¿Cómo llenar una región entre dos curvas en Tikz?

Question

Básicamente, estoy intentando reproducir una versión modificada de esta figura:

Esto es lo que he hecho hasta ahora y me gustaría rellenar la región entre u1 y u2 con el mismo color de degradado:

\documentclass{article}
\usepackage{tikz}

\usepackage{tikz-3dplot}

\usepackage[active,tightpage]{preview}  
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{2mm}

\begin{document}

\tdplotsetmaincoords{60}{110}

\pgfmathsetmacro{\rvec}{.8}
\pgfmathsetmacro{\thetavec}{30}
\pgfmathsetmacro{\phivec}{60}

\begin{tikzpicture}[scale=5,tdplot_main_coords]

\coordinate (O1) at (0,0,0);

\tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}

\draw[] (O1) -- (1,0,0) node[anchor=north east]{$x_1$};
\draw[] (O1) -- (0,1,0) node[anchor=north west]{$x_2$};
\draw[] (O1) -- (0,0,1) node[anchor=south]{$x_3$};

\draw[thick,-stealth] (O1) -- (.25,0,0) node[anchor=north west]{$\mathbf{e}_1$};
\draw[thick,-stealth] (O1) -- (0,.25,0) node[anchor=north west]{$\mathbf{e}_2$};
\draw[thick,-stealth] (O1) -- (0,0,.25) node[anchor=north east]{$\mathbf{e}_3$};

\coordinate (O2) at (0,0.5,0.5);

\draw[thick,-stealth,color=black] (O1) -- (O2);

\draw[thick,-stealth] (O2) -- (0,0.85,0.25) node[anchor=south west]{$\mathbf{g}_1$};
\draw[thick,-stealth] (O2) -- (0,.75,0.75) node[anchor=west]{$\mathbf{g}_2$};
\draw[thick,-stealth] (O2) -- (0,0.35,0.75) node[anchor= west,pos=0.9]{$\mathbf{g}_3$};

\draw [draw=black, >=latex, thick] (O2) to [bend left=8] coordinate [pos=7/8] (q2n) (0,0.9,0.1) coordinate (q2) node [right] {$u_1$ curve};
\draw [draw=black, >=latex, thick] (O2) to [bend left=8] coordinate [pos=7/8] (q3n) (0,1,0.8) coordinate (q3) node [right] {$u_2$ curve};
\draw [draw=black, >=latex, thick] (O2) to [bend right=8] coordinate [pos=7/8] (q1n) (0,0.2,0.8) coordinate (q1) node [anchor=south] {$u_3$ curve};

\end{tikzpicture}

\end{document}

Answer 1

Un sombreado bastará. Aquí se utiliza un sombreado horizontal personalizado para (intentar) evitar líneas nítidas en el borde de la región sombreada. El éxito de esto puede depender del espectador. Además, el sombreado se coloca sobre una capa de fondo para que no cubra las líneas:

\documentclass[tikz,border=5]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{backgrounds}
\begin{document}
\tdplotsetmaincoords{60}{110}

\pgfmathsetmacro{\rvec}{.8}
\pgfmathsetmacro{\thetavec}{30}
\pgfmathsetmacro{\phivec}{60}

\pgfdeclarehorizontalshading{fade out}{100bp}{
color(0bp)=(gray);color(60bp)=(white);
color(90bp)=(white);color(100bp)=(white)}
\begin{tikzpicture}[scale=5, tdplot_main_coords, line cap=round]

\coordinate (O1) at (0,0,0);

\tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}

\draw (O1) -- (1,0,0) node [below left]  {$x_1$};
\draw (O1) -- (0,1,0) node [below right] {$x_2$};
\draw (O1) -- (0,0,1) node [above]       {$x_3$};

\draw [thick, -stealth] (O1) -- (.25,0,0) node [below right] {$\mathbf{e}_1$};
\draw [thick, -stealth] (O1) -- (0,.25,0) node [below right] {$\mathbf{e}_2$};
\draw [thick, -stealth] (O1) -- (0,0,.25) node [below left]  {$\mathbf{e}_3$};

\coordinate (O2) at (0,0.5,0.5);

\draw [thick, -stealth] (O1) -- (O2);

\draw [thick, -stealth] (O2) -- (0,0.85,0.25) 
  node [above right] {$\mathbf{g}_1$};
\draw [thick, -stealth] (O2) -- (0,0.75,0.75) 
  node [right] {$\mathbf{g}_2$};
\draw [thick, -stealth] (O2) -- (0,0.35,0.75) 
  node [right, pos=0.9] {$\mathbf{g}_3$};

\draw [draw=black, >=latex, thick] (O2) to [bend left=8]  
  coordinate [pos=7/8] (q2n) (0,0.9,0.1) 
  coordinate (q2) node [right] {$u_1$ curve};
\draw [draw=black, >=latex, thick] (O2) to [bend left=8]  
  coordinate [pos=7/8] (q3n) (0,1,0.8) 
  coordinate (q3) node [right] {$u_2$ curve};
\draw [draw=black, >=latex, thick] (O2) to [bend right=8] 
  coordinate [pos=7/8] (q1n) (0,0.2,0.8) 
  coordinate (q1) node [anchor=south] {$u_3$ curve};

\begin{pgfonlayer}{background}
\shade [shading=fade out, shading angle=-15] 
  (O2) to [bend left=8] (q3) to [bend left=8] (q2) to [bend right=8] (O2);
\end{pgfonlayer}
\end{tikzpicture}
\end{document}