Singular minimal translation graphs in Euclidean spaces
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we consider the problem of finding the hypersurface $M^{n}$ in the Euclidean $\left( n+1\right)$-space $\mathbb{R}^{n+1}$ that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper halfspace $\mathbb{R}_{+}^{3}\left( \mathbf{u} \right) $ with lowest gravity center, for a fixed unit vector $\mathbf{u}\in \mathbb{R}^{3}$. We first state that a singular minimal cylinder $M^{n}$ in $\mathbb{R}^{n+1}$ is either a hyperplane or a $\alpha $-catenary cylinder. It is also shown that this result remains true when $M^{n}$ is a translation hypersurface and $\mathbf{u}$ is a horizantal vector. As a further application, we prove that a singular minimal translation graph in $\mathbb{R }^{3}$ of the form $z=f(x)+g(y+cx),$ $c\in \mathbb{R-\{}0\},$ with respect to a certain horizantal vector $\mathbf{u}$ is either a plane or a $\alpha $- catenary cylinder.
Abstract
In this paper, we consider the problem of finding the hypersurface $M^{n}$ in the Euclidean $\left( n+1\right)$-space $\mathbb{R}^{n+1}$ that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper halfspace $\mathbb{R}_{+}^{3}\left( \mathbf{u} \right) $ with lowest gravity center, for a fixed unit vector $\mathbf{u}\in \mathbb{R}^{3}$. We first state that a singular minimal cylinder $M^{n}$ in $\mathbb{R}^{n+1}$ is either a hyperplane or a $\alpha $-catenary cylinder. It is also shown that this result remains true when $M^{n}$ is a translation hypersurface and $\mathbf{u}$ is a horizantal vector. As a further application, we prove that a singular minimal translation graph in $\mathbb{R }^{3}$ of the form $z=f(x)+g(y+cx),$ $c\in \mathbb{R-\{}0\},$ with respect to a certain horizantal vector $\mathbf{u}$ is either a plane or a $\alpha $- catenary cylinder.
참고문헌 (32)
- M. E. Aydin, "Translation hypersurfaces and Tzitzeica translation hypersurfaces of the Euclidean space", Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci, 16(4), pp.477~483 (2015)
- B.-Y. Chen, "Geometry of Submanifolds" (1973), Marcel Dekker, Inc
- B.-Y. Chen, "On some geometric properties of h-homogeneous production functions in microeconomics", Kragujevac J. Math, 35(3), pp.343~357 (2011)
- B.-Y. Chen, "Pseudo-Riemannian Geometry, δ-Invariants and Applications" (2011), World Scientific Publishing Co. Pte. Ltd
- U. Dierkes, "A Bernstein result for energy minimizing hypersurfaces", Calculus of Variations and Partial Differential Equations, 1(1), pp.37~54 (1993), Springer Science and Business Media LLC[DOI]
- U. Dierkes, "Geometric Analysis and Nonlinear Partial Differential Equations", pp.177~193 (2003), Springer
- F. Dillen, "A generalization of the translation surfaces of Scherk", pp.107~109 (1991), KUL
- W. Goemans, "Translation and homothetical lightlike hypersurfaces of a semi-Euclidean space", Kuwait J. Sci. Engrg, 38(2A), pp.35~42 (2011)
- A. Gray, "Modern differential Geometry of Curves and Surfaces with Mathematica, second edition" (1998), CRC Press
- Thomas Hasanis, "Translation surfaces with non-zero constant mean curvature in Euclidean space", Journal of Geometry, 110(2), pp.8~ (2019), Springer Science and Business Media LLC[DOI]
- "Translation surfaces in Euclidean space with constant Gaussian curvature" (2018)[URL]
- Thomas Hasanis, "Classification and Construction of Minimal Translation Surfaces in Euclidean Space", Results in Mathematics, 75(1), pp.22~ (2020), Springer Science and Business Media LLC[DOI]
- Seoung Dal Jung, "Weingarten Affine Translation Surfaces in Euclidean 3-Space", Results in Mathematics, 72(4), pp.1839~1848 (2017), Springer Science and Business Media LLC[DOI]
- Barnabé P. Lima, "Generalized translation hypersurfaces in Euclidean space", Journal of Mathematical Analysis and Applications, 470(2), pp.1129~1135 (2019), Elsevier BV[DOI]
- Huili Liu, "Translation surfaces with constant mean curvature in 3-dimensional spaces", Journal of Geometry, 64(1-2), pp.141~149 (1999), Springer Science and Business Media LLC[DOI]
- Huili Liu, "Affine translation surfaces with constant mean curvature in Euclidean 3-space", Journal of Geometry, 108(2), pp.423~428 (2017), Springer Science and Business Media LLC[DOI]
- Huili Liu, "Affine translation surfaces in Euclidean 3-space", Proceedings of the Japan Academy, Series A, Mathematical Sciences, 89(9), pp.111~113 (2013), Project Euclid[DOI]
- R. L´opez, "Constant Mean Curvature Surfaces with Boundary, Springer Monographs in Mathematics" (2013), Springer
- Rafael López, "Separation of variables in equations of mean-curvature type", Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 146(5), pp.1017~1035 (2016), Cambridge University Press (CUP)[DOI]
- Rafael López, "Invariant singular minimal surfaces", Annals of Global Analysis and Geometry, 53(4), pp.521~541 (2018), Springer Science and Business Media LLC[DOI]
- "Compact singular minimal surfaces with boundary" (2018)[URL]
- Rafael López, "The one dimensional case of the singular minimal surfaces with density", Geometriae Dedicata, 200(1), pp.303~320 (2019), Springer Science and Business Media LLC[DOI]
- Rafael López, "The two-dimensional analogue of the Lorentzian catenary and the Dirichlet problem", Pacific Journal of Mathematics, 305(2), pp.693~719 (2020), Mathematical Sciences Publishers[DOI]
- Rafael L\'opez, "TRANSLATION AND HOMOTHETICAL SURFACES IN EUCLIDEAN SPACE WITH CONSTANT CURVATURE", 대한수학회지, 52(3), pp.523~535 (2015), 대한수학회
- Marilena Moruz, "Minimal translation hypersurfaces in E 4", Journal of Mathematical Analysis and Applications, 439(2), pp.798~812 (2016), Elsevier BV[DOI]
- Marian Ioan Munteanu, "Minimal Translation Hypersurfaces in Euclidean Space", Mediterranean Journal of Mathematics, 13(5), pp.2659~2676 (2016), Springer Science and Business Media LLC[DOI]
- K. Seo, "Translation hypersurfaces with constant curvature in space forms", Osaka J. Math, 50(3), pp.631~641 (2013)
- G. Aydın S¸ekerci, "On the translation hypersurfaces with Gauss map G satisfying ∆G = AG", Miskolc Math. Notes, 20(2), pp.1215~1225 (2019)
- B. van Brunt, "The Calculus of Variations, Universitext" (2004), Springer-Verlag
- Dan Yang, "On affine translation surfaces in affine space", Journal of Mathematical Analysis and Applications, 440(2), pp.437~450 (2016), Elsevier BV[DOI]
- D. Yang, "A note on minimal translation graphs in Euclidean space", Mathematics, 7(10), pp.889~ (2019)
- Dae Won Yoon, "Weighted minimal affine translation surfaces in Euclidean space with density", International Journal of Geometric Methods in Modern Physics, 15(11), pp.1850196~ (2018), World Scientific Pub Co Pte Ltd[DOI]