A monotonicity formula for the extrinsic biharmonic map heat flow
Abstract
We investigate new properties of the biharmonic heat kernel on Euclidean space and derive a localized entropy-like quantity $\Psi$ for the extrinsic biharmonic map heat flow $u:\mathbb R^n\times[0,T) \to N$, where $N$ is a smooth closed manifold. For general $n$, we derive a formula for the $R$-derivative of $\Psi$. For $1\leq n\leq 3$, we show that $\Psi$ is monotone in the sense $\Psi(R_1) \leq \Psi(R_2) + c \cdot (R_2 - R_1)$ for $R_1\leq R_2$, where $c$ only depends on $u_0$, $n$ and $N$. Here, $R_i$ denote (sometimes up to a constant) the parabolic time-distances to a fixed $t_0\in (0,T)$ via $t_i=t_0- R_i^4$, $i=1,2$. For $n=4$, we obtain a similar result under a smallness condition on $u_0$.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Geometry and complex manifolds
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Communications in Analysis and Geometry
- Volume
- 34
- Issue
- 1
- Pages
- 185–228
- Publication date
- 1 January 2026
- Language
- EN