2. The capillary prototype

The point of departure was not a ready-made formula, but a constrained search problem in our stable-capillary work. Let \(M^n\) be a capillary minimal hypersurface, let \(\nu\) be its unit normal, and write

Here \(e_2\) is the unit vector orthogonal to \(e_1\) in the two-plane containing the prescribed normal directions, and we write

We wanted one nonnegative scalar function \(g=g(\nu)\) satisfying all three of the following requirements:

• a strictly positive Schoen-type inequality

for some \(c\gt{}0\);

• the exact zero set

• the Robin boundary condition

Here and below, \(\eta\) denotes the outer unit conormal of \(\partial M\) in \(M\).

2.1 The boundary condition as a design equation

The Robin condition was not an afterthought. It was one of the main filters on every candidate. Along \(\partial M\),

Consequently, if a one-variable squared profile \(U=p^2(\nu_1)\) is positive on the boundary, then direct substitution into (2.2) shows that its square-root profile satisfies

This logarithmic-derivative constraint, evaluated at \(\nu_1=\cos\theta\), was used to generate the one-variable candidates.

There was also an elementary but decisive closure property. The operator \(B_\theta\) is linear, and hence

Thus squared profiles could be designed additively without introducing a second boundary operator.

2.2 What was tried before the final family

Before reaching the final capillary family, we tried many alternatives. We began with the natural two-pole detector

It has exactly the desired two zeros and satisfies \(B_\theta(g_{\mathrm{pair}}^2)=0\). On the other hand, the one-pole background

satisfies the exceptionally clean identities

but it does not detect the two prescribed normals. Thus one candidate had the right zero set, while the other had the cleaner differential inequality.

We next tried to interpolate between these advantages through the geometric mixture

as well as truncated functions, power profiles, and profiles \(p(\nu_1)=\exp(q(\nu_1))\) chosen to satisfy (2.4). Working initially in the acute-angle range \(0\lt{}\theta\lt{}\pi/2\), we tested, among others:

We also tested the direct power \(1-\cos^{2-a}\theta\,\nu_1^a\). The calculations explained why these formulas were not the final choice: the multiplicative mixtures created an unfavorable cross-gradient term, while the one-variable modifiers could not simultaneously provide the exact two-pole zero set and a uniform interior margin.

At this stage, hand calculations were repeatedly converted into symbolic and numerical tests. For a chosen profile and parameter, we varied the normal direction and the trace-free second fundamental form, then computed the smallest value of the normalized expression in (2.1). The experiments were especially useful for rejecting a profile when its minimum became negative and for identifying a small-parameter regime that was worth proving analytically.

2.3 The additive route to the final family

The addition rule for \(B_\theta\) led directly to the final family. In addition to \(g_{\mathrm{pair}}\), define \(D(\nu)=1-\nu_1^2\). The three building blocks are \(D(\nu)\), \(g_{\mathrm{pair}}^2(\nu)\), and \(\nu_2^2\). Using (2.3), a direct computation gives \(B_\theta(D)=B_\theta(g_{\mathrm{pair}}^2)=B_\theta(\nu_2^2)=0\). Consequently, every function of the form

satisfies \(B_\theta(U_\alpha)=0\).

At either prescribed normal \(p_\pm\), we have \(D(p_\pm)=\nu_2^2(p_\pm)=\sin^2\theta\) and \(g_{\mathrm{pair}}(p_\pm)=0\). Thus \(U_\alpha(p_\pm)=0\) precisely when \(\alpha_1+\alpha_3=0\). Taking \(\alpha_3=-\alpha_1\), and using

we obtain

If \(\alpha_2\gt{}0\) and \(C_\alpha=\alpha_1+\alpha_2\sin^2\theta\gt{}0\), this expression is nonnegative on the unit sphere and vanishes exactly when \(\nu_1=\cos\theta\) and \(\nu_2=\pm\sin\theta\), that is, precisely at \(p_+\) and \(p_-\). Dividing by the harmless positive constant \(C_\alpha\), the family can be written using the more transparent parameter \(k=\frac{\alpha_2}{C_\alpha}\gt{}0\) as

Its first part confines the normal to the two-plane containing \(p_\pm\), but cannot distinguish points inside that plane. The second part selects the two prescribed normals, at the cost of introducing the difficult mixed terms. The parameter \(k\) controls their size: the margin left by the first part absorbs them when \(k\) is in the appropriate range.

We then tested this family numerically with very small values of \(k\). The experiments suggested that a positive lower bound should indeed exist, although the observed margin decreased more rapidly as \(\theta\) became small. This numerical evidence identified a plausible regime, but did not prove uniform positivity. The subsequent rigorous calculation verified analytically that the required positive lower bound exists.

2.4 The design blueprint from the capillary problem

The capillary construction suggested more than one successful formula. It provided a general architecture for the next stage of the search. At this point we regarded

only as a prototype, with the following division of roles.

• The background \(S\). This term should force the normal into a preferred low-dimensional set, usually a low-dimensional span. It should also provide the robust part of the differential inequality. By itself, however, it need not distinguish the prescribed normal directions inside that set.

• The detector \(P\). This term should separate the individual prescribed directions left indistinguishable by \(S\), and ultimately produce the desired finite zero set. Its derivatives are typically much harder to control than those of the background.

• The small parameter \(k\). The coefficient of \(P\) should be chosen small enough that the complicated terms introduced by the detector can be absorbed by the margin supplied by \(S\). The parameter is therefore part of the design, rather than a final cosmetic adjustment.

• Freedom in the outer form. The general problem no longer carries the capillary Robin boundary condition. We were therefore not restricted to taking the test function to be exactly a square root of \(S+kP\). Powers such as \((S+kP)^\gamma\), product forms, normalized products, and related compositions were all legitimate candidates, provided that they retained the correct zero set and differential positivity.

This became the blueprint for the subsequent experiments: first choose a simple \(S\) that enforces low-dimensionality, then vary the detector \(P\), the parameter \(k\), and the outer power or product structure. Symbolic and numerical tests could then identify which versions preserved a positive margin and, equally importantly, where each proposed design failed.