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Journals / International Electronic Journal of Geometry / 2026 / Volume: 14 Issue: 1

Recent Developments in Wintgen Inequality and Wintgen Ideal Submanifolds

Bang-yen Chen(Michigan State University)0000-0002-1270-094X
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Publication Date
2021-04-15
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Wintgen proved in [Sur l’inégalité de Chen-Willmore.
📊
Key Findings & Data:Paris 288, 993–995 (1979)] that the Gauss curvature $G$ and the normal curvature $K^D$ of a surface in the Euclidean 4-space $E^4$ satisfy $$G+|K^D|\leq \Vert H\Vert ^2,$$ where $\Vert H\Vert ^2$ is the squared mean curvature.
💡
Conclusion & Impact:In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.
🔬Design:Survey & Empirical Study
📍Scope:Comprehensive Sample
⚙️Variables:Gauss curvature · normal curvature · squared mean curvature

Abstract

P. Wintgen proved in [Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993–995 (1979)] that the Gauss curvature $G$ and the normal curvature $K^D$ of a surface in the Euclidean 4-space $E^4$ satisfy $$G+|K^D|\leq \Vert H\Vert ^2,$$ where $\Vert H\Vert ^2$ is the squared mean curvature. A surface $M^{2}$ in $E^4$ is called a {Wintgen ideal} surface if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in $E^4$ form an important family of surfaces; namely, surfaces with circular ellipse of curvature. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms, which was well-known as the DDVV conjecture. Later, the DDVV conjecture was proven by Z. Lu and by Ge and Z. Tang independently. In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.

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Cite This Article

Chen, B. (2021). Recent Developments in Wintgen Inequality and Wintgen Ideal Submanifolds. International Electronic Journal of Geometry, Volume: 14 Issue: 1. https://doi.org/10.36890/iejg.838446

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