On Gaussian curvature flow

The current article is to study the solvability of Nirenberg problem on S2 through the so-called Gaussian curvature flow. We aim to propose a unified method to treat the problem for candidate functions without sign restriction and non-degenerate assumption. As a first step, we reproduce the followin...

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Bibliographic Details
Published in:Journal of Differential Equations 2021-09, Vol.294, p.178-250
Main Authors: Chen, Xuezhang, Li, Mingxiang, Li, Zirui, Xu, Xingwang
Format: Article
Language:English
Subjects:
Blow-up analysis
Concentration and compactness
Gaussian curvature flow
Nirenberg problem
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Summary:The current article is to study the solvability of Nirenberg problem on S2 through the so-called Gaussian curvature flow. We aim to propose a unified method to treat the problem for candidate functions without sign restriction and non-degenerate assumption. As a first step, we reproduce the following statement: suppose the critical points of a smooth function f with positive critical values are non-degenerate. Then the required solution exists, if the difference between the number of the local maximum points with positive values and the number of the saddle points with positive critical values as well as negative Laplace is not equal to 1. This statement has been proved for nearly thirty years through different methods.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2021.05.048