Energy reduction for Fourth order Willmore energy

TL;DR

The paper defines a fourth-order Willmore-type energy $\mathcal{E}^{(\mu,\nu)}$ for closed four-dimensional submanifolds in $\mathbb{R}^n$, incorporating extrinsic $L$-based terms to address unboundedness. It proves an inversion-based energy identity $\mathcal{E}^{(\mu,\nu)}(\hat{\Phi})=\mathcal{E}^{(\mu,\nu)}(\Phi)-\mathcal{E}^{(\mu,\nu)}(\mathbb{S}^4)\cdot \mathrm{card}\,\Phi^{-1}(0)$ and interprets the energy loss via a Gauss–Bonnet–Chern framework, with a leading neck-energy contribution controlled by a triharmonic interpolation in a connected-sum construction. The main result is a connected-sum energy reduction inequality: for two non-round embeddings $\Phi_1,\Phi_2$, there exists a connected-sum immersion $\Phi:\Sigma_1\#\Sigma_2\to \mathbb{R}^n$ with topological type $\Sigma_1\#\Sigma_2$ such that $\mathcal{E}^{(\mu,\nu)}(\Phi)<\mathcal{E}^{(\mu,\nu)}(\Phi_1)+\mathcal{E}^{(\mu,\nu)}(\Phi_2)-8\pi^2$ for all real $\mu,\nu$. The construction hinges on an explicit inversion expansion, a careful neck interpolation via a tri-harmonic equation, and a bilinear-form analysis to guarantee positive energy interaction, thus extending Bauer–Kuwert’s two-dimensional result to four dimensions. This provides a framework toward minimizing conformal four-manifolds in ambient Euclidean spaces via a robust energy functional built from intrinsic and extrinsic conformal invariants.

Abstract

We introduce a fourth-order Willmore-type problem for closed four-dimensional submanifolds immersed in $\mathbb{R}^n$ and establish a connected sum energy reduction for the general fourth-order Willmore energy, analogous to the seminal result of Bauer and Kuwert \cite{Bauer-Kuwert03}.

Theorems & Definitions (45)

• Theorem 1.1: Simon's Criterion
• Theorem 1.2: Connected Sum Energy Reduction
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Case-Graham-Kuo
• proof : Proof of Theorem \ref{['Main 2']}
• Remark 3.1
• Lemma 3.2: GBC formula on $\hat{\Sigma}$
• proof
• Lemma 4.1