On Gaussian curvature flow. (English) Zbl 1479.58023
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 \(d\mu(t)=e^{2u(t)}d\mu_0\).
(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 \(d\mu(t)=e^{2u(t)}d\mu_0\).
Reviewer: Thomas J. Bartsch (Gießen)
MSC:
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35K55 | Nonlinear parabolic equations |
| 58J05 | Elliptic equations on manifolds, general theory |
Keywords:
Nirenberg problem; concentration and compactness; Gaussian curvature flow; blow-up analysisReferences:
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