Beijing Institute of Mathematical Sciences and Applications

Abderrahim Mesbah , Xiaoming John Zhang

Sandra Carolina García Martínez

Monday, September 14, 2026 2:00 PM - 3:00 PM

A3-4-101

Zoom 242 742 6089 (BIMSA)

The submanifolds of finite type are those whose isometric immersion in the ambient space is constructed with a finite number eigenfunctions of their Laplacian $\Delta$. This notion was introduced by B.Y. Chen during the late 1970s for Euclidean submanifolds, since then, it have been study by many authors in ambient different spaces (see [3, 2, 1]). This concept was originally defined in terms of the Laplacian, however, since this operator can be seen as the first one of a sequence of operators $L_{0} = \Delta$, $L_{1}, \dots, L_{n - 1}$, where $n$ is the dimension of submanifold and $L_{k}$ stands for the linearized operator of the first variation of the $(k + 1)$-th mean curvature arising from normal variations (see [10]). It is therefore natural to generalize this notion of finite type submanifold for any operator $L_{k}$ and seek new results, as well as, compare them with the classical ones. Following this approach, Ramirez and Lucas in [8, 9] analyzed the case $n = 2$ and $k = 1$ in the non-flat Riemannian space forms, that is, $L_{1} - 2$ type surfaces immersed into $\mathbb{H}^{3}$ and $\mathbb{S}^{3}$, where $L_{1} = \square$ is the well known Cheng-Yau operator (see [6]). This talk aims to show some classification results that we have obtained jointly with Ramirez and Lucas in the non-Riemannian case (see [7]), which is more interesting, since in this setting the shape operator may be non-diagonalizable.

References

1. Alías, L.J., Ferrández, A., Lucas, P. 2-type surfaces in $S_{1}^{3}$ and $H_{3}^{1}$. Tokyo J. Math. 17, 447-454 (1994)
2. M. Barros and O.J. Garay. 2-type surfaces in $S^{3}$, Geom. Dedicata 24 (1987), 329-336
3. B.Y. Chen. Total Mean Curvature and Submanifolds of Finite Type. World Scientific Publisher, Singapore and New Jersey, 1984.
4. B.Y. Chen. A report on submanifolds of finite type, Soochow J. Math. 22 (1996), 117-337.
5. B.Y. Chen. Some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math. 45 (2014), 87-108.
6. S.Y. Cheng and S.T. Yau (1977), Hypersurfaces with constant scalar curvature, Math. Ann. 225, 195-204.
7. García-Martínez, S.C., Lucas P., Ramírez-Ospina, H.F. $L_{1} - 2$-Type Surfaces in 3-Dimensional De Sitter and Anti De Sitter Spaces, Bull. Malays. Math. Sci. Soc. (2023) 46:139.
8. Lucas, P., Ramirez-Ospina, H.F. Hyperbolic surfaces of $L_{1} - 2$-type. Bull. Iran. Math. Soc. 43(6), 1769-1779 (2017)
9. Lucas, P., Ramirez-Ospina, H.F. Surfaces in S3 of $L_{1} - 2$ type. Bull. Malays. Math. Sci. Soc. 41(4), 1759-1771 (2018)
10. R. Reilly. Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diff. Geom. 8 (1973), 465-477

Sandra Carolina García Martínez is an Associate Professor of Mathematics at the National University of Colombia, Bogotá Campus, where she also serves as Co-Leader of the Research Group in Differential Geometry and Geometric Analysis. She holds a Master’s degree in Mathematical Sciences from the University of Valle, Colombia, as well as a Master’s degree in Advanced Mathematics and a Ph.D. in Mathematics from the University of Murcia, Spain. She subsequently completed a postdoctoral fellowship at IME-USP, University of São Paulo, Brazil. Her research interests primarily focus on Differential Geometry and Geometric Analysis, particularly on geometric applications of maximum principles for trace-type operators, the classification of finite-type $(L_k)$ submanifolds, and the study of hypersurfaces with constant mean or scalar curvature in different ambient spaces. He is visiting BIMSA with a support from the "ICMRA Visiting Scholars Program".