Conformal metrics of constant scalar curvature with unbounded volumes

TL;DR

This paper constructs a conformal metric on the sphere for dimensions n≥25 that admits a sequence of scalar curvature one metrics with unbounded volume growth.

Contribution

It demonstrates the existence of conformal metrics with constant scalar curvature and unbounded volume on high-dimensional spheres, a novel geometric construction.

Findings

Existence of conformal metrics with scalar curvature 1 and unbounded volume for n≥25.

Construction method applicable to high-dimensional spheres.

Sequence of metrics with diverging volumes.

Abstract

For $n\geq 25$, we construct a smooth metric $\tilde{g}$ on the standard $n$-dimensional sphere $\mathbb{S}^n$ such that there exists a sequence of smooth metrics $\{\tilde{g}_k\}_{k\in\mathbb{N}}$ conformal to $\tilde g$ where each $\tilde g_k$ has scalar curvature $R_{\tilde{g}_k}\equiv 1$ and their volumes $\text{Vol}(\mathbb{S}^n,\tilde{g}_k)$ tend to infinity as $k$ approaches infinity.