Inventiones mathematicae

, Volume 219, Issue 1, pp 1–37 | Cite as

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature



The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.



The author wishes to thank Rick Schoen, Brian White, Leon Simon, Rafe Mazzeo, Or Hershkovits and Christos Mantoulidis for stimulating conversations. He also wishes to thanks the referee for greatly improving the exposition. Part of this work was carried out when the author was visiting the University of California, Irvine. He wants to thank Department of Mathematics, UCI, for their hospitality.


  1. 1.
    Aleksandrov, A.D.: A theorem on triangles in a metric space and some of its applications. Trudy Mat. Inst. Steklov. 38, 5–23 (1951) MathSciNetGoogle Scholar
  2. 2.
    Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95, 417–491 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Allard, W.K.: On the first variation of a varifold: boundary behavior. Ann. Math. (2) 101, 418–446 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ambrozio, L.: Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds. J. Geom. Anal. 25(2), 1001–1017 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom. 18(4), 821–830 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54(1), 13–35 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54(1), 37–74 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cai, M., Galloway, G.J.: Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature. Commun. Anal. Geom. 8(3), 565–573 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Colding, T.H., Naber, A.: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math. (2) 176(2), 1173–1229 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cheeger, J., Naber, A.: Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent. Math. 191(2), 321–339 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    De Philippis, G., Maggi, F.: Regularity of free boundaries in anisotropic capillarity problems and the validity of Young’s law. Arch. Ration. Mech. Anal. 216(2), 473–568 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Finn, R.: Equilibrium capillary surfaces. In: Byrne, C. (ed.) Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer, Berlin (1986) Google Scholar
  14. 14.
    Gromov, M.: Dirac and Plateau billiards in domains with corners. Cent. Eur. J. Math. 12(8), 1109–1156 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124, 281–311 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lieberman, G.M.: Hölder continuity of the gradient at a corner for the capillary problem and related results. Pac. J. Math. 133(1), 115–135 (1988)zbMATHCrossRefGoogle Scholar
  17. 17.
    Lieberman, G.M.: Optimal Hölder regularity for mixed boundary value problems. J. Math. Anal. Appl. 143(2), 572–586 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, C., Mantoulidis, C.: Positive scalar curvature with skeleton singularities. Math. Ann. 374(1–2), 99–131 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169(3), 903–991 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Li, M., Zhou, X.: A maximum principle for free boundary minimal varieties of arbitrary codimension. Commun. Anal. Geom. (to appear) Google Scholar
  21. 21.
    Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Micallef, M., Moraru, V.: Splitting of 3-manifolds and rigidity of area-minimising surfaces. Proc. Am. Math. Soc. 143(7), 2865–2872 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mahmoudi, F., Mazzeo, R., Pacard, F.: Constant mean curvature hypersurfaces condensing on a submanifold. Geom. Funct. Anal. 16(4), 924–958 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Mazzeo, R., Pacard, F.: Foliations by constant mean curvature tubes. Commun. Anal. Geom. 13(4), 633–670 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Nunes, I.: Rigidity of area-minimizing hyperbolic surfaces in three-manifolds. J. Geom. Anal. 23(3), 1290–1302 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ros, A., Souam, R.: On stability of capillary surfaces in a ball. Pac. J. Math. 178(2), 345–361 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Simon, L.: Regularity of capillary surfaces over domains with corners. Pac. J. Math. 88(2), 363–377 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Simon, L.: A strict maximum principle for area minimizing hypersurfaces. J. Differ. Geom. 26(2), 327–335 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62(1), 79–125 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Shi, Y., Tam, L.-F.: Scalar curvature and singular metrics. Pac. J. Math. 293(2), 427–470 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sturm, K.-T.: A curvature-dimension condition for metric measure spaces. C. R. Math. Acad. Sci. 342(3), 197–200 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Solomon, B., White, B.: A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals. Indiana Univ. Math. J. 38(3), 683–691 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Schoen, R., Yau, S.T.: On the structure of manifolds with positive scalar curvature. Manuscr. Math. 28(1–3), 159–183 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Jean, E.: Taylor, boundary regularity for solutions to various capillarity and free boundary problems. Commun. Partial Differ. Equ. 2(4), 323–357 (1977)zbMATHCrossRefGoogle Scholar
  37. 37.
    White, B.: The maximum principle for minimal varieties of arbitrary codimension. Commun. Anal. Geom. 18(3), 421–432 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Ye, R.: Foliation by constant mean curvature spheres. Pac. J. Math. 147(2), 381–396 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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