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Lipschitz-Volume Rigidity and Sobolev Coarea Inequality for Metric Surfaces

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Abstract

We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study the regularity properties of such a map under different geometric assumptions. Our proof relies on a coarea inequality for continuous Sobolev functions on metric surfaces that we establish, and which generalizes a recent result of Esmayli–Ikonen–Rajala.

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References

  1. Ball, K.: An elementary introduction to modern convex geometry. In: Flavors of Geometry. Mathematical Sciences Research Institute Publications, vol. 31, pp. 1–58. Cambridge University Press, Cambridge (1997)

  2. Basso, G., Creutz, P., Soultanis, E.: Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces. J. Reine Angew. Math. 805, 213–239 (2023)

    MathSciNet  Google Scholar 

  3. Basso, G., Marti, D., Wenger, S.: Geometric and analytic structures on metric spaces homeomorphic to a manifold. arXiV Preprint (2023). arXiv:2303.13490

  4. Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5), 731–799 (1995)

    Article  MathSciNet  Google Scholar 

  5. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)

  6. Burago, D., Ivanov, S.: Boundary rigidity and filling volume minimality of metrics close to a flat one. Ann. Math. (2) 171(2), 1183–1211 (2010)

    Article  MathSciNet  Google Scholar 

  7. Creutz, P., Soultanis, E.: Maximal metric surfaces and the Sobolev-to-Lipschitz property. Calc. Var. Partial Differ. Equ. 59(5), Paper No. 177, 34 (2020)

  8. Daverman, R.J.: Decompositions of Manifolds. Pure and Applied Mathematics, p. 124. Academic Press, Orlando (1986)

  9. Del Nin, G., Perales, R.: Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into \(\mathbb{R} ^n\). J. Math. Anal. Appl. 526(1), 127297 (2023)

    Article  Google Scholar 

  10. Eilenberg, S., Harrold, O.G., Jr.: Continua of finite linear measure. I. Am. J. Math. 65, 137–146 (1943)

    Article  MathSciNet  Google Scholar 

  11. Eriksson-Bique, S., Poggi-Corradini, P.: On the sharp lower bound for duality of modulus. Proc. Am. Math. Soc. 150, 2955–2968 (2022)

    Article  MathSciNet  Google Scholar 

  12. Esmayli, B., Hajłasz, P.: The Coarea inequality. Ann. Fenn. Math. 46(2), 965–991 (2021)

    Article  MathSciNet  Google Scholar 

  13. Esmayli, B., Ikonen, T., Rajala, K.: Coarea inequality for monotone functions on metric surfaces. Trans. Am. Math. Soc. 376, 7377–7406 (2023)

    MathSciNet  Google Scholar 

  14. Evans, L.C., Gariepy, R.F., Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)

  15. Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)

    Article  MathSciNet  Google Scholar 

  16. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, p. 153. Springer, New York (1969)

  17. Gray, A.: Tubes, 2nd edn. Progress in Mathematics, vol. 221. Birkhäuser, Basel (2004). With a preface by Miquel, V.

  18. Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. New Mathematical Monographs, vol. 27. Cambridge University Press, Cambridge (2015)

  19. Ikonen, T.: Uniformization of metric surfaces using isothermal coordinates. Ann. Fenn. Math. 47(1), 155–180 (2022)

    Article  MathSciNet  Google Scholar 

  20. Karmanova, M.B.: Area and co-area formulas for mappings of the Sobolev classes with values in a metric space Sibirsk. Mat. Zh. 48(4), 778–788 (2007)

    MathSciNet  Google Scholar 

  21. Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)

  22. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane. Grundlehren der mathematischen Wissenschaften, 2nd edn, p. 126. Springer, New York (1973)

  23. Li, N.: Lipschitz-volume rigidity in Alexandrov geometry. Adv. Math. 275, 114–146 (2015)

    Article  MathSciNet  Google Scholar 

  24. Li, N.: Lipschitz-volume rigidity and globalization. In: Proceedings of the International Consortium of Chinese Mathematicians 2018, pp. 311–322. International Press, Boston (2020)

  25. Li, N.: Lipschitz-volume rigidity on limit spaces with Ricci curvature bounded from below. Differ. Geom. Appl. 35, 50–55 (2014)

    Article  MathSciNet  Google Scholar 

  26. Lytchak, A., Stefan, W.: Area minimizing discs in metric spaces. Arch. Ration. Mech. Anal. 223(3), 1123–1182 (2017)

    Article  MathSciNet  Google Scholar 

  27. Lytchak, A., Wenger, S.: Intrinsic structure of minimal discs in metric spaces. Geom. Topol. 22(1), 591–644 (2018)

    Article  MathSciNet  Google Scholar 

  28. Malý, J., Swanson, D., Ziemer, W.P.: The co-area formula for Sobolev mappings. Trans. Am. Math. Soc. 355(2), 477–492 (2003)

    Article  MathSciNet  Google Scholar 

  29. Meier, D., Wenger, S.: Quasiconformal almost parametrizations of metric surfaces. arXiv Preprint (2021). arXiv:2106.01256

  30. Moore, R.L.: Concerning triods in the plane and the junction points of plane continua. Proc. Natl Acad. Sci. U.S.A. 14(1), 85–88 (1928)

    Article  Google Scholar 

  31. Ntalampekos, D., Romney, M.: Polyhedral approximation and uniformization for non-length surfaces. arXiv Preprint (2022). arXiv:2206.01128

  32. Ntalampekos, D., Romney, M.: Polyhedral approximation of metric surfaces and applications to uniformization. Duke. Math. J. 172(9), 1673–1734 (2023)

    Article  MathSciNet  Google Scholar 

  33. Ntalampekos, D.: Monotone Sobolev functions in planar domains: level sets and smooth approximation. Arch. Ration. Mech. Anal. 238, 1199–1230 (2020)

    Article  MathSciNet  Google Scholar 

  34. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften, p. 299. Springer, Berlin (1992)

  35. Rajala, K.: Uniformization of two-dimensional metric surfaces. Invent. Math. 207(3), 1301–1375 (2017)

    Article  MathSciNet  Google Scholar 

  36. Rajala, K., Romney, M.: Reciprocal lower bound on modulus of curve families in metric surfaces. Ann. Acad. Sci. Fenn. Math. 44(2), 681–692 (2019)

    Article  MathSciNet  Google Scholar 

  37. Storm, P.A.: Rigidity of minimal volume Alexandrov spaces. Ann. Acad. Sci. Fenn. Math. 31(2), 381–389 (2006)

    MathSciNet  Google Scholar 

  38. Whyburn, G.T.: Analytic Topology. American Mathematical Society Colloquium Publications, vol. 28. American Mathematical Society, New York (1942)

  39. Wilder, R.L.: Topology of Manifolds. American Mathematical Society Colloquium Publications, vol. 32. American Mathematical Society, Providence (1963)

  40. Willard, S.: General Topology. Addison-Wesley, Reading (1970)

    Google Scholar 

  41. Williams, M.: Geometric and analytic quasiconformality in metric measure spaces. Proc. Am. Math. Soc. 140(4), 1251–1266 (2012)

    Article  MathSciNet  Google Scholar 

  42. Züst, R.: Lipschitz rigidity of Lipschitz manifolds among integral current spaces. arXiv Preprint (2023). arXiv:2302.07587

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Acknowledgements

The paper was written while the first-named author was visiting Stony Brook University. She wishes to thank the department for their hospitality. The authors would like to thank Stefan Wenger and the anonymous referee for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700, Fribourg, Switzerland

    Damaris Meier

  2. Mathematics Department, Stony Brook University, Stony Brook, NY, 11794, USA

    Dimitrios Ntalampekos

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  1. Damaris Meier
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  2. Dimitrios Ntalampekos
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Correspondence to Dimitrios Ntalampekos.

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Damaris Meier is partially supported by UniFr Doc.Mobility Grant DM-22-10. Dimitrios Ntalampekos is partially supported by NSF Grants DMS-2000096 and DMS-2246485, and by the Simons Foundation.

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Meier, D., Ntalampekos, D. Lipschitz-Volume Rigidity and Sobolev Coarea Inequality for Metric Surfaces. J Geom Anal 34, 128 (2024). https://doi.org/10.1007/s12220-024-01577-x

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  • DOI: https://doi.org/10.1007/s12220-024-01577-x

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