The Scalar Curvature Equation on $S^3$

TL;DR

This paper establishes existence results for solutions to the scalar curvature equation on the 3-sphere, assuming the prescribed curvature function is positive, Morse, and meets an index condition.

Contribution

It provides new existence theorems for the scalar curvature problem on $S^3$ under specific Morse and index conditions, expanding previous understanding.

Findings

Existence of solutions under positive Morse curvature functions.

Solutions are characterized by index-count conditions.

Advances in geometric analysis on $S^3$.

Abstract

We give existence results for solutions of the prescribed scalar curvature equation on $S^3$, when the curvature function is a positive Morse function and satisfies an index-count condition.