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

Zheng-Chao Han; YanYan Li

doi:10.3934/dcds.2010.28.607

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2010, Volume 28, Issue 2: 607-615. Doi: 10.3934/dcds.2010.28.607 open access

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

Zheng-Chao Han^1, and
YanYan Li^2,

1.
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854
2.
Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Rd., Piscataway, NJ 08854

Received: December 31, 2009

Revised: February 28, 2010

Published: June 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J60; 58J05; 53A30.

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