I am wondering if there is an analog of the following theorem by Morgan and White: Suppose $g_E$ is the Euclidean metric on $R^3$. If $\gamma$ is a closed $C^{k,\alpha}$ curve in $(R^3,g_E)$, that bounds a unique area minimizing surface. Then every closed curve sufficiently closed to $\gamma$ in $C^{k,\alpha}$ also bounds a unique area minimizing surface.

Specifically, I am wondering what happens if we fix a curve, say a circle in $R^3$, and let the metric change. Intuitively, when the metric is closed to $g_E$, one expects that $\gamma$ still bounds a unique minimal surface. Is there any theorem that settled this question? More generally, if I consider the set of all circles on $R^3$, do I expect that when the metric $g$ is closed to $g_E$, all of them bound a unique area minimizing surface?

Any insight or reference would be very appreciated.

I am wondering if there is an analog of the following theorem by Morgan and White: Suppose $g_E$ is the Euclidean metric on $\mathbf R^3$. If $\gamma$ is a closed $C^{k,\alpha}$ curve in $(\mathbf R^3,g_E)$ that bounds a unique area minimizing surface, then every closed curve sufficiently closed to $\gamma$ in $C^{k,\alpha}$ also bounds a unique area minimizing surface.

Specifically, I am wondering what happens if we fix a curve, say a circle in $\mathbf R^3$, and let the metric change. Intuitively, when the metric is closed to $g_E$, one expects that $\gamma$ still bounds a unique minimal surface. Is there any theorem that settled this question? More generally, if I consider the set of all circles on $\mathbf R^3$, do I expect that when the metric $g$ is closed to $g_E$, all of them bound a unique area minimizing surface?

Any insight or reference would be very appreciated.