Minimal Surfaces

Similar to the shortest paths, we can study a family of surface patches \(\sigma^{\tau}:U\rightarrow\mathbb R^3\) and let \(\sigma = \sigma^0\). And the family must be smooth, a.k.a the map

\[M: \{(u,v,\tau): (u,v) \in U, \tau\in(-\delta, \delta)\}\rightarrow \mathbb R^3, M(u,v,\tau):=\sigma^{\tau}(u,v)\]

is smooth. Then define the surface variation of the family as \(\varphi: U\rightarrow\mathbb R^4 := \dot\sigma^{\tau}\mid_{\tau = 0}\)

For some simplse closed curve \(\pi: (a,b)\rightarrow U\) where the curve and its interior is in \(U\), then \(\gamma^\tau := \sigma^\tau\circ \pi\) is also a closed curve in \(\sigma^\tau\), we can then define the area function of the enclosed region as

\[A(\tau) = \int_{int(\pi)}dA_{\sigma^\tau}\]

Then, if a family of surfaces has a fixed boundary curve \(\gamma\), which means \(\forall \tau, \gamma^\tau = \gamma\), then we have that \(\varphi(u,v) = \vec 0\) for \((u,v)\) on the curve \(\pi\).

Theorem If the surface variation \(\varphi^tau\) vanishes along the boundary curve \(\pi\), then

\[A'(0) = -2\int_{int(\pi)} H(EG-F^2)^{1/2} (\varphi\cdot\mathbf N) dudv\]

Then, similar to how we define shortest path. Intuitively, a surface \(\Sigma\) is a minimal surface if \(\Sigma\) has the leastarea among all surfaces with the same boundary curve. Which means its surface patches \(\sigma = \sigma^0, A'(0) = 0\). Therefore, this formula is possible only if \(H=0\) for all points of \(U\).

Therefore, a minimal surface is a surface whose mean curvature is zero everywhere.

Gaussian Curvature of minimal surfaces

Theorem Gaussian curvature of a minimal surface is \(\leq 0\) everywhere, and \(=0\) IFF the surface is a open subset of a plane.

proof. By minimal surface, we have that \(H = \kappa_1 + \kappa_2 = 0\) for principal curvatures \(\kappa_1,\kappa_2\), thus \(K=\kappa_1\kappa_2 = -\kappa_1^2 \geq 0\).
If \(K = 0\), then at least one of \(\kappa_1,\kappa_2 = 0\), by minimal surface, both of them must be 0.

Parallel Surfaces

For an oriented surface \(\Sigma\) and some \(\lambda \in \mathbb R\). The parallel surface \(\Sigma^{\lambda}\) is

\[\Sigma^\lambda = \{\mathbf p + \lambda \mathbf N_p:p\in\Sigma\}\]

a.k.a. translate the surface \(\Sigma\) along the unit normal for a distance of \(\lambda\).

Theorem \(\sigma^\lambda_u\times \sigma^\lambda_v = (1-\lambda\kappa_1)(1-\lambda\kappa_2) (\sigma_u\times\sigma_v)\)

proof. \(\sigma^\lambda = \sigma + \lambda \mathbf N\), if we write derivatives of the normal in the basis of \(\sigma_u,\sigma_v\),

\[\sigma^\lambda_u = \sigma_u + \lambda \mathbf N_u = \sigma_u + \lambda(-a\sigma_u - b\sigma_v) = (1-\lambda a)\sigma_u - \lambda b \sigma_v\]

\[\sigma_v^\lambda = (1-\lambda d)\sigma_v - \lambda c\sigma_u\]

Therefore, the cross product is

\[\begin{align*} \sigma^\lambda_u \times \sigma^\lambda_v &= ((1-\lambda a)\sigma_u - \lambda b \sigma_v) \times ((1-\lambda d)\sigma_v - \lambda c\sigma_u)\\ &= (1-\lambda a - \lambda d + \lambda^2 (ad-bc))(\sigma_u\times \sigma_v)\\ &= (1-\lambda tr(W) + \lambda^2 \det(W))(\sigma_u\times \sigma_v)\\ &= (1 - \lambda(\kappa_1 + \kappa_2) + \lambda^2\kappa_1\kappa_2)(\sigma_u\times \sigma_v)\\ &= (1-\lambda\kappa_1)(1-\lambda\kappa_2) (\sigma_u\times\sigma_v) \end{align*}\]

Theorem Let \(\sigma\) be a minimal surface patch, let \(U\) be some region s.t. the bounded area of the region on \(\sigma\) is finite, a.k.a. \(A_\sigma(U) < \infty\). Let \(\lambda \neq 0\) and assume that the principal curvatures \(\kappa\) of \(|\lambda \kappa| < 1\) everywhere, so that the parallel surface \(\sigma^\lambda\) is a regular surface patch. Then, \(A_{\sigma^\lambda}(U) \leq A_{\sigma}(U)\) and the equality holds IFF \(\sigma(U)\) is an open subset of a plane.

\[\begin{align*} A_{\sigma^\lambda}(U) &= \iint_U \|\sigma_u^\lambda \times \sigma_v^\lambda\|dudv\\ &= \iint_U |1-\lambda\kappa_1||1-\lambda\kappa_2| \|\sigma_u\times\sigma_v\|dudv\\ &= \iint_U |1-\lambda\kappa_1||1 + \lambda\kappa_1| \|\sigma_u\times\sigma_v\|dudv &\kappa_1 = -\kappa_2\\ &= \iint_U |1-\lambda^2 \kappa_1^2| \|\sigma_u\times\sigma_v\|dudv\\ &\leq \iint_U \|\sigma_u\times\sigma_v\|dudv &|1-\lambda^2 \kappa_1^2| < 1\\ &= A_{\sigma}(U) \end{align*}\]

The equality IFF \(|1-\lambda^2 \kappa_1^2| = 1\) for all possible \(\lambda\), implying \(\kappa_1 = -\kappa_2 = 0\) IFF \(\sigma\) is an open subset of plane.

Compact Surface

A set of \(\mathbb R^3\) is compact iff closed and bounded, and it attains its maximum and minimum for any continuous function \(f:\mathbb R^3\rightarrow\mathbb R\). Since a surface is a subset of \(\mathbb R^3\), this also applies to surfaces.

Theorem If \(\Sigma\) is compact, then exists \(p\in Sigma\) where \(K(p) > 0\), \(K\) is the Gaussian curvature.

proof. Let \(f(x) = \|x\|^2\), \(f\) is continuous, take \(p\in\Sigma\) s.t. \(f(p)\) attains the maximum on \(\Sigma\). Since \(f(p)\) attains its maximum, it implies that the surface must be contained in the sphere center at 0 of radius \(\|p\|\). Thus, the Gaussian curvature at \(p\) is at least the Gaussian curvature of the sphere \(K(p) \geq \|p\|^{-2} > 0\).

Corollary There is no compact minimal surface.

proof. Minimal surface means \(K\leq 0\) for all points on the surface, contradicting with the fact that \(K > 0\) for some point on the compact surface.

Isometry of Minimal Surfaces

Theorem Applying an isometry or dilation on a minimal surface gives another minimal surface. proof. The first fundamental form does not change after the isometry, and the second fundamental form differs by a scale (\(\times a\) for dilation, \(a=\pm 1\) for isometry). hence \(H = a0 = 0\).

Theorem Applying a local isometry does not necessarily gives another minimal surface.
proof. Consider from a plane to a cylinder.

Examples of Minimal Surfaces

Catenoid and Surfaces of Revolution

For catenoid \(\sigma(u,v) = (\cosh u \cos v, \cosh u\sin u, u)\), we can compute its first and second fundamental form

\[E=G=\cosh^2 u, F=0, L=-1, M=0,N=1\]

thus, its mean curvature is

\[H = \frac{LG-2MF+NE}{2(EG-F^2)} = 0\]

Catenoid is a minimal surface.

Note that the theorem only applies to closed region bounded by a curve, for example

For \(a > 0, b = \cosh a\). The surface bounded by \(|z| < a\) of two circles of radius \(b\) has the area

\[A = \int_0^{2\pi}\int_{-a}^a \sqrt{EG-F^2} dudv = \int_0^{2\pi}\int_{-a}^a \cosh^2 u dudv = 2\pi(a+\sinh a\cosh a)\]

However, this two circles also bounds the area of two circles of radius \(b\), where the total area is \(2\pi b = 2\pi \cosh a\).

Theorem Any minimal surface of revolution is an open subset of a plane or a catenoid.

Helicoid and Ruled Surfaces

A helicoid is obtained by rotating a straight line at a constant speed, and move along an axis perpendicular to the line at constant speed.

By isometry, we define a helicoid of a line on the x-axis at a constant angular speed \(\omega\), along the z-axis at constant speed \(a\). Thus the parameterization is

\[\sigma(u,v) = (u\cos(\omega v), u\sin(\omega v), av)\]

The first and second fundamental form is

\[E= 1, F=0, G=(u^2 + a^2), L = N = 0, M = \frac{a^2}{\sqrt{a^2 + u^2} }\]

Thus, the mean curvature is \(H = 0\) since \(L = F = N = 0\)

Theorem Any minimal ruled surface is an open subset of a plane or a helicoid.

Catalan's Surface

The Catalan's surface can be parameterized as

\[\sigma(u,v) = (u-\sin u \cosh v, 1 - \cos u\cosh v, -4 \sin\frac{u}{2}\sinh\frac{v}{2})\]

The first fundamental form is

\[(\cosh v + 1)(\cosh v - \cos u)(du^2 + dv^2)\]

Since \(E=G\), to make \(H = 0\), we need \(L=-N\), a.k.a \(\sigma_{uu} = -\sigma_{vv}\), which can be verified.

Then, take \(u = 0, \gamma(v) = (0, 1 + \cosh v, 0)\) is a straight line, it is a geodesic.

Take \(u = \pi, \gamma(v) = (\pi - \cosh v, 1, -4 \sinh(v/2))\) is the parabola \(z^2 = 8(y-2)\), using the geodesic euqation, we can verify this is a geodesic.

Take \(v = 0, \gamma(u) = (u-\sin u, 1-cos u, 0)\) is a cycloid, using the geodesic euqation, we can verify this is a geodesic.

Source code

import plotly.graph_objects as go
from numpy import sin, cos, sinh, cosh, pi, mgrid

u, v = mgrid[-2 * pi: 2 * pi:20j, -0.5*pi:0.5*pi:10j]

fig = go.Figure(data=[go.Surface(
    x=u - sin(u) * cosh(v), 
    y=1 - cos(u) * cosh(v), 
    z=-4*sin(u/2) * sinh(v/2)
    )])
fig.update_traces(showscale=False)
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0))

with open("../assets/catalan.json", "w") as f:
    f.write(fig.to_json())