Tesis por compendio de publicaciones / It is well known that Takahashi's Theorem [7] characterizes the submanifolds in the Euclidean space whose coordinate functions are eigenfunctions of the Laplacian associated to the same nonzero eigenvalue: they are minimal submanifolds in a hypersphere. Later on, many authors have obtained different extensions of Takahashi's Theorem. One of these extensions is given by Dillen-Pas-Verstraelen in [2]. In that work, the authors study surfaces in the 3-dimensional space whose immersion ψ satisfy Δψ=Aψ+b, where Δ denotes the Laplacian operator, A is a 3x3 real matrix and b is a constant vector. They obtain that the only surfaces satisfying that equation are minimal ones, spheres and circular cylinders. After that different authors have studied this condition in the case of hypersurfaces Mn immersed in pseudo-Euclidean spaces Rn+1 for any index t≥0, and showed that Mn must be an open part of a minimal Rn+1 surfaces, a totally umbilical hypersurface or a standard pseudo-Riemannian product. Recently, that equation has been extended to operators different to the Laplacian one. In fact, Alías and Gürbüz study in [2] hypersurfaces in the Euclidean space Rn+1 whose position vector ψ satisfies Lkψ=Aψ+b, where Lk is the linealized differential operator associated to the mean curvature of order k+1, for k=0, 1,..., n-1 (note that for k=0 we obtain the Laplacian operator). Those authors show that the only hypersurfaces satisfying the above condition are k-minimal hypersurfaces, hyperspheres and generalized cylinders (for appropriate radii and dimensions). In view of that result for operators Lk, we study the same condition but for hypersurfaces immersed in pseudo-Euclidean spaces Rn+1 for any index t≥0, and show (in papers [5] and [6]) that the only hypersurfaces in the pseudo-Euclidean spaces satisfying that condition are k-minimal hypersurfaces, hyperspheres and generalized cylinders (for appropriate radii and dimensions). After solving the problem for hypersurfaces in pseudo-Euclidean spaces, we study the condition Lkψ=Aψ+b for hypersurfaces immersed in pseudo-Riemannian space forms, for arbitrary index t≥0 and nonzero constant curvature. We show (in papers [3] and [4]), that the only hypersurfaces satisfying that condition are k-minimal hypersurfaces, totally umbilical hypersurfaces, standard pseudo-Riemannian products and some quadratic hypersurfaces. In conclusion, the results obtained in this Thesis extend completely to pseudo- Euclidean spaces and pseudo-Riemannian space forms of nonzero constant curvature the results previously obtained in [2]. References [1] L.J. Alías and N. Gürbüz. An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113-127. [2] F. Dillen, J. Pas and L. Verstraelen. On surfaces of finite type in Euclidean 3-space, Kodai Math. J. 13 (1990), 10-21. [3] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in non-flat Lorentzian space forms satisfying Lkψ=Aψ+b , Taiwanese J. Math. 16 (2012), 1173-1203. [4] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in non-flat pseudo-Euclidean space form satisfying a linear condition in the linearized operator of a higher order mean curvatures, Taiwanese J. Math. 17 (2013), 15-45. [5] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ=Aψ+b , Geom. Dedicata 153 (2011), 151-175. [6] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in pseudo-Euclidean space satisfying a linear condition on the linearized operator of a higher order mean curvatures, Diff. Geom. and its Appl. 13 (2013), 175-189. [7] T. Takahashi. Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385.
| Identifer | oai:union.ndltd.org:TDX_UM/oai:www.tdx.cat:10803/134934 |
| Date | 08 May 2014 |
| Creators | Ramírez Ospina, Héctor Fabián |
| Contributors | Lucas Saorín, Pascual, Universidad de Murcia. Departamento de Matemáticas |
| Publisher | Universidad de Murcia |
| Source Sets | Universidad de Murcia |
| Language | Spanish |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/publishedVersion |
| Format | 20 p., application/pdf |
| Source | TDR (Tesis Doctorales en Red) |
| Rights | info:eu-repo/semantics/openAccess, ADVERTENCIA. El acceso a los contenidos de esta tesis doctoral y su utilización debe respetar los derechos de la persona autora. Puede ser utilizada para consulta o estudio personal, así como en actividades o materiales de investigación y docencia en los términos establecidos en el art. 32 del Texto Refundido de la Ley de Propiedad Intelectual (RDL 1/1996). Para otros usos se requiere la autorización previa y expresa de la persona autora. En cualquier caso, en la utilización de sus contenidos se deberá indicar de forma clara el nombre y apellidos de la persona autora y el título de la tesis doctoral. No se autoriza su reproducción u otras formas de explotación efectuadas con fines lucrativos ni su comunicación pública desde un sitio ajeno al servicio TDR. Tampoco se autoriza la presentación de su contenido en una ventana o marco ajeno a TDR (framing). Esta reserva de derechos afecta tanto al contenido de la tesis como a sus resúmenes e índices. |
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