Continuous $HG$-deformations of surfaces with boundary in Euclidean space

A. I. Bodrenko

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2014), pp. 6-13.

Résumé

The properties of continuous deformations of surfaces with boundary in Euclidean $3$-space preserving its Grassmannian image and mean curvature are studied in this article. We determine the continuous $HG$-deformation for simply connected oriented surface $F$ with boundary $\partial F$ in Euclidean $3$-space. We derive the differential equations of $G$-deformations of surface $F$. We prove the lemma where we derive auxiliary properties of functions characterizing $HG$-deformations of surface $F$. Then on the surface $F$ we introduce conjugate isothermal coordinate system which simplifies the form of equations of $G$-deformations. From the system of differential equations characterizing $G$-deformations of surface $F$ in conjugate isothermal coordinate system we go to the nonlinear integral equation and resolve it by the method of successive approximations. We derive the equations of $HG$-deformations of surface $F$. We get the formulas of change $\Delta(g_{ij})$ and $\Delta(b_{ij})$ of coefficients $g_{ij}$ and $b_{ij}$ of the first and the second fundamental forms of surface $F$, respectively, for deformation $\{F_t\}$. Then, using formulas of $\Delta(g_{ij})$ and $\Delta(b_{ij})$, we find the conditions characterizing $HG$-deformations of two-dimensional surface $F$ in Euclidean space $E^3$. We show that finding of $HG$-deformations of surface $F$ brings to the following boundary-value problem $(A)$: $$ \partial_{\overline{z}}\dot{w}+A\dot{w}+B\overline{\dot{w}}+E(\dot{w})=\dot{\Psi}, \qquad Re\{\overline{\lambda}\dot{w}\}=\dot{\varphi} \quad \rm{on} \quad \partial F, $$ where $A$, $B$, $\lambda$, $\dot{\Psi}$, $\dot{\varphi}$ are given functions of complex variable, $\dot{w}$ is unknown function of complex variable, operator $E(\dot{w})$ has implicit form. Prior to resolving boundary-value problem $(A)$ we find the solution of the following boundary-value problem for generalized analytic functions: $$\partial_{\bar{z}}\dot{w}+A\dot{w}+B\bar{\dot{w}}=\dot{\Psi}, \quad Re\{\overline{\lambda}\dot{w}\}=\dot{\varphi} \quad \rm{on} \quad \partial F. $$ Then we use the theory of Fredholm operator of index zero and the theory of Volterra operator equation. Using the method of successive approximations and the principle of contractive mapping, we obtain solution of boundary-value problem $(A)$ and the proof of theorem 1, the main result of this article.

Détail

Keywords: deformation of surface, mean curvature, Gaussian curvature, $G$-deformation, continuous deformation.

Bibliographie

[1] A.\;I. Bodrenko, “Some properties of continuous almost $ARG$-deformations”, Russian Mathematics, 1996, no. 2 (405), 13–16 | MR | Zbl

[2] A.\;I. Bodrenko, “Continuous $MG$-deformations of surfaces with boundary in Euclidean space”, Science Journal of Volgograd State University. Mathematics. Physics, 2013, no. 1 (18), 6–23

[3] A.\;I. Bodrenko, “On properties of almost AR-deformations of hypersurfaces with condition of generalized sliding”, OP Surveys in Applied and Industrial Mathematics, 18:5 (2011), 745

[4] A.\;I. Bodrenko, “On defective index of tubular hypersurfaces in Euclidean space”, OP Surveys in Applied and Industrial Mathematics, 13:4 (2006), 616–617

[5] A.\;I. Bodrenko, “Surfaces with recurrent second fundament form in $E^3$”, OP Surveys in Applied and Industrial Mathematics, 11:2 (2004), 300–301

[6] A.\;I. Bodrenko, “Continuous $MG$-deformations of surfaces in Euclidean space”, Torus Actions: Topology, Geometry and Number Theory, Abstracts of the International open chinese-russian conference (02–07 September, 2013, Khabarovsk, Russia), Pacific National University Publ., Khabarovsk, 2013, 13–14 | MR

[7] A.\;I. Bodrenko, “Some properties of continuous almost $ARG$-deformations”, Russian Mathematics, 40:2 (1996), 11–14 | MR | Zbl