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Creators/Authors contains: "Heslin, Patrick"

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  1. ; ; (, The Journal of Geometric Analysis)
    Abstract We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order$$q\in [0,\infty )$$ q ∈ [ 0 , ∞ ) . We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if$$q>1/2$$ q > 1 / 2 . Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if$$q>3/2$$ q > 3 / 2 , whereas if$$q<3/2$$ q < 3 / 2 then finite-time blowup may occur. The geodesic completeness for$$q>3/2$$ q > 3 / 2 is obtained by proving metric completeness of the space of$$H^q$$ H q -immersed curves with the distance induced by the Riemannian metric. 
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  2. ; ; ; (, Mathematische Annalen)
    Abstract We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach 
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