Radial prescribing scalar curvature on $RP^n$

TL;DR

This paper proves that all smooth, positive, radial, non-negative functions on $RP^n$ can be realized as scalar curvatures within the standard conformal class, resolving previous nonexistence conjectures.

Contribution

The paper demonstrates the existence of prescribed scalar curvatures on $RP^n$ for all suitable radial functions, using a novel quotient approach to simplify the proof.

Findings

All smooth radial non-negative functions positive at the pole are prescribing scalar curvatures.

Counterexamples to nonexistence in prior work are invalid on $RP^n$.

The quotient method simplifies existence proofs in scalar curvature problems.

Abstract

The study of radial prescribing scalar curvature of X.Xu and P.C.Yang [2] in 1993 showed a nonexistence result on $S^2$. Later in 1995, W.Chen and C.Li [2] generalized the nonexistence result to higher dimensions. G.Bianchi and E.Egnell [1] suggested that there may exist some non-negative smooth radial function which cannot be scalar curvature in the standard conformal class of $RP^n.$ However, in our paper, we prove that all smooth radial non-negative smooth functions which are positive on the pole could be prescribing scalar curvatures. We consider the quotient $\frac{v_\lambda}{V_\lambda}$ rather than the difference $v_\lambda-V_\lambda$ as in [1]. This trick yields a concise argument of the existence. Consequently, their counter examples stated in [1] cannot be true on $RP^n.$