Research Article
Year 2024,
Volume: 17 Issue: 1, 34 - 43, 23.04.2024
Abstract
The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by
N. Kuiper. The special examples of this type are generalized Beltrami
submanifolds and toroidals submanifold. The second named authour and at. all
recently have considered $3-$dimensional rotational embedded submanifolds in
$\mathbb{E}^{5}$. They gave some basic curvature properties of this type of
submaifolds. Self-similar flows emerge as a special solution to the mean
curvature flow that preserves the shape of the evolving submanifold. In
this article we consider self-similar submanifolds in Euclidean spaces. We
obtained some results related with self-shrinking rotational submanifolds in
Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and
sufficient conditions for these type of submanifolds to be homothetic
solitons for their mean curvature flows.
References
- [1] Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differential Geom. 23, 175-196 (1986).
- [2] Andrews, B.: Classification of limiting shapes for isotropic curve flows. J. Amer. Math. Soc. 16, 443-459 (2003).
- [3] Arezzo, C., Sun, J.: Self-shrinkers for the mean curvature flow in arbitrary codimension. Math. Z. 274, 993-1027 (2013).
- [4] Arslan, K., Bayram (Kılıç), B., Bulca, B., Öztürk, G.: Rotation submanifolds in Euclidean spaces. Int. J. Geom. Meth. Mod. Phy. 16, 1-12 (2019).
- [5] Cao, H.D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879-889 (2013).
- [6] Castro I., Lerma A.M.: The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow. Int. Math. Res. Not. 16, 1515-1152 (2014).
- [7] Castro, I., Lerma A.M.: Homothetic solitons for the inverse mean curvature flow. Results in Math. 71, 1109-1125 (2017).
- [8] Chen, B. Y.: Geometry of Submanifolds. Dekker, New York (1973).
- [9] Chen, B. Y.: Topics in differential geometry associated with position vector fields on Euclidean submanifolds. Arab J. Math. Sci. 23, 1-17 (2017).
- [10] Chen, B.Y., Deshmukd, S.: Classification of Ricci solitons on Euclidean hypersurfaces. Int. J. Math. 25, 1-22 (2014).
- [11] Cheng, Q.M., Li, Z., Wei, G.: Complete self-shrinkers with constant norm of the second fundamental form. Mathematische Zeitschrift. 300,995-1018 (2022).
- [12] Cheng, Q.M., Peng, Y.: Complete self-shrinkers of the mean curvature flow. Calc. Var. 52, 497-506 (2015).
- [13] Cooper, A.A.: Mean Curvature Flow in Higher Codimension. Ph.D. Thesis. Michigan State University, Graduate Program in Mathematics, USA.(2011).
- [14] Drugan, G., Lee, H., Nguyen, X.H.: A survey of closed self-shrinkers with symmetry. Results in Math. 32, 73-32 (2018).
- [15] Drugan, G., Lee, H., Wheeler, G.: Solitons for the inverse mean curvature flow. Pasific J. Math. 284, 309-326 (2016).
- [16] Ecker, K.: Regularity theory for mean curvature flow. Birkhäuser Inc., Boston, (2004).
- [17] Ecker, K., Huisken, G.: Mean curvature evolution of entire gaphs. Annals of Math. 130, 453-471 (1989).
- [18] Etemoğlu, E., Arslan, K., Bulca, B.: Self similar surfaces in Euclidean spaces. Selçuk J. Appl. Math. 14, 71-81 (2013).
- [19] Gorkavyy G., Nevmerzhytska, O.: Pseudo-spherical submanifolds with degenerate Bianchi transformation. Results. Math. 60, 103-116 (2011).
- [20] Guan Z., Li, F.: Self-shrinker type submanifolds in the Euclidean space. Bul. Iranian Math. Soc. 47, 101-110 (2021).
- [21] Guo, S. H.: Self shrinkers and singularity models of the main curvature flow. Ph.D. Thesis, The State University of New Jersey, Graduate Program in Mathematics, USA, (2017).
- [22] Halldorsson, P.H.: Self-similar solutions to the mean curvature flow in Euclidean and Minkowski space. Ph.D. Thesis. Masschusetts Institute of Technology, Department of Mathematics, USA, (2013).
- [23] Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59, 353-437 (2001).
- [24] Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differential Geom. 80, 433-451 (2008).
- [25] Hussey, C.: Classification and analysis of low index mean curvature flow self-shrinkers. Ph.D. Thesis. The Johns Hopkins University,Department of Mathematics, USA, (2012).
- [26] Joyse, D., Lee, Y., Tsui M.P.: Self-similar solutions and translating solutions for Lagrangian mean curvature flow. J. Differential Geom. 84, 127-161 (2010).
- [27] Kuiper, N.H.: Minimal total absolute curvature for immersions. Invent. Math. 10, 209-238 (1970).
- [28] Montegazza, C.: Lecture notes on mean curvature flow, Birkhauser (2011).
- [29] Peng, Y.: Complete self-shrinkers of mean curvature flow. Ph.D. Thesis. Saga University, Graduate School of Science and Engineering, Department of Science and Advanced Technology, Japan (2013).
- [30] Schulze, F.: Introduction to mean curvature flow. Lecture Notes, University College London, (2017).
- [31] Sigal, I. M.: Lectures on mean curvature flow and stability. Lecture Notes, Dept. of Mathematics, Univ. of Toronto, (2014).
- [32] Smoczyk, K.: Mean curvature flow in higher codimension: Introduction and survey. Global Differential Geometry, Springer Verlag, Berlin,Heidelberg, (2012).
Year 2024,
Volume: 17 Issue: 1, 34 - 43, 23.04.2024
Abstract
References
- [1] Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differential Geom. 23, 175-196 (1986).
- [2] Andrews, B.: Classification of limiting shapes for isotropic curve flows. J. Amer. Math. Soc. 16, 443-459 (2003).
- [3] Arezzo, C., Sun, J.: Self-shrinkers for the mean curvature flow in arbitrary codimension. Math. Z. 274, 993-1027 (2013).
- [4] Arslan, K., Bayram (Kılıç), B., Bulca, B., Öztürk, G.: Rotation submanifolds in Euclidean spaces. Int. J. Geom. Meth. Mod. Phy. 16, 1-12 (2019).
- [5] Cao, H.D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879-889 (2013).
- [6] Castro I., Lerma A.M.: The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow. Int. Math. Res. Not. 16, 1515-1152 (2014).
- [7] Castro, I., Lerma A.M.: Homothetic solitons for the inverse mean curvature flow. Results in Math. 71, 1109-1125 (2017).
- [8] Chen, B. Y.: Geometry of Submanifolds. Dekker, New York (1973).
- [9] Chen, B. Y.: Topics in differential geometry associated with position vector fields on Euclidean submanifolds. Arab J. Math. Sci. 23, 1-17 (2017).
- [10] Chen, B.Y., Deshmukd, S.: Classification of Ricci solitons on Euclidean hypersurfaces. Int. J. Math. 25, 1-22 (2014).
- [11] Cheng, Q.M., Li, Z., Wei, G.: Complete self-shrinkers with constant norm of the second fundamental form. Mathematische Zeitschrift. 300,995-1018 (2022).
- [12] Cheng, Q.M., Peng, Y.: Complete self-shrinkers of the mean curvature flow. Calc. Var. 52, 497-506 (2015).
- [13] Cooper, A.A.: Mean Curvature Flow in Higher Codimension. Ph.D. Thesis. Michigan State University, Graduate Program in Mathematics, USA.(2011).
- [14] Drugan, G., Lee, H., Nguyen, X.H.: A survey of closed self-shrinkers with symmetry. Results in Math. 32, 73-32 (2018).
- [15] Drugan, G., Lee, H., Wheeler, G.: Solitons for the inverse mean curvature flow. Pasific J. Math. 284, 309-326 (2016).
- [16] Ecker, K.: Regularity theory for mean curvature flow. Birkhäuser Inc., Boston, (2004).
- [17] Ecker, K., Huisken, G.: Mean curvature evolution of entire gaphs. Annals of Math. 130, 453-471 (1989).
- [18] Etemoğlu, E., Arslan, K., Bulca, B.: Self similar surfaces in Euclidean spaces. Selçuk J. Appl. Math. 14, 71-81 (2013).
- [19] Gorkavyy G., Nevmerzhytska, O.: Pseudo-spherical submanifolds with degenerate Bianchi transformation. Results. Math. 60, 103-116 (2011).
- [20] Guan Z., Li, F.: Self-shrinker type submanifolds in the Euclidean space. Bul. Iranian Math. Soc. 47, 101-110 (2021).
- [21] Guo, S. H.: Self shrinkers and singularity models of the main curvature flow. Ph.D. Thesis, The State University of New Jersey, Graduate Program in Mathematics, USA, (2017).
- [22] Halldorsson, P.H.: Self-similar solutions to the mean curvature flow in Euclidean and Minkowski space. Ph.D. Thesis. Masschusetts Institute of Technology, Department of Mathematics, USA, (2013).
- [23] Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59, 353-437 (2001).
- [24] Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differential Geom. 80, 433-451 (2008).
- [25] Hussey, C.: Classification and analysis of low index mean curvature flow self-shrinkers. Ph.D. Thesis. The Johns Hopkins University,Department of Mathematics, USA, (2012).
- [26] Joyse, D., Lee, Y., Tsui M.P.: Self-similar solutions and translating solutions for Lagrangian mean curvature flow. J. Differential Geom. 84, 127-161 (2010).
- [27] Kuiper, N.H.: Minimal total absolute curvature for immersions. Invent. Math. 10, 209-238 (1970).
- [28] Montegazza, C.: Lecture notes on mean curvature flow, Birkhauser (2011).
- [29] Peng, Y.: Complete self-shrinkers of mean curvature flow. Ph.D. Thesis. Saga University, Graduate School of Science and Engineering, Department of Science and Advanced Technology, Japan (2013).
- [30] Schulze, F.: Introduction to mean curvature flow. Lecture Notes, University College London, (2017).
- [31] Sigal, I. M.: Lectures on mean curvature flow and stability. Lecture Notes, Dept. of Mathematics, Univ. of Toronto, (2014).
- [32] Smoczyk, K.: Mean curvature flow in higher codimension: Introduction and survey. Global Differential Geometry, Springer Verlag, Berlin,Heidelberg, (2012).
There are 32 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebraic and Differential Geometry |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | April 5, 2024 |
| Publication Date | April 23, 2024 |
| Acceptance Date | March 24, 2024 |
| Published in Issue | Year 2024 Volume: 17 Issue: 1 |
