Title: Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of
\n general degree
\nAuthors: Melanie Rupflin
\nCategories: math.AP math.DG
\nMSC-class: 53C43, 58E20, 30C70, 26D10,
\n\\\\
\n As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4\\pi \\vert
\ndeg(v)\\vert$ with equality if and only if $v$ is a rational map one might ask
\nwhether maps with small energy defect $\\delta_v=E(v)-4\\pi \\vert deg(v)\\vert$
\nare necessarily close to a rational map. While such a rigidity statement turns
\nout to be false for maps of general degree, we will prove that any map $v$ with
\nsmall energy defect is essentially given by a collection of rational maps that
\ndescribe the behaviour of $v$ at very different scales and that the
\ncorresponding distance is controlled by a quantitative rigidity estimate of the
\nform $dist^2\\leq C \\delta_v(1+\\vert\\log\\delta_v\\vert)$ which is indeed sharp.
\n\\\\ ( https:\/\/arxiv.org\/abs\/2305.17045 , 45kb)<\/p>\n","protected":false},"excerpt":{"rendered":"