Journal of Evolution Equations

, Volume 18, Issue 4, pp 1595–1632 | Cite as

The Yamabe flow on incomplete manifolds

  • Yuanzhen ShaoEmail author


This article is concerned with developing an analytic theory for second-order nonlinear parabolic equations on singular manifolds. Existence and uniqueness of solutions in an \(L_p\)-framework are established by maximal regularity tools. These techniques are applied to the Yamabe flow. It is proven that the Yamabe flow admits a unique local solution within a class of incomplete initial metrics.


Singular parabolic equations Maximal \(L_p\)-regularity Incomplete Riemannian manifolds Geometric evolution equations The Yamabe flow 

Mathematics Subject Classification

35K55 35K67 35R01 53C21 53C44 58J99 


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  1. 1.
    H. Amann, Linear and Quasilinear Parabolic Problems: Volume I. Abstract Linear Theory. Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA (1995).Google Scholar
  2. 2.
    H. Amann, Function spaces on singular manifolds, Math. Nachr. 286, no. 5–6, 436–475 (2013).MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. Amann, Anisotropic function spaces on singular manifolds. arXiv.1204.0606.
  4. 4.
    H. Amann, Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems. Recent Developments of Mathematical Fluid Mechanics, H. Amann, Y. Giga, H. Kozono, H. Okamoto, M. Yamazaki (Eds.), pages 43–77. Birkhäuser, Basel, 2016.Google Scholar
  5. 5.
    H. Amann, Uniformly regular and singular Riemannian manifolds. Elliptic and parabolic equations, 1–43, Springer Proc. Math. Stat., 119, Springer, Cham, 2015.Google Scholar
  6. 6.
    W. Arendt, A. Grabosch, G. Greiner, U. Groh, H.P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck, One-Parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, 1184. Springer-Verlag, Berlin, 1986.Google Scholar
  7. 7.
    E. Bahuaud, E.B. Dryden, B. Vertman, Mapping properties of the heat operator on edge manifolds. Math. Nachr. 288, no. 2–3, 126–157 (2015).MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Bahuaud, B. Vertman, Yamabe flow on manifolds with edges. Math. Nachr. 287 , no. 2–3, 127–159 (2014).MathSciNetCrossRefGoogle Scholar
  9. 9.
    E. Bahuaud, B. Vertman, Long-time existence of the edge Yamabe flow. arXiv:1605.03935.
  10. 10.
    S. Brendle, A generalization of the Yamabe flow for manifolds with boundary. Asian J. Math. 6, no. 4, 625–644 (2002).MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Clément, S. Li, Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3, Special Issue, 17–32 (1993/94).Google Scholar
  12. 12.
    R.R. Coifman, R. Rochberg, G. Weiss, Applications of transference: the \(L_p\) version of von Neumann’s inequality and the Littlewood-Paley-Stein theory. Linear spaces and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977), pp. 53–67. Internat. Ser. Numer. Math., Vol. 40, Birkhäuser, Basel, 1978.Google Scholar
  13. 13.
    S. Coriasco, E. Schrohe, J. Seiler, Bounded imaginary powers of differential operators on manifolds with conical singularities. Math. Zeitschrift 244, 235–269 (2003).MathSciNetCrossRefGoogle Scholar
  14. 14.
    M.G. Cowling, Harmonic analysis on semigroups. Ann. of Math. (2) 117, no. 2, 267–283 (1983).Google Scholar
  15. 15.
    R. Denk, M. Hieber, J. Prüss, \(\mathscr {R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166, no. 788, 2003.Google Scholar
  16. 16.
    E. DiBenedetto, D.J. Diller, About a singular parabolic equation arising in thin film dynamics and in the Ricci flow for complete \(R^2\). Partial differential equations and applications, 103-119, Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, 1996.Google Scholar
  17. 17.
    M. Disconzi, Y. Shao, G. Simonett, Some remarks on uniformly regular Riemannian manifolds. Math. Nachr. 289, no. 2–3, 232–242 (2016).MathSciNetCrossRefGoogle Scholar
  18. 18.
    X.T. Duong, \({\cal{H}}^{\infty }\) functional calculus of second order elliptic partial differential operators on Lp spaces. Miniconference on Operators in Analysis (Sydney, 1989), 91–102, Proc. Centre Math. Anal. Austral. Nat. Univ., 24, Austral. Nat. Univ., Canberra, 1990.Google Scholar
  19. 19.
    J. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. of Math. (2) 136, no. 1, 1–50 (1992).Google Scholar
  20. 20.
    G. Giesen, P. Topping, Ricci flow of negatively curved incomplete surfaces. Calc. Var. Partial Differential Equations 38, no. 3–4, 357–367 (2010).MathSciNetCrossRefGoogle Scholar
  21. 21.
    G. Giesen, P. Topping, Existence of Ricci flows of incomplete surfaces. Comm. Partial Differential Equations 36, no. 10, 1860–1880 (2011).MathSciNetCrossRefGoogle Scholar
  22. 22.
    G. Giesen, P. Topping, Ricci flows with bursts of unbounded curvature. arXiv:1302.5686v2.
  23. 23.
    J. Isenberg, R. Mazzeo, N. Sesum, Ricci flow in two dimensions. Surveys in geometric analysis and relativity, 259–280, Adv. Lect. Math. (ALM), 20, Int. Press, Somerville, MA, 2011.Google Scholar
  24. 24.
    J. Isenberg, R. Mazzeo, N. Sesum, Ricci flow on asymptotically conical surfaces with nontrivial topology. J. Reine Angew. Math. 676, 227–248 (2013).MathSciNetzbMATHGoogle Scholar
  25. 25.
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.zbMATHGoogle Scholar
  26. 26.
    V. A. Kondratev, Boundary value problems for elliptic equations in domains with conical or angular points. (Russian) Trudy Moskov. Mat. Obšč. 16 209–292 (1967).Google Scholar
  27. 27.
    R. Lauter, J. Seiler, Pseudodifferential analysis on manifolds with boundary-a comparison of b-calculus and cone algebra. Approaches to singular analysis (Berlin, 1999), 131–166, Oper. Theory Adv. Appl., 125, Birkhäuser, Basel, 2001.Google Scholar
  28. 28.
    L. Ma, Y. An, The maximum principle and the Yamabe flow. Partial differential equations and their applications (Wuhan, 1999), 211–224, World Sci. Publ., River Edge, NJ, 1999.Google Scholar
  29. 29.
    L. Ma, L. Cheng, A. Zhu, Extending Yamabe flow on complete Riemannian manifolds. Bull. Sci. Math. 136, no. 8, 882–891 (2012).MathSciNetCrossRefGoogle Scholar
  30. 30.
    C.-I. Martin, B.-W. Schulze, Parameter-dependent edge operators. Ann. Global Anal. Geom. 38, no. 2, 171–190 (2010).MathSciNetCrossRefGoogle Scholar
  31. 31.
    R. Mazzeo, Elliptic theory of differential edge operators. I. Comm. Partial Differential Equations 16, no. 10, 1615–1664 (1991).Google Scholar
  32. 32.
    R. Mazzeo and B. Vertman, Analytic torsion on manifolds with edges, Adv. Math. 231(2), 1000–1040 (2012).MathSciNetCrossRefGoogle Scholar
  33. 33.
    R. Mazzeo, B. Vertman, Elliptic theory of differential edge operators, II: Boundary value problems. Indiana Univ. Math. J. 63, no. 6, 1911–1955 (2014).MathSciNetCrossRefGoogle Scholar
  34. 34.
    R. Mazzeo, Y.A. Rubinstein, N. Sesum, Ricci flow on surfaces with conic singularities. Anal. PDE 8, no. 4, 839–882 (2015).MathSciNetCrossRefGoogle Scholar
  35. 35.
    A. McIntosh, Operators which have an \(H_{\infty }\) functional calculus. Miniconference on operator theory and partial differential equations (North Ryde, 1986), 210–231, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, 1986.Google Scholar
  36. 36.
    R.B. Melrose, Transformation of boundary problems. Acta Math. 147, no. 3–4, 149–236 (1981).MathSciNetCrossRefGoogle Scholar
  37. 37.
    R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics, 4. A K Peters, Ltd., Wellesley, MA, 1993.Google Scholar
  38. 38.
    V. Nazaikinskii, A.Yu. Savin, B.-W. Schulze, B. Yu. Sternin, Elliptic Theory on Singular Manifolds. Differential and Integral Equations and Their Applications, 7. Chapman Hall/CRC, Boca Raton, FL, 2006.Google Scholar
  39. 39.
    R. S. Phillips, Semi-groups of positive contraction operators. Czechoslovak Math. J. 12 (87) 294–313, 1962.MathSciNetzbMATHGoogle Scholar
  40. 40.
    J. Prüss, Maximal regularity for evolution equations in \(L_p\)-spaces. Conf. Semin. Mat. Univ. Bari No. 285 (2002), 1–39 (2003).Google Scholar
  41. 41.
    J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics. Birkhäuser Verlag. 2016.CrossRefGoogle Scholar
  42. 42.
    N. Roidos, E. Schrohe, The Cahn-Hilliard equation and the Allen-Cahn equation on manifolds with conical singularities. Comm. Partial Differential Equations 38:5, 925–943 (2013).MathSciNetCrossRefGoogle Scholar
  43. 43.
    N. Roidos, E. Schrohe, Bounded imaginary powers of cone differential operators on higher order Mellin-Sobolev spaces and applications to the Cahn-Hilliard equation J. Differential Equations 257, 611–637 (2014).MathSciNetCrossRefGoogle Scholar
  44. 44.
    N. Roidos, E. Schrohe, Existence and maximal \(L^{p}\)-regularity of solutions for the porous medium equation on manifolds with conical singularities. Comm. Partial Differential Equations 41, no. 9, 1441–1471 (2016).MathSciNetCrossRefGoogle Scholar
  45. 45.
    B.-W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics. Mathematical Topics, 4. Akademie Verlag, Berlin, 1994.Google Scholar
  46. 46.
    B.-W. Schulze, Boundary Value Problems and Edge Pseudo-differential Operators. Microlocal analysis and spectral theory (Lucca, 1996), 165–226, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490, Kluwer Acad. Publ., Dordrecht, 1997.Google Scholar
  47. 47.
    B.-W. Schulze, J. Seiler, The edge algebra structure of boundary value problems. Ann. Global Anal. Geom. 22, no. 3, 197–265 (2002).MathSciNetCrossRefGoogle Scholar
  48. 48.
    B.-W. Schulze, J. Seiler, Edge operators with conditions of Toeplitz type. J. Inst. Math. Jussieu 5 , no. 1, 101–123 (2006).MathSciNetCrossRefGoogle Scholar
  49. 49.
    Y. Shao, G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds. J. Evol. Equ. 1, no. 14, 211–248, (2014).MathSciNetCrossRefGoogle Scholar
  50. 50.
    Y. Shao, Continuous maximal regularity on singular manifolds and its applications. Evol. Equ. Control Theory 5, no. 2, 303–335 (2016).MathSciNetCrossRefGoogle Scholar
  51. 51.
    Y. Shao, Singular parabolic equations of second order on manifolds with singularities. J. Differential Equations 260, no. 2, 1747–1800 (2016).MathSciNetCrossRefGoogle Scholar
  52. 52.
    P. Topping, Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics. J. Eur. Math. Soc. (JEMS) 12, no. 6, 1429–1451 (2010).MathSciNetCrossRefGoogle Scholar
  53. 53.
    P. Topping, Uniqueness of Instantaneously Complete Ricci flows. Geometry and Topology 19-3, 1477–1492 (2015).MathSciNetCrossRefGoogle Scholar
  54. 54.
    B. Vertman, Ricci flow on singular manifolds. arXiv:1603.06545v2.
  55. 55.
    H. Yin, Ricci flow on surfaces with conical singularities. J. Geom. Anal. 20, no. 4, 970–995 (2010).MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

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