An Invitation to Biharmonic and Biconservative Submanifolds

TL;DR

This note surveys two principal generalizations of minimal submanifolds, biharmonic and biconservative submanifolds, using the bienergy functional and its bitension field $\tau_2$ together with the stress‑bienergy tensor $S_2$. It discusses Chen's conjecture and the Generalized Chen conjecture, summarizing known results in $\mathbb{R}^n$ and in spheres and highlighting open classification problems in higher codimension and in space forms. A key theme is the link between holomorphic data and mean curvature on surfaces: the generalized Hopf function $Q$ satisfies holomorphy precisely when $|\mathbf{H}|$ is constant on biconservative surfaces. The text also reports existence results for closed non‑CMC biconservative surfaces in $\mathbb{S}^3$ and outlines open problems in homogeneous spaces, illustrating active directions for future work.

Abstract

This note is based on a lecture delivered by the author at the Second Conference on Differential Geometry, held in Fez in October 2024. It offers an accessible introduction to biharmonic and biconservative submanifolds, exploring the motivations for their study and highlighting some key facts and open problems in the field.

Figures (7)

• Figure 1: The inclusions between minimal, biharmonic, and biconservative submanifolds.
• Figure 2: The solution to Question 3.
• Figure 3: Inclusions between the families of minimal, biharmonic, CMC, and biconservative surfaces in a $3$-dimensional space form $N^3(c)$.
• Figure 4: Plot of the profile curve of a biconservtive surface in ${\mathbb R}^3$.
• Figure 5: The graph of $Q(u)$ according to the value of the curvature $c$.

Theorems & Definitions (18)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4: see MR3441516 and MR3693945 for a generalized version
• Theorem 2.5
• Proposition 2.6