documentclass[svgnames]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.8}

begin{document}
    begin{tikzpicture}
    begin{axis}[
            axis lines=center,
            axis equal,
            domain=0:360,
            y domain=0:1.25,
            y axis line style=stealth-,
            y label style={at={(0.325,0.15)}},
            xmax=1.75,ymax=1.5,zmax=1.5,
            xlabel = $varphi_1$,
            ylabel=$varphi_2$,
            zlabel=$U_k(rho)$,
            ticks=none,
        ]

        addplot3 [surf,shader=flat,draw=black,fill=white,z buffer=sort] ({sin(x)*y}, {cos(x)*y}, {(y^2-1)^2});
        end{axis}
        fill[DarkBlue] (3.05,3.4) circle (0.1) coordinate (center);
        fill[DarkRed] (4.7,2) circle (0.1) coordinate (minimum);
        draw[LightGray,line width=0.8,shorten <=5,shorten >=5] (center) edge[out=-15,in=150,->] (minimum);
    end{tikzpicture}
end{document}