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Dergiler / International Electronic Journal of Geometry / 2021 / Cilt: 14 - Sayı: 1

Recent Developments in Wintgen Inequality and Wintgen Ideal Submanifolds

Bang-Yen CHEN
Sayfa
6–45
Yayın Tarihi
2022-10-18
DOI
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Tang independently.In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Wintgen proved in [Sur l’inégalité de Chen-Willmore.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı:Tang independently.In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.
🔬Tür:Uygulamalı Anket & Alan Araştırması
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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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Bu Makaleye Atıf Yap

CHEN, B. (2022). Recent Developments in Wintgen Inequality and Wintgen Ideal Submanifolds. International Electronic Journal of Geometry, Cilt: 14 - Sayı: 1, 6–45. https://ajindex.com/articles/539584

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