Doctor of Philosophy
School of Mathematics and Applied Statistics
This thesis is divided into two parts.
In Part I we consider closed immersed surfaces in R3 evolving by the geometric polyharmonic heat fl ow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L2 . We further use an ε-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-umbilic embedded stationary surface. This allows us to conclude that any solution with initial L2-norm of the trace-free curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to 3√3V0/4π, where V0; denotes the signed enclosed volume of the initial data.
In Part II we study the anisotropic polyharmonic heat flow (a fl ow of arbitrarily high even order) for closed curves immersed in the Minkowski plane M2, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that endows an anisotropic distance metric on vectors in M. The indicatrix ∂U (where U⊂R2 is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix ~I. This set is the unique convex set that minimises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We prove that under the anisotropic polyharmonic heat ow, closed curves that are initially close to a homothetic rescaling of the isoperimetrix in an averaged L2 sense exists for all time and converge exponentially fast to a homothetic rescaling of the isoperimetrix that has the same enclosed area as the initial immersion.
Parkins, Scott, A selection of higher-order parabolic curvature flows, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2017. https://ro.uow.edu.au/theses1/21