# Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

## TL;DR

This paper constructs examples demonstrating non-uniqueness and lack of compactness in solutions to the constant scalar curvature and boundary mean curvature equation, especially in high dimensions.

## Contribution

It provides explicit counterexamples showing non-uniqueness and non-compactness of solutions in certain high-dimensional warped product manifolds.

## Key findings

- Non-uniqueness of solutions demonstrated
- Counterexamples show failure of compactness in dimensions ≥62
- Warped product manifolds used to illustrate phenomena

## Abstract

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails when the dimension of the manifold is not less than $62$.