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Soap films with gravity and almost-minimal surfaces

F. M. and S. S. have been supported by NSF Grants DMS-1565354, DMS-1361122 and DMS-1262411

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  • Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.

    Mathematics Subject Classification: Primary: 49Q05, 49Q15; Secondary: 53A10.

    Citation:

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  • Figure 1.  On the left, a boundary $ \Gamma $, consisting of three circles, that is accessible from infinity. The acute wedges realizing the inclusions 3 are depicted by dashed lines. Notice that it is not necessary that $ \Gamma $ is contained into a convex set, or into a mean convex set, for the condition to hold. On the right, another set of circles defining a boundary $ \Gamma $ which does not satisfy accessibility from infinity. Indeed, there is no way to touch the smaller circle with an acute wedge containing the larger ones

    Figure 5.  The construction described in Example 7

    Figure 4.  When Γ consists of two parallel disks there are, in addition to the disconnected surface defined by two disks, four minimal surfaces, two of them singular, all composed by joining pieces of catenoids

    Figure 3.  Using Gauss' capillarity energy to formulate Plateau's problem. Minimization of $ \sigma\,\mathcal{H}^2(M) $ among surfaces with $ \partial M = \Gamma $ is replaced by minimizing the capillarity energy among regions contained in the complement of a $ \delta $-neighborhood of $ \Gamma $. Equilibrium configurations with volume $ \varepsilon\ll\delta\,\mathcal{H}^2(S)\ll1 $ arise as normal neighborhoods of minimal surfaces spanned by $ \Gamma $. Here $ S $ denotes the boundary of $ E $ away from the wire frame

    Figure 2.  The derivation of 12, after [16, Section Ⅰ.4]

    Figure 6.  Bubbling is possible even when $\Gamma$ is accessible from infinity if a weak notion of deficit is used. Here $M_j$ is the surface of revolution obtained by rotating the one-dimensional profile on the right, $B_{\varepsilon _j}(\Gamma_1)$ denotes an $\varepsilon _j$-neighborhood of the circle $\Gamma_1$, and $M_j^*$ is the part of $M_j$ lying outside $B_{\varepsilon _j}(\Gamma_1)$. We take $\varepsilon _j$ such that $M_j$ intersects $\partial B_{\varepsilon _j}(\Gamma_1)$ in three circles, and so that the $H_{M_j}$ is uniformly small on $M_j\setminus M_j^*$. The limit surface counts one copy of $K$, and two copies of the disk filling $\Gamma_1$

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