Resumen de Hipersuperficies en los espacios forma pseudo-riemannianos satisfaciendo L_K\PSI=A \PSI+B

Hector Fabian Ramirez Ospina

• Resumen Como es bien conocido, el clásico teorema de Takahashi [7] caracteriza las subvariedades del espacio euclídeo cuyas funciones coordenadas son propias para el laplaciano asociadas al mismo valor propio: son subvariedades minimales en una hiperesfera. Con posterioridad, numerosos autores rescataron el teorema de Takahashi y probaron diferentes extensiones del mismo. Una de esas extensiones es la planteada por Dillen-Pas-Verstraelen en [2]. En su trabajo estudiaron las superficies del espacio 3-dimensional cuya inmersión ? satisfacía ??=A?+b, donde ? es el operador laplaciano, A una matriz 3x3 y b un vector constante, probando que las únicas que satisfacían dicha condición eran las minimales, las esferas y los cilindros circulares. Posteriormente diferentes autores estudiaron esta misma condición en el caso de hipersuperfcies Mn inmersas en espacios pseudo-euclideanos Rn+1 de cualquier índice t?0, probando que Mn debe ser un trozo abierto de una hipersuperficie minimal, una hipersuperficie totalmente umbilical o un producto estándar pseudo-riemanniano. Esta ecuación ha sido recientemente generalizada al considerar otros operadores distintos al laplaciano. Concretamente, en [2] los autores Alías-Gürbüz estudian las hipersuperficies del espacio euclidiano Rn+1 cuyo vector de posición ? satisface Lk?=A?+b donde Lk es el operador linealizado de la curvatura media de orden k+1, para k=0, 1,..., n-1 (notemos que para k=0 se obtiene el operador laplaciano usual). El resultado obtenido en esta ocasión, afirma que las únicas hipersuperficies satisfaciendo dicha ecuación son las k-minimales, las hiperesferas y ciertos cilindros generalizados. A la vista del este primer resultado para operadores Lk, nos planteamos el estudio de esta misma condición para hipersuperficies inmersas en el espacios pseudo-euclídianos Rn+1 de cualquier índice t?0, y logramos demostrar en los artículos [5] y [6] que las únicas hipersuperficies en estos espacios pseudo-euclidianos satisfaciendo dicha ecuación son las k-minimales, las hipersuperficies totalmente umbilicales y ciertos cilindros generalizados. Llegados a este punto, nos planteamos un nuevo objetivo, el estudio de la condición Lk?=A?+b para hipersuperficies inmersas en espacios forma pseudo-riemannianos de cualquier índice t?0 y de curvatura constante positiva y negativa. En este nuevo estudio logramos demostrar en los artículos [3] y [4], que las únicas hipersuperficies inmersas en los espacios forma pseudo-riemannianos de curvatura constante no cero satisfaciendo dicha ecuación son las k-minimales, las totalmente umbilicales, los productos estándar pseudo-riemannianos y ciertas hipersuperficies cuadráticas. En conclusión, los resultados presentados en esta investigación extienden completamente a los espacios forma pseudo-euclídeos de curvatura constante cero, positiva y negativa, el resultado obtenido inicialmente por Alías y Gürbüz en [2]. Referencias [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 Lk?=A?+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 Lk?=A?+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. Abstract 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 ? satisfy ??=A?+b, where ? 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 ? satisfies Lk?=A?+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 Lk?=A?+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 Lk?=A?+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 Lk?=A?+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.