A Monotonicity Formula for the Extrinsic Biharmonic Map Heat Flow
Elena Mäder-Baumdicker, Nils Neumann
TL;DR
The paper introduces a monotonicity framework for the extrinsic biharmonic map heat flow by exploiting the backward biharmonic heat kernel and a localized, scale-invariant pre-entropy $\Psi$. It proves that shrinking solitons are critical points of this functional and derives a differential inequality for $\partial_R\Psi$ with a leading nonnegative term, yielding dimension-dependent monotonicity results for $n\le 4$, including a clean one-dimensional case and conditional results in higher dimensions. The analysis separates the linear (scalar bi-heat) case from the nonlinear extrinsic case, establishing sharp a priori estimates and cut-off constructions to control error terms, and showing how small-energy conditions in dimension four restore a stronger monotonicity. This provides a first parabolic monotonicity-type tool for fourth-order flows and sets the stage for regularity and singularity investigations in biharmonic map heat flows.
Abstract
We explore novel properties of the biharmonic heat kernel on Euclidean space and derive an entropy type quantity for the extrinsic biharmonic map heat flow which exhibits monotonicity behaviors for $n\leq 4$.