nuclear Phys. B, 1999, 546(1-2): 52-64.

[77] Guven J. Conformally invariant bending energy for hypersurfaces. J. Phys. A, 2005, 38(37): 7943-7955.

[78] Han Q, Lin F H.Elliptic Partial Differential Equations, 2nd edition. Courant Lecture Notes, Vol. 1. Courant Institute of Mathematical Sciences, New York; American Mathematical So-ciety, Providence, RI, 2011.

[79] Heinonen J, Kilpeläinen T. BLD-mappings in W2,2 are locally invertible. Math. Ann., 2000, 318(2): 391-396.

[80] Hélein F. Willmore immersions and loop groups. J. Differential Geom., 1998, 50(2): 331-385.

[81] Hélein F. Harmonic Maps, Conservation Laws and Moving Frames, Translated from the 1996 French Original, with a Foreword by James Eells, 2nd edition. Cambridge Tracts in Mathematics, Vol. 150. Cambridge University Press, Cambridge, 2002.

[82] Heller L, Ndiaye C B. First explicit constrained Willmore minimizers of non-rectangular conformal class. Adv. Math., 2021, 386: 107804.

[83] Hencl S, Koskela P. Lectures on Mappings of Finite Distortion. Lecture Notes in Mathemat-ics,Vol. 2096. Springer, Cham, 2014.

[84] Henningson M, Skenderis K. The holographic Weyl anomaly. J. High Energy Phys., 1998, 7: 23.

[85] Hirsch J, Kusner R, Mäder-Baumdicker E. Geometry of complete minimal surfaces at in-finity and the Willmore index of their inversions. Calc. Var. Partial Differential Equations, 2024, 63(8): 190.

[86] Hirsch J, Mäder-Baumdicker E. On the index of Willmore spheres. J. Differential Geom., 2023, 124(1): 37-79.

[87] Hsu L, Kusner R, Sullivan J. Minimizing the squared mean curvature integral for surfaces in space forms. Experiment. Math., 1992, 1(3): 191-207.

[88] Hughes J F. Another proof that every Eversion of the sphere has a quadruple point. Amer. J. Math., 1985, 107(2): 501-505.

[89] Hummel C. Gromov’s compactness theorem for pseudo-holomorphic curves. Progress in Mathematics,Vol. 151. Birkhäuser Verlag, Basel, 1997.

[90] Hunt R A. On L(p,q) spaces. Enseign. Math., 1966, 12(2): 249-276.

[91] Imayoshi Y, Taniguchi M. An Introduction to Teichm üller Spaces, translated and revised from the Japanese by the authors. Springer-Verlag, Tokyo, 1992.

[92] Iwaniec T, Kopiecki R. Stability in the differential equations for quasiregular mappings. In: Analytic Functions (Proceedings of the Seventh Conference held at Kozubnik, 1979), 203-214. Lecture Notes in Mathematics, Vol. 798. Springer, Berlin, 1980.

[93] Jones P W. Extension Theorems for BMO. Indiana Univ. Math. J., 1980, 29(1): 41-66.

[94] Jost J. Compact Riemann Surfaces: an Introduction to Contemporary Mathematics, 3rd edition. Universitext. Springer-Verlag, Berlin, 2006.

[95] Jost J. Riemannian Geometry and Geometric Analysis, 7th edition. Universitext. Springer, Cham, 2017.

[96] Jost J, Li-Jost X Q.Calculus of variations. Cambridge Studies in Advanced Mathematics, Vol. 64. Cambridge University Press, Cambridge, 1998.

[97] Keller L G A, Mondino A, Rivière T. Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint. Arch. Ration. Mech. Anal., 2014, 212(2): 645-682.

[98] Klingenberg W. Lectures on Closed Geodesics. Grundlehren der Mathematischen Wis-senschaften, Vol. 230. Springer-Verlag, Berlin-New York, 1978.

[99] Kohsaka Y, Nagasawa T. On the existence of solutions of the Helfrich flow and its center manifold near spheres. Differential Integral Equations, 2006, 19(2): 121-142.

[100] Kusner R. Comparison surfaces for the Willmore problem. Pacific. J., 1989, 138(2): 317-345.

[101] Kusner R, Lü Y, Wang P. The Willmore problem for surfaces with symmetry.

[102] Kusner R, Mondino A, Schulze F. Willmore Bending Energy on the Space of Surfaces. MSRI Emissary, Spring, Berlin,

[103] Kuwert E, Li Y. W2,2-conformal immersions of a closed Riemann surface into Rn. Comm. Anal. Geom., 2012, 20(2): 313-340.

[104] Kuwert E, Schätzle R. The Willmore flow with small initial energy. J. Differential Geom., 2001, 57(3): 409-441.

[105] Kuwert E, Schatzle R. Minimizers of the Willmore functional under fixed conformal class. J. Differential Geom., 2013, 93(3): 471-530.

[106] Lamm T, Nguyen H T. Branched Willmore spheres. J. Reine Angew. Math., 2015, 701: 169-194.

[107] Lan T.Analysis of Surface Energies Depending on the First and Second Fundamental Form. Master thesis, ETH Z ürich, March, 2023.

[108] Laurain P, Martino D.Huber theorem revisited in dimensions 2 and 4. Preprint,

[109] Laurain P, Rivière T. Energy quantization of Willmore surfaces at the boundary of the mod-uli space. Duke Math. J., 2018, 167(11): 2073-2124.

[110] Laurain P, Rivière T. Optimal estimate for the gradient of Green’s function on degenerating surfaces and applications. Comm. Anal. Geom., 2018, 26(4): 887-913.

[111] Lee J.Introduction to Smooth Manifolds, 2nd edition. Graduate Texts in Mathematics, Vol. 218. Springer, New York, 2013.

[112] Li P, Yau S. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue on compact surfaces. Invent. Math., 1982, 69(2): 269-291.

[113] Li Y. Some remarks on Willmore surfaces embedded in R3. J. Geom. Anal., 2016, 26(3): 2411-2424.

[114] Li Y, Luo Y, Tang H. On the moving frame of a conformal map from 2-disk into Rn. Calc. Var. Partial Differential Equations, 2013, 46(1-2): 31-37.

[115] Li Y, Yin H, Zhou Y. 3-circle Theorem for Willmore surfaces I I-degeneration of the complex structure. Preprint,

[116] Loewner C, Nirenberg L. Partial differential equations invariant under conformal or pro-jective transformations. In: Contributions to Analysis (a Collection of Papers Dedicated to Lipman Bers), 245-272. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974.

[117] Luo Y, Sun J. Remarks on a Bernstein type theorem for entire Willmore graphs in R3. J. Geom. Anal., 2014, 24(3): 1613-1618.

[118] Ma X, Pedit F, Wang P.Classification of Willmore 2-spheres in Sn.

[119] Ma X, Wang C, Wang P. Classification of Willmore two-spheres in the 5-dimensional sphere. J. Differential Geom., 2017, 106(2): 245-281.

[120] Ma X. Adjoint transform of Willmore surfaces in Sn. Manuscripta Math., 2006, 120(2): 163-179.

[121] Ma X, Wang P. Willmore 2-spheres in Sn:a survey. In: Geometry and Topology of Mani-folds (Shanghai, 2014), 211-233. Springer Proceedings in Mathematics & Statistics, Vol. 154. Springer, Tokyo, 2016.

[122] Mäder-Baumdicker E, Seidel J. The Willmore energy landscape of spheres and avoidable singularities of the Willmore flow. Preprint,

[123] Maldacena J. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys., 1998, 2(2): 231-252.

[124] Marque N. Moduli spaces of Willmore immersions. Doctoral thesis, Université Paris Cité, December 2019.

[125] Marque N. Conformal Gauss map geometry and application to Willmore surfaces in model spaces. Potential Anal., 2021, 54(2): 227-271.

[126] Marque N. Minimal bubbling for Willmore surfaces. Int. Math. Res. Not. IMRN, 2021, 23: 17708-17765.

[127] Marque N, Martino D. Some global properties of umbilic points of Willmore immersions in the 3-sphere. Preprint, https://arxiv.org/abs/2506.10720, 2025.

[128] Marques F C, Neves A. Min-max theory and the Willmore conjecture. Ann. Math., 2014, 179(2): 683-782.

[129] Martino D. A duality theorem for a four dimensional Willmore energy. Preprint, https://arxiv.org/abs/2308.11433, 2024.

[130] Martino D. Classification of branched Willmore spheres. Preprint, Proc. Amer. Math. Soc.(Accepted), https://arxiv.org/abs/2306.07965, 2024.

[131] Martino D. Energy quantization for Willmore surfaces with bounded index. J. Eur. Math. Soc., 2025, published online first, DOI 10.4171/JEMS/1629.

[132] Martino D, Rivière T. Construction of harmonic coordinates for weak immersions. Preprint, https://arxiv.org/abs/2510.10601, 2025.

[133] Martino D, Rivière T. Weak immersions with second fundamental form in a critical Sobolev space. Preprint, https://arxiv.org/abs/2510.10594, 2025.

[134] Martio O, Väisäla¨ J. Elliptic equations and maps of bounded length distortion. Math. Ann., 1988, 282(3): 423-443.

[135] Max N, Banchoff T. Every sphere eversion has a quadruple point. In: Contributions to Analysis and Geometry (Baltimore, MD, 1980), 191-209. Johns Hopkins University Press, Baltimore, MD, 1981.

[136] Mayer U F, Simonett G. Self-intersections for Willmore flow. In: Evolution Equations: Ap-plications to Physics, Industry, Life Sciences and Economics (Levico Terme, 2000), 341-348. Progress in Nonlinear Differential Equations and Their Applications, Vol. 55. Birkhäuser Verlag, Basel, 2003.

[137] Maz’ya V. Sobolev Spaces: with Applications to Elliptic Partial Differential Equations, 2nd, revised and augmented edition. Grundlehren der mathematischen Wissenschaften, Vol. 342. Springer, Heidelberg, 2011.

[138] Meyers N G. An Lp-estimate for the gradient of solutions of second order elliptic diver-gence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1963, 17: 189-206.

[139] Michael J H, Simon L M. Sobolev and mean-value inequalities on generalized submanifolds of Rn. Comm. Pure Appl. Math., 1973, 26: 361-379.

[140] Michelat A. Morse index estimates of min-max Willmore surfaces. Preprint, https://arxiv.org/abs/1808.07700, 2018.

[141] Michelat A. Morse theoretical aspects of the Willmore energy. Doctoral thesis, Diss. ETH No. 26223, 2019.

[142] Michelat A. On the Morse index of Willmore spheres in S3. Comm. Anal. Geom., 2020, 28(6): 1337-1406.

[143] Michelat A. Morse index stability of branched Willmore immersions. Preprint, https://arxiv.org/abs/1808.07700, 2025.

[144] Michelat A, Rivière T. A viscosity method for the min-max construction of closed geodesics. ESAIM Control Optim. Calc. Var., 2016, 22(4): 1282-1324.

[145] Michelat A, Rivière T. The classification of branched Willmore spheres in the 3-sphere and the 4-sphere. Ann. Sci. éc. Norm. Supér., 2022, 55(5): 1199-1288.

[146] Michelat A, Rivière T. Pointwise expansion of degenerating immersions of finite total cur-vature. J. Geom. Anal., 2023, 33: 24.

[147] Michelat A, Rivière T. Morse index stability of Willmore immersions. Memoirs of the European Mathematical Society, to appear, memoir combining the two articles https://arxiv.org/abs/2306.04608 and https://arxiv.org/abs/2306.04609, 2025.

[148] Mingione G. Gradient potential estimates. J. Eur. Math. Soc. (JEMS), 2011, 13(2): 459-486.

[149] Mondino A, Nguyen H T. Global conformal invariants of submanifolds. Ann. Inst. Fourier (Grenoble), 2018, 68(6): 2663-2695.

[150] Montiel S, Ros A. Minimal immersions of surfaces by the first eigenfunctions and confor-mal area. Invent. Math., 1986, 83(1): 153-166.

[151] Montiel S. Willmore two-spheres in the four-sphere. Trans. Amer. Math. Soc., 2000, 352(10): 4469-4486.

[152] Moore J D. Introduction to Global Analysis: Minimal Surfaces in Riemannian Manifolds. Graduate Studies in Mathematics, Vol.187. American Mathematical Society, Providence, RI, 2017.

[153] Morrey C B Jr. Multiple Integrals in the Calculus of Variations, reprint of the 1966 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008.

[154] M üller S, ˇSver´ak V. On surfaces of finite total curvature. J. Differential Geom., 1995, 42(2): 229-258.

[155] Nag S. The Complex Analytic Theory of Teichm üller Spaces. Canadian Mathematical Soci-ety Series of Monographs and Advanced Texts. Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.

[156] Ndiaye C B, Schätzle R M. New examples of conformally constrained Willmore minimizers of explicit type. Adv. Calc. Var., 2015, 8(4): 291-319.

[157] Noether E. Invariante Variationsprobleme. Nachr. König. Gesellsch. Wiss. Göttingen Math-Phys. Kl., 1918, 235-257.

[158] Palmer B. The conformal Gauss map and the stability of Willmore surfaces. Ann. Global Anal. Geom., 1991, 9(3): 305-317.

[159] Palmurella F, Rivière T. The parametric approach to the Willmore flow. Adv. Math., 2022, 400: 108257.

[160] Palmurella F, Rivière T. The parametric Willmore flow. J. Reine Angew. Math., 2024, 811: 1-91.

[161] Pinkall U. Hopf tori in S3. Invent. Math., 1985, 81(2): 379-386.

[162] Plotnikov P I. Isothermal coordinates of W2,2 immersions: a counterexample. Tr. Mat. Inst. Steklova, 2024, 327: 265-282.

[163] Poisson S D. Mémoire sur les surfaces élastiques. Mem. Cl. Sei. Math. Phys., Inst. de France, 1814, 167-225.

[164] Pozzetta M. On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves. ESAIM Control Optim. Calc. Var., 2021, 27: S2.

[165] Reimann H M. Functions of bounded mean oscillation and quasiconformal mappings. Comment. Math. Helv., 1974, 49: 260-276.

[166] Reshetnyak Y G. Space Mappings with Bounded Distortion, Translated from the Russian by McFaden H H. Translations of Mathematical Monographs, Vol.73. American Mathematical Society, Providence, RI, 1989.

[167] Rivière T. Analysis aspects of Willmore surfaces. Invent. Math., 2008, 174(1): 1-45.

[168] Rivière T. Sequences of smooth global isothermic immersions. Comm. Partial Differential Equations, 2013, 38(2): 276-303.

[169] Rivière T. Variational principles for immersed surfaces with L2-bounded second funda-mental form. J. Reine Angew. Math., 2014, 695: 41-98.

[170] Rivière T. Critical weak immersed surfaces within sub-manifolds of the Teichm üller space. Adv. Math., 2015, 283: 232-274.

[171] Rivière T. Weak Immersions of Surfaces with L2-bounded Second Fundamental Form. In: Geometric analysis, 303-384. IAS/Park City Mathematics Series, Vol. 22. American Mathe-matical Society, Providence, RI, 2016.

[172] Rivière T. A viscosity method in the min-max theory of minimal surfaces. Publ. Math. Inst. Hautes études Sci., 2017, 126: 177-246.

[173] Rivière T. Willmore minmax surfaces and the cost of the sphere eversion. J. Eur. Math. Soc.(JEMS), 2021, 23(2): 349-423.

[174] Rupp F. The volume-preserving Willmore flow. Nonlinear Anal., 2023, 230: 113220.

[175] Rupp F. The Willmore flow with prescribed isoperimetric ratio. Comm. Partial Differential Equations, 2024, 49(1-2): 148-184.

[176] Rupp F, Scharrer C, Schlierf M. Gradient flow dynamics for cell membranes in the Canham-Helfrich model. Preprint, 2024, https://arxiv.org/abs/2408.07493.

[177] Scharrer C. Embedded Delaunay tori and their Willmore energy. Nonlinear Anal., 2022, 223: 113010.

[178] Scharrer C, West A. Energy quantization for constrained Willmore surfaces. Preprint, https://arxiv.org/abs/2505.19898, 2025.

[179] Scharrer C, West A. On the minimization of the Willmore energy under a constraint on total mean curvature and area. Arch. Ration. Mech. Anal., 2025, 249(2): 17.

[180] Schätzle R M. The umbilic set of Willmore surfaces. Preprint, https://arxiv.org/abs/1710.06127, 2017.

[181] Schoen R, Uhlenbeck K K. Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom., 1983, 18(2): 253-268.

[182] Schygulla J. Willmore minimizers with prescribed isoperimetric ratio. Arch. Ration. Mech. Anal., 2012, 203(3): 901-941.

[183] Simon L. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom., 1993, 1(2): 281-326.

[184] Simonett G. The Willmore flow near spheres. Differential Integral Equations, 2001, 14(8): 1005-1014.

[185] Stein E M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory In-tegrals, with the Assistance of Timothy S. Murphy. Princeton Mathematical Series, Vol. 43, Princeton University Press, Princeton, NJ, 1993.

[186] Thomsen G. über konforme Geometrie I: Grundlagen der konformen Flächentheorie. Abh. Math. Sem. Univ. Hamburg, 1924, 3(1): 31-56.

[187] Topping P. The optimal constant in Wente’s L∞ estimate. Comment. Math. Helv., 1997, 72(2): 316-328.

[188] Tromba A J. Teichm üller Theory in Riemannian Geometry, lecture notes prepared by Jochen Denzler. Lectures in Mathematics ETH Z ürich. Birkhäuser Verlag, Basel, 1992.

[189] Uhlenbeck K K. Connections with Lp bounds on curvature. Comm. Math. Phys., 1982, 83(1): 31-42.

[190] Weiner J L. et al. On a problem of Chen, Willmore, Indiana Univ. Math. J., 1978, 27(1): 19-35.

[191] Wente H C. An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl., 1969, 26: 318-344.

[192] Willmore T J. Note on embedded surfaces. An. Sti. Univ. “Al. I. Cuza” Iasi Sect. I a Mat.(N.S.), 1965, 11B: 493-496.

[193] Wu N, Yan Z. Energy reduction for Fourth order Willmore energy. Preprint, https://arxiv.org/abs/2504.08105, 2025.

[194] Wu Y. Classification of Willmore surfaces with vanishing Gaussian curvature. J. Geom. Anal., 2023, 33(7): 209.

[195] Xia Q, Shen Y. Weierstrass type representation of Willmore surfaces in Sn. Acta Math. Sin.(Engl. Ser.), 2004, 20(6): 1029-1046.

[196] Zhang Y. Graham-Witten’s conformal invariant for closed four-dimensional submanifolds. J. Math. Study, 2021, 54(2): 200-226.

[197] Ziemer W P. Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, Vol. 120. Springer-Verlag, New York, 1989.

[198] Zwiebach B. A First Course in String Theory, with a Foreword by David Gross, 2nd edition. Cambridge University Press, Cambridge, 2009.