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Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

Journal Article


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Abstract


  • In this paper we establish a gap phenomenon for immersed surfaces

    with arbitrary codimension, topology and boundaries that satisfy one

    of a family of systems of fourth-order anisotropic geometric partial differential

    equations. Examples include Willmore surfaces, stationary solitons for the surface

    diffusion flow, and biharmonic immersed surfaces in the sense of Chen. On

    the boundary we enforce either umbilic or flat boundary conditions: that the

    tracefree second fundamental form and its derivative or the full second fundamental

    form and its derivative vanish. For the umbilic boundary condition we

    prove that any surface with small L2-norm of the tracefree second fundamental

    form or full second fundamental form must be totally umbilic, that is, a piece

    of a round sphere or flat plane. We prove that the stricter smallness condition

    allows consideration for a broader range of differential operators. For the flat

    boundary condition we prove the same result with weaker hypotheses, allowing

    more general operators, and a stronger conclusion: only a piece of a flat plane

    is allowed. The method used relies only on the smallness assumption and thus

    holds without requiring the imposition of additional symmetries. The result

    holds in the class of surfaces with any genus and irrespective of the number or

    shape of the boundaries.

Publication Date


  • 2015

Citation


  • Wheeler, G. E. (2015). Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary. Proceedings of the American Mathematical Society, 143 (4), 1719-1737.

Scopus Eid


  • 2-s2.0-84923233859

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=4462&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3445

Has Global Citation Frequency


Number Of Pages


  • 18

Start Page


  • 1719

End Page


  • 1737

Volume


  • 143

Issue


  • 4

Place Of Publication


  • United States

Abstract


  • In this paper we establish a gap phenomenon for immersed surfaces

    with arbitrary codimension, topology and boundaries that satisfy one

    of a family of systems of fourth-order anisotropic geometric partial differential

    equations. Examples include Willmore surfaces, stationary solitons for the surface

    diffusion flow, and biharmonic immersed surfaces in the sense of Chen. On

    the boundary we enforce either umbilic or flat boundary conditions: that the

    tracefree second fundamental form and its derivative or the full second fundamental

    form and its derivative vanish. For the umbilic boundary condition we

    prove that any surface with small L2-norm of the tracefree second fundamental

    form or full second fundamental form must be totally umbilic, that is, a piece

    of a round sphere or flat plane. We prove that the stricter smallness condition

    allows consideration for a broader range of differential operators. For the flat

    boundary condition we prove the same result with weaker hypotheses, allowing

    more general operators, and a stronger conclusion: only a piece of a flat plane

    is allowed. The method used relies only on the smallness assumption and thus

    holds without requiring the imposition of additional symmetries. The result

    holds in the class of surfaces with any genus and irrespective of the number or

    shape of the boundaries.

Publication Date


  • 2015

Citation


  • Wheeler, G. E. (2015). Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary. Proceedings of the American Mathematical Society, 143 (4), 1719-1737.

Scopus Eid


  • 2-s2.0-84923233859

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=4462&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3445

Has Global Citation Frequency


Number Of Pages


  • 18

Start Page


  • 1719

End Page


  • 1737

Volume


  • 143

Issue


  • 4

Place Of Publication


  • United States