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Fourth order geometric evolution equations

Journal Article


Abstract


  • In this thesis the chief objects of study are hypersurface flows of fourth order,

    with the speed of the flow varying from the Laplacian of the mean curvature, to

    the more general constrained flows which include a function of time in the speed,

    and satisfy various conditions. Our aim is to instigate a study of the regularity of

    these flows, answering questions of local and global existence, and some preliminary

    singularity analysis. Among our results are positive lower bounds for smooth

    and regular existence, classification of stationary solutions, interior estimates, and

    blowup asymptotics. Applying these results to a certain class of constrained surface

    diffusion flows, we obtain long time existence and exponential convergence to spheres

    for initial surfaces with small L2 norm of tracefree curvature. We present one

    application of this theorem, using it to deduce the isoperimetric inequality with optimal

    constant for 2-surfaces satisfying the above smallness condition. The long time

    existence theorem can be thought of as a stability of spheres result, as the smallness

    condition is an averaged distance from a standard round sphere to the initial manifold

    in L2. This strengthens a related earlier result specialized to the surface diffusion

    and Willmore flows T17U, where the distance is small in C2;, obtained through a

    completely different method. Our techniques have more in common with T9–11U,

    from which we have drawn much inspiration. The results throughout this thesis are

    new contributions for both surface diffusion flow, which has been considered by many

    authors T1–8, 12, 16, 17U, and the constrained flows, which have only recently been

    considered T13–15, 18, 19U.

Publication Date


  • 2010

Citation


  • Wheeler, G. (2010). Fourth order geometric evolution equations. Bulletin of the Australian Mathematical Society, 82 (3), 523-524.

Scopus Eid


  • 2-s2.0-85022354345

Number Of Pages


  • 1

Start Page


  • 523

End Page


  • 524

Volume


  • 82

Issue


  • 3

Place Of Publication


  • United Kingdom

Abstract


  • In this thesis the chief objects of study are hypersurface flows of fourth order,

    with the speed of the flow varying from the Laplacian of the mean curvature, to

    the more general constrained flows which include a function of time in the speed,

    and satisfy various conditions. Our aim is to instigate a study of the regularity of

    these flows, answering questions of local and global existence, and some preliminary

    singularity analysis. Among our results are positive lower bounds for smooth

    and regular existence, classification of stationary solutions, interior estimates, and

    blowup asymptotics. Applying these results to a certain class of constrained surface

    diffusion flows, we obtain long time existence and exponential convergence to spheres

    for initial surfaces with small L2 norm of tracefree curvature. We present one

    application of this theorem, using it to deduce the isoperimetric inequality with optimal

    constant for 2-surfaces satisfying the above smallness condition. The long time

    existence theorem can be thought of as a stability of spheres result, as the smallness

    condition is an averaged distance from a standard round sphere to the initial manifold

    in L2. This strengthens a related earlier result specialized to the surface diffusion

    and Willmore flows T17U, where the distance is small in C2;, obtained through a

    completely different method. Our techniques have more in common with T9–11U,

    from which we have drawn much inspiration. The results throughout this thesis are

    new contributions for both surface diffusion flow, which has been considered by many

    authors T1–8, 12, 16, 17U, and the constrained flows, which have only recently been

    considered T13–15, 18, 19U.

Publication Date


  • 2010

Citation


  • Wheeler, G. (2010). Fourth order geometric evolution equations. Bulletin of the Australian Mathematical Society, 82 (3), 523-524.

Scopus Eid


  • 2-s2.0-85022354345

Number Of Pages


  • 1

Start Page


  • 523

End Page


  • 524

Volume


  • 82

Issue


  • 3

Place Of Publication


  • United Kingdom