@article{kusner_mcgrath_2023, title={On the Canham Problem: Bending Energy Minimizers for any Genus and Isoperimetric Ratio}, volume={247}, ISSN={["1432-0673"]}, DOI={10.1007/s00205-022-01833-w}, abstractNote={Building on work of Mondino–Scharrer, we show that among closed, smoothly embedded surfaces in $${\mathbb {R}}^3$$ of genus g and given isoperimetric ratio v, there exists one with minimum bending energy $${\mathcal {W}}$$ . We do this by gluing $$g+1$$ small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation $$\Delta (\Delta +2)\varphi =0$$ on the $$(g+1)$$ -punctured sphere $${\mathbb {S}}^2$$ to construct a comparison surface of genus g with arbitrarily small isoperimetric ratio $$v\in (0, 1)$$ and $${\mathcal {W}}< 8\pi $$ .}, number={1}, journal={ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}, author={Kusner, Robert and McGrath, Peter}, year={2023}, month={Feb} }
@article{mcgrath_meekins_2023, title={Two-point functions and constant mean curvature surfaces in R3}, volume={16}, ISSN={["1944-4184"]}, DOI={10.2140/involve.2023.16.467}, number={3}, journal={INVOLVE, A JOURNAL OF MATHEMATICS}, author={Mcgrath, Peter and Meekins, Everett}, year={2023}, pages={467–482} }
@article{hoisington_mcgrath_2022, title={Symmetry and isoperimetry for Riemannian surfaces}, volume={61}, ISSN={["1432-0835"]}, DOI={10.1007/s00526-021-02117-z}, abstractNote={For a domain $$\Omega $$ in a geodesically convex surface, we introduce a scattering energy $$\mathcal {E}(\Omega )$$ , which measures the asymmetry of $$\Omega $$ by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving $$\mathcal {E}(\Omega )$$ and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new of the sharp Sobolev inequality for Riemannian surfaces.}, number={1}, journal={CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS}, author={Hoisington, Joseph Ansel and McGrath, Peter}, year={2022}, month={Feb} }
@article{mcgrath_2020, title={Bases for Second Order Linear ODEs}, volume={127}, ISSN={["1930-0972"]}, DOI={10.1080/00029890.2020.1803626}, abstractNote={The usual analysis of the constant coefficient ODE ay″+by′+cy=0 —which concludes that ert is a solution when r=(−b±b2−4ac)/(2a) —fails to produce a basis for the solution space when b2−4ac=0 . Redu...}, number={9}, journal={AMERICAN MATHEMATICAL MONTHLY}, author={McGrath, Peter}, year={2020}, month={Oct}, pages={849–849} }