The Nirenberg problem of prescribed Gauss curvature on S2S^2

  • Michael T. Anderson

    Stony Brook University, USA
The Nirenberg problem of prescribed Gauss curvature on $S^2$ cover
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Abstract

We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on S2S^{2} conformal to the round metric. A key tool is to employ the smooth Cheeger–Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures KK contained in naturally defined stable regions. We prove that in such stable regions, the map uKgu \to K_{g}, g=e2ug+1g = e^{2u}g_{+1} is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on S2S^{2}.

In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger–Moser–Aubin–Onofri.

Cite this article

Michael T. Anderson, The Nirenberg problem of prescribed Gauss curvature on S2S^2. Comment. Math. Helv. 96 (2021), no. 2, pp. 215–274

DOI 10.4171/CMH/512