Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 091, 29 pages      arXiv:2305.01453
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

Luca Benatti a, Mattia Fogagnolo b and Lorenzo Mazzieri c
a) Università degli Studi di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
b) Università di Padova, via Trieste 63, 35121 Padova, Italy
c) Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy

Received May 03, 2023, in final form October 23, 2023; Published online November 10, 2023

We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in $3$-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1$ < $p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.

Key words: Penrose inequality; positive mass theorem; isoperimetric mass; nonlinear potential theory; nonlinear potential theory.

pdf (602 kb)   tex (42 kb)  


  1. Agostiniani V., Fogagnolo M., Mazzieri L., Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature, Invent. Math. 222 (2020), 1033-1101, arXiv:1812.05022.
  2. Agostiniani V., Fogagnolo M., Mazzieri L., Minkowski inequalities via nonlinear potential theory, Arch. Ration. Mech. Anal. 244 (2022), 51-85, arXiv:1906.00322.
  3. Agostiniani V., Mantegazza C., Mazzieri L., Oronzio F., Riemannian Penrose inequality via nonlinear potential theory, arXiv:2205.11642.
  4. Agostiniani V., Mazzieri L., Monotonicity formulas in potential theory, Calc. Var. Partial Differential Equations 59 (2020), 6, 32 pages, arXiv:1606.02489.
  5. Agostiniani V., Mazzieri L., Oronzio F., A Green's function proof of the positive mass theorem, arXiv:2108.08402.
  6. Arnowitt R., Deser S., Misner C.W., Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961), 997-1006.
  7. Bartnik R., The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661-693.
  8. Benatti L., Monotonicity formulas in nonlinear potential theory and their geometric applications, Ph.D. Thesis, Università degli studi di Trento, 2022.
  9. Benatti L., Fogagnolo M., Mazzieri L., The asymptotic behaviour of $p$-capacitary potentials in asymptotically conical manifolds, Math. Ann., to appear, arXiv:2207.08607.
  10. Benatti L., Fogagnolo M., Mazzieri L., Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature, arXiv:2101.06063.
  11. Benatti L., Fogagnolo M., Mazzieri L., On the isoperimetric Riemannian Penrose inequality, arXiv:2212.10215.
  12. Bray H., Miao P., On the capacity of surfaces in manifolds with nonnegative scalar curvature, Invent. Math. 172 (2008), 459-475, arXiv:0707.3337.
  13. Bray H.L., Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), 177-267, arXiv:math.DG/9911173,.
  14. Chan P.-Y., Chu J., Lee M.-C., Tsang T.-Y., Monotonicity of the $p$-Green functions, arXiv:2202.13832.
  15. Chodosh O., Eichmair M., Shi Y., Yu H., Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds, Comm. Pure Appl. Math. 74 (2021), 865-905, arXiv:1606.04626.
  16. Chruściel P., Boundary conditions at spatial infinity from a Hamiltonian point of view, in Topological Properties and Global Structure of Space-Time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B: Phys., Vol. 138, Springer, Boston, MA, 1986, 49-59, arXiv:1312.0254.
  17. DiBenedetto E., $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850.
  18. Evans L.C., A new proof of local $C^{1,\alpha }$ regularity for solutions of certain degenerate elliptic p.d.e., J. Differential Equations 45 (1982), 356-373.
  19. Fan X.-Q., Shi Y., Tam L.-F., Large-sphere and small-sphere limits of the Brown-York mass, Comm. Anal. Geom. 17 (2009), 37-72, arXiv:0711.2552.
  20. Fogagnolo M., Mazzieri L., Minimising hulls, $p$-capacity and isoperimetric inequality on complete Riemannian manifolds, J. Funct. Anal. 283 (2022), 109638, 49 pages, arXiv:2012.09490.
  21. Fogagnolo M., Mazzieri L., Pinamonti A., Geometric aspects of $p$-capacitary potentials, Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), 1151-1179, arXiv:1803.10679.
  22. Heinonen J., Kilpeläinen T., $A$-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat. 26 (1988), 87-105.
  23. Heinonen J., Kilpeläinen T., Martio O., Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006.
  24. Holopainen I., Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes (1990), 1-45.
  25. Holopainen I., Volume growth, Green's functions, and parabolicity of ends, Duke Math. J. 97 (1999), 319-346.
  26. Huisken G., An isoperimetric concept for the mass in general relativity, available at
  27. Huisken G., Ilmanen T., The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), 353-437.
  28. Jauregui J.L., ADM mass and the capacity-volume deficit at infinity, arXiv:2002.08941.
  29. Jauregui J.L., Lee D.A., Lower semicontinuity of mass under $C^0$ convergence and Huisken's isoperimetric mass, J. Reine Angew. Math. 756 (2019), 227-257, arXiv:1602.00732.
  30. Kotschwar B., Ni L., Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 1-36, arXiv:0711.2291.
  31. Ladyzhenskaya O.A., Ural'tseva N.N., Linear and quasilinear elliptic equations, Academic Press, New York, 1968.
  32. Lewis J.L., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), 849-858.
  33. Lieberman G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219.
  34. Lou H., On singular sets of local solutions to $p$-Laplace equations, Chinese Ann. Math. Ser. B 29 (2008), 521-530.
  35. Mari L., Rigoli M., Setti A.G., On the $1/H$-flow by $p$-Laplace approximation: new estimates via fake distances under Ricci lower bounds, Amer. J. Math. 144 (2022), 779-849, arXiv:1905.00216.
  36. Moser R., The inverse mean curvature flow and $p$-harmonic functions, J. Eur. Math. Soc. 9 (2007), 77-83.
  37. Moser R., The inverse mean curvature flow as an obstacle problem, Indiana Univ. Math. J. 57 (2008), 2235-2256.
  38. Munteanu O., Wang J., Comparison theorems for 3D manifolds with scalar curvature bound, Int. Math. Res. Not. 2023 (2023), 2215-2242, arXiv:2105.12103.
  39. Ni L., Mean value theorems on manifolds, Asian J. Math. 11 (2007), 277-304, arXiv:math.DG/0608608.
  40. Oronzio F., ADM.
  41. Serrin J., Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302.
  42. Tolksdorf P., On the Dirichlet problem for quasilinear equations, Comm. Partial Differential Equations 8 (1983), 773-817.
  43. Trudinger N.S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747.
  44. Ural'tseva N.N., Degenerate quasilinear elliptic systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184-222.
  45. Valtorta D., On the $p$-Laplace operator on Riemannian manifolds, Ph.D. Thesis, Università degli Studi di Milano, 2013, arXiv:1212.3422.
  46. Wang X., Zhang L., Local gradient estimate for $p$-harmonic functions on Riemannian manifolds, Comm. Anal. Geom. 19 (2011), 759-771, arXiv:1010.2889.
  47. Xia C., Yin J., The anisotropic $p$-capacity and the anisotropic Minkowski inequality, Sci. China Math. 65 (2022), 559-582, arXiv:2012.13933.
  48. Xiao J., The $p$-harmonic capacity of an asymptotically flat 3-manifold with non-negative scalar curvature, Ann. Henri Poincaré 17 (2016), 2265-2283.
  49. Zhu S., The comparison geometry of Ricci curvature, in Comparison Geometry (Berkeley, CA, Math. Sci. Res. Inst. Publ., Vol. 30, Cambridge University Press, Cambridge, 1997, 221-262.

Previous article  Next article  Contents of Volume 19 (2023)