On the finite time blow-up of biharmonic map flow in dimension four

TL;DR

This paper proves that under specific initial conditions, the biharmonic map flow in four dimensions inevitably develops singularities in finite time, highlighting limitations in the flow's long-term behavior.

Contribution

It establishes finite time blow-up results for biharmonic map flow in four dimensions for certain initial data, advancing understanding of flow singularities.

Findings

Finite time blow-up occurs for certain initial conditions.

Blow-up behavior is proven specifically in four-dimensional cases.

Results highlight limitations of biharmonic map flow in higher dimensions.

Abstract

In this paper, we show that for certain initial values, the (extrinsic) biharmonic map flow in dimension four must blow up in finite time.