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

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.