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On the local solvability of the Nirenberg problem on $\mathbb S^2$

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  • We present some results on the local solvability of the Nirenberg problem on $\mathbb S^2$.More precisely, an $L^2(\mathbb S^2)$ function near $1$ is the Gauss curvature of an$H^2(\mathbb S^2)$ metric on the round sphere $\mathbb S^2$, pointwise conformal to the standardround metric on $\mathbb S^2$, provided its $L^2(\mathbb S^2)$ projection into thethe space of spherical harmonics of degree $2$ satisfy a matrix invertibility condition,and the ratio of the $L^2(\mathbb S^2)$ norms ofits $L^2(\mathbb S^2)$ projections into the the space of spherical harmonics of degree $1$vs the space of spherical harmonics of degrees other than $1$ is sufficiently small.
    Mathematics Subject Classification: Primary: 35J60; 58J05; 53A30.

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