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A fourth order linear parabolic equation on conical surfaces

  • School of Mathematical Sciences, University of Science and Technology of China, Heifei 230026, China

Cite this:
https://doi.org/10.52396/JUST-2021-0076
  • Received Date: 17 March 2021
  • Rev Recd Date: 10 June 2021
  • Publish Date: 30 June 2021
    Abstract

  • Abstract

    A parabolic equation of fourth order on surfaces with conical singularities is considered. By the analysis of energy and approximations, the existence and uniqueness of the solution of this equation in a special space that has some approximation property are proved. Finally, it's proved that the property is equivalent to the finiteness of energy for some functions when β∈(-1,0).

    Abstract

    A parabolic equation of fourth order on surfaces with conical singularities is considered. By the analysis of energy and approximations, the existence and uniqueness of the solution of this equation in a special space that has some approximation property are proved. Finally, it's proved that the property is equivalent to the finiteness of energy for some functions when β∈(-1,0).
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    • References(13)
  • [1]
    Eugenio C. Extremal Kähler metrics. In: Seminar on Differential Geometry. Princeton, NJ: Princeton University Press, 1982: 259-290.
    [2]
    Chruściel P T. Semi-global existence and convergence of solutions of the Robinson Trautman (2-dimensional Calabi) equation. Comm. Math. Phys., 1991, 137(2): 289-313.
    [3]
    Chen X X. Calabi flow in Riemann surfaces revisited: A new point of view. International Mathematics Research Notices, 2001, 2001(6): 275-297.
    [4]
    Struwe M. Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2002, 1(2): 247-274.
    [5]
    Li H Z, Wang B, Zheng K. Regularity scales and convergence of the Calabi flow. Journal of Geometric Analysis, 2018, 28(3): 2050-2101.
    [6]
    Yin H. Ricci flow on surfaces with conical singularities. J. Geom. Anal., 2010, 20(4): 970-995.
    [7]
    Mazzeo R, Rubinstein Y, Sesum N. Ricci flow on surfaces with conic singularities. Anal. PDE, 2015, 8(4): 839-882.
    [8]
    Phong D H, Song J, Sturm J, et al. The Ricci flow on the sphere with marked points. J. Differential Geom., 2020, 114(1): 117-170.
    [9]
    Daniel R. Ricci flow on cone surfaces. Port. Math.,2018, 75(1): 11-65.
    [10]
    Yin H. Analysis aspects of Ricci flow on conical surfaces. https://arxiv.org/abs/1605.08836.
    [11]
    Zheng K. Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability. https://arxiv.org/abs/1803.09506.
    [12]
    Zheng K. Geodesics in the space of Kähler cone metrics II: Uniqueness of constant scalar curvature Kähler cone metrics. Communications on Pure and Applied Mathematics, 2019, 72(12): 2621-2701.
    [13]
    Eidelman S D, Zhitarashu N V. Parabolic Boundary Value Problems. Basel, Switzerland: Birkhäuser Verlag, 1998.
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    [1]
    Eugenio C. Extremal Kähler metrics. In: Seminar on Differential Geometry. Princeton, NJ: Princeton University Press, 1982: 259-290.
    [2]
    Chruściel P T. Semi-global existence and convergence of solutions of the Robinson Trautman (2-dimensional Calabi) equation. Comm. Math. Phys., 1991, 137(2): 289-313.
    [3]
    Chen X X. Calabi flow in Riemann surfaces revisited: A new point of view. International Mathematics Research Notices, 2001, 2001(6): 275-297.
    [4]
    Struwe M. Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2002, 1(2): 247-274.
    [5]
    Li H Z, Wang B, Zheng K. Regularity scales and convergence of the Calabi flow. Journal of Geometric Analysis, 2018, 28(3): 2050-2101.
    [6]
    Yin H. Ricci flow on surfaces with conical singularities. J. Geom. Anal., 2010, 20(4): 970-995.
    [7]
    Mazzeo R, Rubinstein Y, Sesum N. Ricci flow on surfaces with conic singularities. Anal. PDE, 2015, 8(4): 839-882.
    [8]
    Phong D H, Song J, Sturm J, et al. The Ricci flow on the sphere with marked points. J. Differential Geom., 2020, 114(1): 117-170.
    [9]
    Daniel R. Ricci flow on cone surfaces. Port. Math.,2018, 75(1): 11-65.
    [10]
    Yin H. Analysis aspects of Ricci flow on conical surfaces. https://arxiv.org/abs/1605.08836.
    [11]
    Zheng K. Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability. https://arxiv.org/abs/1803.09506.
    [12]
    Zheng K. Geodesics in the space of Kähler cone metrics II: Uniqueness of constant scalar curvature Kähler cone metrics. Communications on Pure and Applied Mathematics, 2019, 72(12): 2621-2701.
    [13]
    Eidelman S D, Zhitarashu N V. Parabolic Boundary Value Problems. Basel, Switzerland: Birkhäuser Verlag, 1998.
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