On Gaussian curvature flow
type: Item
instance of: scholarly article
title: On Gaussian curvature flow
zbMATH Open document ID: 1479.58023
author: Xuezhang Chen
author: Mingxiang Li
author: Zirui Li
author: Xingwang Xu
published in: Journal of Differential Equations
publication date: 2021-08-06T00:00:00Z
review text: The authors study the Nirenberg problem on \(S^2\): Given a smooth function \(f:S^2\to\mathbb{R}\) which is positive somewhere, does there exist a metric \(g\) on \(S^2\) conformally equivalent to the standard metric \(g_0\) and having Gaussian curvature \(f\). The main theorem states that a solution exists under the following assumptions. (a) Critical points of \(f\) with \(f>0\) are isolated. (b) \(|\nabla f(x)|^2+|\Delta f(x)|^2 \ne 0\) whenever \(f(x)>0\). (c) Let \(p\) be the number of local maximum points of \(f\) with positive values and let \(q\) be the number of saddle points of \(f\) with positive values and negative Laplace, then \(p-q\ne1\). The proof is based on a normalized Gaussian curvature flow given by \[ \partial_tu=\frac{\int_{S^2} fK(t) d\mu(t)}{\int_{S^2} f^2 d\mu(t)}f-K(t),\quad g(t)=e^{2u(t)}g_0 \] where \(K(t)\) is the Gaussian curvature of \(g(t)\) and \(d\mu(t)=e^{2u(t)}d\mu_0\).
reviewed by: Thomas J. Bartsch
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 7379523
zbMATH Keywords: Nirenberg problem
zbMATH Keywords: concentration and compactness
zbMATH Keywords: Gaussian curvature flow
zbMATH Keywords: blow-up analysis
MaRDI profile type: Publication
full work available at URL: j.jde.2021.05.048
OpenAlex ID: W3172224670
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DOI: 10.1016/J.JDE.2021.05.048
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description: scientific article; zbMATH DE number 7379523