I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case.

They prove that if $K:S^2\rightarrow \mathbb{R}$ is a positive smooth function satisfying some nondegeneracy condition, then $K$ can be obtained as Gaussian curvature from a conformal change of metric.

The idea of the proof is to consider the one parameter family of maps: $$ K_s=sK+(1-s) $$ and looking for zeroes of the maps $$ \psi_s(w)=w-\Delta^{-1}\Psi_s(w) $$

where $\Psi_s(w)=1-K_se^{2w}$.

They apply the Leray-Schauder degree theory and some apriori estimates on the $C^{2,\alpha}$ norm of zeroes of $\psi_s$ for $s\geq s_0$, for any given $s_0>0$

It is not clear to me, however, how to apply degree theory in this context. More precisely, from what I know, Leray-Schauder degree theory can by applied to maps $Id-K:\overline{U}\rightarrow X$ where $X$ is a Banach space, $U$ is a bounded open set in $X$, and $K$ is a compact map such that $0\notin(I-K)(\partial U)$

In the case at hand, it is not clear me how to choose $X$ and $U$.

The authors of the article consider the set $X=\{w\in C^{2,\alpha}(S^2)| \frac{1}{4\pi}\int Ke^{2w}=1\}$ and the bounded set $\Omega_C=\{w\in X| |w|_{C^{2,\alpha}}< C\}$ where $C$ is given by the apriori estimates.

I have a few doubts about this choice:

  • $\Delta^{-1}\Psi_s(w)$ is not well defined if $w\in X$, since for this to make sense I need that $\frac{1}{4\pi}\int K_se^{2w}=1$
  • X is not a Banach space, and therefore it is not clear to me how to apply degree theory.

Is there something I am missing?



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