## Geometry, analysis, and morphogenesis: Problems and prospects

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Marta Lewicka and L. Mahadevan
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## Abstract:

The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics, and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, which brings together analysis, geometry, and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discuss open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.## References

- Amit Acharya, Marta Lewicka, and Mohammad Reza Pakzad,
*The metric-restricted inverse design problem*, Nonlinearity**29**(2016), no. 6, 1769–1797. MR**3502228**, DOI 10.1088/0951-7715/29/6/1769 - S. Al Mosleh, A. Gopinathan, C. Santangelo,
*Growth of form in thin elastic structures*, Soft Matter,**14**(2018), no. 41, 8361–8371. - Marino Arroyo and Antonio DeSimone,
*Shape control of active surfaces inspired by the movement of euglenids*, J. Mech. Phys. Solids**62**(2014), 99–112. MR**3131810**, DOI 10.1016/j.jmps.2013.09.017 - B. Audoly and A. Boudaoud,
*Self-similar structures near boundaries in strained systems*, Phys. Rev. Lett.,**91**, 086105, (2004). - Peter Bella and Robert V. Kohn,
*Metric-induced wrinkling of a thin elastic sheet*, J. Nonlinear Sci.**24**(2014), no. 6, 1147–1176. MR**3275221**, DOI 10.1007/s00332-014-9214-9 - Peter Bella and Robert V. Kohn,
*Coarsening of folds in hanging drapes*, Comm. Pure Appl. Math.**70**(2017), no. 5, 978–1021. MR**3628880**, DOI 10.1002/cpa.21643 - Martine Ben Amar and Alain Goriely,
*Growth and instability in elastic tissues*, J. Mech. Phys. Solids**53**(2005), no. 10, 2284–2319. MR**2167636**, DOI 10.1016/j.jmps.2005.04.008 - M. Ben Amar, M. M. Müller, and M. Trejo,
*Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity*, New J. Physics,**14**(2012), 085014. - H. Ben Belgacem, S. Conti, A. DeSimone, and S. Müller,
*Rigorous bounds for the Föppl-von Kármán theory of isotropically compressed plates*, J. Nonlinear Sci.**10**(2000), no. 6, 661–683. MR**1799395**, DOI 10.1007/s003320010007 - Hafedh Ben Belgacem, Sergio Conti, Antonio DeSimone, and Stefan Müller,
*Energy scaling of compressed elastic films—three-dimensional elasticity and reduced theories*, Arch. Ration. Mech. Anal.**164**(2002), no. 1, 1–37. MR**1921161**, DOI 10.1007/s002050200206 - K. Bertoldi, V. Vitelli, J. Christensen, and M. Van Hecke,
*Flexible mechanical metamaterials*, Nature Reviews Materials,**2**(2017), no. 11, 1–11. - Kaushik Bhattacharya, Marta Lewicka, and Mathias Schäffner,
*Plates with incompatible prestrain*, Arch. Ration. Mech. Anal.**221**(2016), no. 1, 143–181. MR**3483893**, DOI 10.1007/s00205-015-0958-7 - M. K. Blees, A. W. Barnard, P. A. Rose, S. P. Roberts, K. L. McGill, P. Y. Huang, A. R. Ruyack, J. W. Kevek, B. Kobrin, D. A. Muller, and P. L. McEuen,
*Graphene kirigami*, Nature,**524**(2015), no. 7564, 204–207. - J. W. Boley, W. Van Rees, C. Lissandrello, M. Horenstein, R. Truby, A. Kotikian, J. Lewis, and L. Mahadevan,
*Shape-shifting structured lattices via multimaterial 4D printing*, Proceedings of the National Academy of Sciences,**116**(2019), no. 42, 20856–20862. - Ju. F. Borisov,
*$C^{1,\,\alpha }$-isometric immersions of Riemannian spaces*, Dokl. Akad. Nauk SSSR**163**(1965), 11–13 (Russian). MR**0192449** - Alberto Bressan and Marta Lewicka,
*A model of controlled growth*, Arch. Ration. Mech. Anal.**227**(2018), no. 3, 1223–1266. MR**3744385**, DOI 10.1007/s00205-017-1183-3 - S. Callens and A. Zadpoor,
*From flat sheets to curved geometries: Origami and kirigami approaches*, Materials Today,**21**(2018), no. 3, 241–264. - Wentao Cao and László Székelyhidi,
*Very weak solutions to the two-dimensional Monge-Ampére equation*, Sci. China Math.**62**(2019), no. 6, 1041–1056. MR**3951880**, DOI 10.1007/s11425-018-9516-7 - E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan,
*Conical dislocations in crumpling*, Nature,**401**(1999), no. 6748, 46–49. - E. Cerda and L. Mahadevan,
*Conical surfaces and crescent singularities in crumpled sheets*, Physical Review Letters,**80**(1998), no. 11, 2358. - E. Cerda and L. Mahadevan,
*Confined developable elastic surfaces: cylinders, cones and the*, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.*Elastica***461**(2005), no. 2055, 671–700. MR**2121930**, DOI 10.1098/rspa.2004.1371 - E. Cerda, L. Mahadevan, and J. M. Pasini,
*The elements of draping*, Proc. Natl. Acad. Sci. USA**101**(2004), no. 7, 1806–1810. MR**2033755**, DOI 10.1073/pnas.0307160101 - E. Cerda, K. Ravi-Chandar, and L. Mahadevan,
*Wrinkling of a stretched elastic sheet*, Nature,**419**(2002), 579. - G. Chaudhary, L. Niu, M. Lewicka, Q. Han, and L. Mahadevan,
*Geometric mechanics of random kirigami*, in preparation, 2021. - Choi, G.P., Dudte, L.H., Mahadevan, L.,
*Programming shape using kirigami tesselations*, Nat. Materials**18**(2019), 999-1004. - Choi, G.P., Dudte, L.H., Mahadevan, L.,
*Compact reconfigurable kirigami*, arXiv:2012.09241, 2020. - Philippe G. Ciarlet,
*Mathematical elasticity. Vol. I*, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR**936420** - Sergio Conti, Camillo De Lellis, and László Székelyhidi Jr.,
*$h$-principle and rigidity for $C^{1,\alpha }$ isometric embeddings*, Nonlinear partial differential equations, Abel Symp., vol. 7, Springer, Heidelberg, 2012, pp. 83–116. MR**3289360**, DOI 10.1007/978-3-642-25361-4_{5} - Sergio Conti and Francesco Maggi,
*Confining thin elastic sheets and folding paper*, Arch. Ration. Mech. Anal.**187**(2008), no. 1, 1–48. MR**2358334**, DOI 10.1007/s00205-007-0076-2 - Gianni Dal Maso,
*An introduction to $\Gamma$-convergence*, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1201152**, DOI 10.1007/978-1-4612-0327-8 - Camillo De Lellis and Dominik Inauen,
*$C^{1, \alpha }$ isometric embeddings of polar caps*, Adv. Math.**363**(2020), 106996, 39. MR**4054053**, DOI 10.1016/j.aim.2020.106996 - Camillo De Lellis, Dominik Inauen, and László Székelyhidi Jr.,
*A Nash-Kuiper theorem for $C^{1,1/5-\delta }$ immersions of surfaces in 3 dimensions*, Rev. Mat. Iberoam.**34**(2018), no. 3, 1119–1152. MR**3850282**, DOI 10.4171/RMI/1019 - Camillo De Lellis and László Székelyhidi Jr.,
*High dimensionality and h-principle in PDE*, Bull. Amer. Math. Soc. (N.S.)**54**(2017), no. 2, 247–282. MR**3619726**, DOI 10.1090/bull/1549 - Erik D. Demaine and Joseph O’Rourke,
*Geometric folding algorithms*, Cambridge University Press, Cambridge, 2007. Linkages, origami, polyhedra. MR**2354878**, DOI 10.1017/CBO9780511735172 - J. Dervaux and M. Ben Amar,
*Morphogenesis of growing soft tissues*, Phys. Rev. Lett.,**101**(2008), 068101. - Julien Dervaux, Pasquale Ciarletta, and Martine Ben Amar,
*Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Föppl-von Kármán limit*, J. Mech. Phys. Solids**57**(2009), no. 3, 458–471. MR**2500622**, DOI 10.1016/j.jmps.2008.11.011 - M. A. Dias, L. H. Dudte, L. Mahadevan, and C. D. Santangelo,
*Geometric mechanics of curved crease origami*, Phys. Rev. Lett.,**109**(2012], 114301. - L. Dudte, E. Vouga, T. Tachi, and L. Mahadevan,
*Programming curvature using origami tessellations*, Nature Materials,**15**(2016), no. 5, 583–588. - K. Efimenko, M. Rackaitis, E. Manias, A. Vaziri, L. Mahadevan, and J. Genzer,
*Self-similar nested wrinkling patterns in skins*, Nature—Materials,**4**(2005), 293–97. - E. Efrati, E. Sharon, and R. Kupferman,
*Elastic theory of unconstrained non-Euclidean plates*, J. Mech. Phys. Solids**57**(2009), no. 4, 762–775. MR**2510285**, DOI 10.1016/j.jmps.2008.12.004 - Gero Friesecke, Richard D. James, Maria Giovanna Mora, and Stefan Müller,
*Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence*, C. R. Math. Acad. Sci. Paris**336**(2003), no. 8, 697–702 (English, with English and French summaries). MR**1988135**, DOI 10.1016/S1631-073X(03)00028-1 - Gero Friesecke, Richard D. James, and Stefan Müller,
*A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity*, Comm. Pure Appl. Math.**55**(2002), no. 11, 1461–1506. MR**1916989**, DOI 10.1002/cpa.10048 - Gero Friesecke, Richard D. James, and Stefan Müller,
*A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence*, Arch. Ration. Mech. Anal.**180**(2006), no. 2, 183–236. MR**2210909**, DOI 10.1007/s00205-005-0400-7 - G. Gemmer, E. Sharon, T. Shearman, and S. Venkataramani,
*Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets*, EPL,**114**(2016), 24003. - John A. Gemmer and Shankar C. Venkataramani,
*Shape selection in non-Euclidean plates*, Phys. D**240**(2011), no. 19, 1536–1552. MR**2842894**, DOI 10.1016/j.physd.2011.07.002 - Gemmer, J., Venkataramani, S.,
*Shape transitions in hyperbolic non-Euclidean plates*, Soft Matter,**34**(2013), 8151-8161. - A. Gladman, E. Matsumoto, R. Nuzzo, L. Mahadevan, and J. Lewis,
*Biomimetic 4D printing*, Nature Materials,**15**(2016), 413–418. - Alain Goriely,
*The mathematics and mechanics of biological growth*, Interdisciplinary Applied Mathematics, vol. 45, Springer, New York, 2017. MR**3585488**, DOI 10.1007/978-0-387-87710-5 - E. Grinspun, A. Hirani, M. Desbrun, and P. Schröder,
*Discrete shells*, In Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation, pp. 62–67, (2003). - Mikhael Gromov,
*Partial differential relations*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR**864505**, DOI 10.1007/978-3-662-02267-2 - Pengfei Guan and Yan Yan Li,
*The Weyl problem with nonnegative Gauss curvature*, J. Differential Geom.**39**(1994), no. 2, 331–342. MR**1267893** - Qing Han and Jia-Xing Hong,
*Isometric embedding of Riemannian manifolds in Euclidean spaces*, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006. MR**2261749**, DOI 10.1090/surv/130 - Q. Han and M. Lewicka,
*Convex integration for the Monge-Ampère systems*, in preparation, 2021. - Q. Han, M. Lewicka, and L. Mahadevan,
*Geodesics and isometric immersions in kirigami*, submitted, 2021. - J. Hong and C. Zuily,
*Isometric embedding of the $2$-sphere with nonnegative curvature in $\textbf {R}^3$*, Math. Z.**219**(1995), no. 3, 323–334. MR**1339708**, DOI 10.1007/BF02572368 - Peter Hornung,
*Continuation of infinitesimal bendings on developable surfaces and equilibrium equations for nonlinear bending theory of plates*, Comm. Partial Differential Equations**38**(2013), no. 8, 1368–1408. MR**3169749**, DOI 10.1080/03605302.2013.795967 - Peter Hornung, Marta Lewicka, and Mohammad Reza Pakzad,
*Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells*, J. Elasticity**111**(2013), no. 1, 1–19. MR**3023590**, DOI 10.1007/s10659-012-9391-4 - Joseph A. Iaia,
*Isometric embeddings of surfaces with nonnegative curvature in $\textbf {R}^3$*, Duke Math. J.**67**(1992), no. 2, 423–459. MR**1177314**, DOI 10.1215/S0012-7094-92-06717-2 - K. D. Irvine and B. I. Shraiman,
*Mechanical control of growth: ideas, facts and challenges*, Development,**144**(2017), no. 23, 4238–4248. - Silvia Jiménez Bolaños and Marta Lewicka,
*Dimension reduction for thin films prestrained by shallow curvature*, Proc. A.**477**(2021), no. 2247, Paper No. 20200854, 24. MR**4247253** - Weimin Jin and Peter Sternberg,
*Energy estimates for the von Kármán model of thin-film blistering*, J. Math. Phys.**42**(2001), no. 1, 192–199. MR**1808773**, DOI 10.1063/1.1316058 - R. Kempaiah and Z. Nie,
*From nature to synthetic systems: shape transformation in soft materials*, J. Mater. Chem. B,**2**(2014), 2357–2368. - Jungwook Kim, James A. Hanna, Myunghwan Byun, Christian D. Santangelo, and Ryan C. Hayward,
*Designing responsive buckled surfaces by halftone gel lithography*, Science**335**(2012), no. 6073, 1201–1205. MR**2934634**, DOI 10.1126/science.1215309 - Yael Klein, Efi Efrati, and Eran Sharon,
*Shaping of elastic sheets by prescription of non-Euclidean metrics*, Science**315**(2007), no. 5815, 1116–1120. MR**2290671**, DOI 10.1126/science.1135994 - Y. Klein, S. Venkataramani, and E. Sharon,
*Experimental study of shape transitions and energy scaling in thin non-Euclidean plates*, Phys. Rev. Lett.,**106**(2011), 118303. - M. Koehl, W. K. Silk, H. Y. Liang, and L. Mahadevan,
*How kelp produce blade shapes suited to different flow regimes: A new wrinkle*, Integ. and Comp. Biol.,**48**(2008), 834–851. - Robert V. Kohn and Ethan O’Brien,
*On the bending and twisting of rods with misfit*, J. Elasticity**130**(2018), no. 1, 115–143. MR**3738374**, DOI 10.1007/s10659-017-9635-4 - Robert V. Kohn and Ethan O’Brien,
*The wrinkling of a twisted ribbon*, J. Nonlinear Sci.**28**(2018), no. 4, 1221–1249. MR**3817781**, DOI 10.1007/s00332-018-9447-0 - Nicolaas H. Kuiper,
*On $C^1$-isometric imbeddings. I, II*, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math.**17**(1955), 545–556, 683–689. MR**0075640**, DOI 10.1016/S1385-7258(55)50075-8 - Raz Kupferman and Cy Maor,
*A Riemannian approach to the membrane limit in non-Euclidean elasticity*, Commun. Contemp. Math.**16**(2014), no. 5, 1350052, 34. MR**3253899**, DOI 10.1142/S0219199713500521 - Raz Kupferman and Jake P. Solomon,
*A Riemannian approach to reduced plate, shell, and rod theories*, J. Funct. Anal.**266**(2014), no. 5, 2989–3039. MR**3158716**, DOI 10.1016/j.jfa.2013.09.003 - Raz Kupferman and Yossi Shamai,
*Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space*, Israel J. Math.**190**(2012), 135–156. MR**2956236**, DOI 10.1007/s11856-011-0187-1 - E. H. Lee,
*Elastic-plastic deformation at finite strains*, ASME J. Appl. Mech.,**36**(1969), 1–6. - H. Le Dret and A. Raoult,
*The membrane shell model in nonlinear elasticity: a variational asymptotic derivation*, J. Nonlinear Sci.**6**(1996), no. 1, 59–84. MR**1375820**, DOI 10.1007/s003329900003 - Marta Lewicka,
*Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models*, Arch. Ration. Mech. Anal.**236**(2020), no. 3, 1677–1707. MR**4076073**, DOI 10.1007/s00205-020-01500-y - Marta Lewicka and Danka Lučić,
*Dimension reduction for thin films with transversally varying prestrain: oscillatory and nonoscillatory cases*, Comm. Pure Appl. Math.**73**(2020), no. 9, 1880–1932. MR**4156611**, DOI 10.1002/cpa.21871 - Marta Lewicka, L. Mahadevan, and Mohammad Reza Pakzad,
*The Föppl-von Kármán equations for plates with incompatible strains*, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.**467**(2011), no. 2126, 402–426. With supplementary data available online. MR**2748099**, DOI 10.1098/rspa.2010.0138 - M. Lewicka, L. Mahadevan, and R. Pakzad,
*Models for elastic shells with incompatible strains*, Proceedings of the Royal Society A,**470**(2014), 20130604. - Marta Lewicka, L. Mahadevan, and Mohammad Reza Pakzad,
*The Monge-Ampère constraint: matching of isometries, density and regularity, and elastic theories of shallow shells*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**34**(2017), no. 1, 45–67 (English, with English and French summaries). MR**3592678**, DOI 10.1016/j.anihpc.2015.08.005 - Marta Lewicka, Maria Giovanna Mora, and Mohammad Reza Pakzad,
*A nonlinear theory for shells with slowly varying thickness*, C. R. Math. Acad. Sci. Paris**347**(2009), no. 3-4, 211–216 (English, with English and French summaries). MR**2538115**, DOI 10.1016/j.crma.2008.12.017 - Marta Lewicka, Maria Giovanna Mora, and Mohammad Reza Pakzad,
*Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**9**(2010), no. 2, 253–295. MR**2731157** - Marta Lewicka, Maria Giovanna Mora, and Mohammad Reza Pakzad,
*The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells*, Arch. Ration. Mech. Anal.**200**(2011), no. 3, 1023–1050. MR**2796137**, DOI 10.1007/s00205-010-0387-6 - Marta Lewicka and Piotr B. Mucha,
*A local and global well-posedness results for the general stress-assisted diffusion systems*, J. Elasticity**123**(2016), no. 1, 19–41. MR**3456950**, DOI 10.1007/s10659-015-9545-2 - Marta Lewicka, Pablo Ochoa, and Mohammad Reza Pakzad,
*Variational models for prestrained plates with Monge-Ampère constraint*, Differential Integral Equations**28**(2015), no. 9-10, 861–898. MR**3360723** - Marta Lewicka and Mohammad Reza Pakzad,
*Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics*, ESAIM Control Optim. Calc. Var.**17**(2011), no. 4, 1158–1173. MR**2859870**, DOI 10.1051/cocv/2010039 - Marta Lewicka and Mohammad Reza Pakzad,
*The infinite hierarchy of elastic shell models: some recent results and a conjecture*, Infinite dimensional dynamical systems, Fields Inst. Commun., vol. 64, Springer, New York, 2013, pp. 407–420. MR**2986945**, DOI 10.1007/978-1-4614-4523-4_{1}6 - Marta Lewicka and Mohammad Reza Pakzad,
*Convex integration for the Monge-Ampère equation in two dimensions*, Anal. PDE**10**(2017), no. 3, 695–727. MR**3641884**, DOI 10.2140/apde.2017.10.695 - Marta Lewicka, Annie Raoult, and Diego Ricciotti,
*Plates with incompatible prestrain of high order*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**34**(2017), no. 7, 1883–1912. MR**3724760**, DOI 10.1016/j.anihpc.2017.01.003 - Haiyi Liang and L. Mahadevan,
*The shape of a long leaf*, Proc. Natl. Acad. Sci. USA**106**(2009), no. 52, 22049–22054. MR**2580802**, DOI 10.1073/pnas.0911954106 - H. Liang and L. Mahadevan,
*Growth, geometry and mechanics of the blooming lily*, Proceedings of the National Academy of Sciences,**108**(2011), 5516–5521. - E. H. Mansfield,
*The bending and stretching of plates*, 2nd ed., Cambridge University Press, Cambridge, 1989. MR**1111482**, DOI 10.1017/CBO9780511525193 - Cy Maor and Asaf Shachar,
*On the role of curvature in the elastic energy of non-Euclidean thin bodies*, J. Elasticity**134**(2019), no. 2, 149–173. MR**3913889**, DOI 10.1007/s10659-018-9686-1 - M. C. Marchetti, J-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha,
*Hydrodynamics of soft active matter*, Reviews of Modern Physics**85**(2013), no. 3, 1143. - Stefan Müller and Heiner Olbermann,
*Conical singularities in thin elastic sheets*, Calc. Var. Partial Differential Equations**49**(2014), no. 3-4, 1177–1186. MR**3168627**, DOI 10.1007/s00526-013-0616-6 - John Nash,
*The imbedding problem for Riemannian manifolds*, Ann. of Math. (2)**63**(1956), 20–63. MR**75639**, DOI 10.2307/1969989 - John Nash,
*$C^1$ isometric imbeddings*, Ann. of Math. (2)**60**(1954), 383–396. MR**65993**, DOI 10.2307/1969840 - U. Nath, B. Crawford, R. Carpenter, E. Coen,
*Genetic control of surface curvature*, Science,**299**(2003), 1404–1407. - S. Nechaev,
*Non-Euclidean geometry in nature*, in Order, Disorder and Criticality (Yurij Holovatch editor), pp. 61–111, (2018). - Sergei Nechaev and Raphaël Voituriez,
*On the plant leaf’s boundary, “jupe à godets” and conformal embeddings*, J. Phys. A**34**(2001), no. 49, 11069–11082. MR**1872981**, DOI 10.1088/0305-4470/34/49/322 - Heiner Olbermann,
*Energy scaling law for the regular cone*, J. Nonlinear Sci.**26**(2016), no. 2, 287–314. MR**3466225**, DOI 10.1007/s00332-015-9275-4 - Heiner Olbermann,
*On a boundary value problem for conically deformed thin elastic sheets*, Anal. PDE**12**(2019), no. 1, 245–258. MR**3842912**, DOI 10.2140/apde.2019.12.245 - A. V. Pogorelov,
*An example of a two-dimensional Riemannian metric that does not admit a local realization in $E_{3}$*, Dokl. Akad. Nauk SSSR**198**(1971), 42–43. MR**0286034** - A. Rafsanjani and K. Bertoldi,
*Buckling-induced kirigami*, Physical Review Letters**118**(2017), no. 8, 084301. - P. Rodriguez, A. Hoger, and A. McCulloch,
*Stress-dependent finite growth in finite soft elatic tissues*, J. Biomechanics**27**(1994), 455–467. - T. Savin, N. A. Kurpios, A. E. Shyer, P. Florescu, H. Liang, L. Mahadevan, and C. J. Tabin,
*On the growth and form of the gut*, Nature**476**(2011), no. 7358, 57–62. - E. Sharon, B. Roman, M. Marder, G. S. Shin, and H. L. Swinney,
*Buckling cascades in free sheets*, Nature**419**(2002), 579–579. - E. Sharon, B. Roman, and H. L. Swinney,
*Geometrically driven wrinkling observed in free plastic sheets and leaves*, Phys. Rev. E**75**(2007), 046211. - E. Sharon and M. Sahaf,
*The mechanics of leaf growth on large scales*, in Plant Biomechanics: From Structure to Function at Multiple Scales (Geitmann, A. and Gril, J. editors), Springer International Publishing, pp. 109–126, (2018). - Toby L. Shearman and Shankar C. Venkataramani,
*Distributed branch points and the shape of elastic surfaces with constant negative curvature*, J. Nonlinear Sci.**31**(2021), no. 1, Paper No. 13, 60. MR**4197392**, DOI 10.1007/s00332-020-09657-2 - B. I. Shraiman,
*Mechanical feedback as a possible regulator of tissue growth*, Proceedings of the National Academy of Sciences**102**(2005), no. 9, 3318–3323. - A. E. Shyer, T. Tallinen, N. L. Nerurkar, Z. Wei, E. S. Gil, D. L. Kaplan, C. Tabin, and L Mahadevan,
*Villification: how the gut gets its villi*, Science**342**(2013), no. 6155, 212–218. - Michael Spivak,
*A comprehensive introduction to differential geometry. Vol. I*, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR**532830** - T. Tallinen, J. Y. Chung, J. S. Biggins, and L Mahadevan,
*Gyrification from constrained cortical expansion*, Proceedings of the National Academy of Sciences**111**(2014), no. 35, 12667–12672. - T. Tallinen, J. Y. Chung, F. Rousseau, N. Girard, J. Lefèvre, and L. Mahadevan,
*On the growth and form of cortical convolutions*, Nature Physics**12**(2016), no. 6, 588–593. - H. Thérien-Aubin, Z. L. Wu, Z. Nie, and E. Kumacheva,
*Multiple shape transformations of composite hydrogel sheets*, Journal of the American Chemical Society**135**(2013), no. 12, 4834–4839. - D’Arcy Wentworth Thompson,
*On Growth and Form*, New edition, Cambridge University Press, Cambridge, England, 1942. MR**0006348** - Ian Tobasco,
*Curvature-driven wrinkling of thin elastic shells*, Arch. Ration. Mech. Anal.**239**(2021), no. 3, 1211–1325. MR**4215193**, DOI 10.1007/s00205-020-01566-8 - I. Tobasco, Y. Timounay, D. Todorova, G. Leggat, J. Paulsen, and E. Katifori,
*Exact solutions for the wrinkle patterns of confined elastic shells*, arXiv:2004.02839, 2020. - A. M. Turing,
*The chemical basis of morphogenesis*, Philos. Trans. Roy. Soc. London Ser. B**237**(1952), no. 641, 37–72. MR**3363444**, DOI 10.1098/rstb.1952.0012 - W. M. van Rees, E. Vouga, and L. Mahadevan,
*Growth patterns for shape-shifting elastic bilayers*, Proceedings of the National Academy of Sciences**114**(2017), no. 44, 11597–11602. - Shankar C. Venkataramani,
*Lower bounds for the energy in a crumpled elastic sheet—a minimal ridge*, Nonlinearity**17**(2004), no. 1, 301–312. MR**2023444**, DOI 10.1088/0951-7715/17/1/017 - W. Warner,
*Topographic mechanics and applications of liquid crystalline solids*, Annual Review of Condensed Matter Physics**11**(2020), 125–145. - Z. Y. Wei, Z. V. Guo, L. Dudte, H. Y. Liang, and L. Mahadevan,
*Geometric mechanics of periodic pleated origami*, Physical Review Letters**110**(2013), 215501. - Z. Wei, J. Jia, J. Athas, C. Wang, S. Raghavan, T. Li, and Z. Nie,
*Hybrid hydrogel sheets that undergo pre-programmed shape transformations*, Soft Matter**10**(2014), 8157–8162. - C. Whitewoods and E. Coen,
*Growth and development of three-dimensional plant form*, Current Biology, R910-18, (2017). - K. K. Yamamoto, T. L. Shearman, E. J. Struckmeyer, J. A. Gemmer, and S. C. Venkataramani,
*Nature’s forms are frilly, flexible, and functional*, Preprint arXiv:2103.10509, 2021. - Kewei Zhang,
*Quasiconvex functions, $\textrm {SO}(n)$ and two elastic wells*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**14**(1997), no. 6, 759–785. MR**1482901**, DOI 10.1016/S0294-1449(97)80132-1

## Additional Information

**Marta Lewicka**- Affiliation: University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, Pennsylvania 15260
- MR Author ID: 619488
- Email: lewicka@pitt.edu
**L. Mahadevan**- Affiliation: School of Engineering and Applied Sciences, and Departments of Physics, and Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 340014
- ORCID: 0000-0002-5114-0519
- Email: lmahadev@g.harvard.edu
- Received by editor(s): April 12, 2021
- Published electronically: May 23, 2022
- Additional Notes: The first author was partially supported by NSF grant DMS 2006439. The second author was partially supported by NSF grants BioMatter DMR 1922321, MRSEC DMR 2011754, and EFRI 1830901
- © Copyright 2022 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**59**(2022), 331-369 - MSC (2020): Primary 35-XX, 49-XX, 53-XX, 74-XX, 92-XX
- DOI: https://doi.org/10.1090/bull/1765
- MathSciNet review: 4437801