Geodesics on Surfaces and Minimal Surfaces

Geometry and Variational Calculus: an Inseparable Link

Differential geometry and variational calculus are closely intertwined: the key geometric objects are defined as extremals of functionals. Geodesics—the shortest curves—are extremals of the length functional. Minimal surfaces are extremals of the area functional. This connection is profound: understanding surface curvature through the lens of variational principles reveals why Einstein could formulate general relativity as geometry, and Dirac—describe spinors via normal bundles.

Geodesics: Definition and Equations

A geodesic on a surface is a curve minimizing the length between two points (locally). Formally: an extremal of the functional $J[\gamma] = \int |\gamma'(t)| dt$.

The geodesic equation in surface coordinates $x^k$:

$ \frac{d^2 x^k}{ds^2} + \Gamma^k_{ij} \frac{dx^i}{ds} \frac{dx^j}{ds} = 0 $

where $s$ is arc length, $\Gamma^k_{ij}$ are Christoffel symbols—computed from the metric tensor $g_{ij}$:

$ \Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}) $

On the plane: $g_{ij} = \delta_{ij} \rightarrow \Gamma^k_{ij} = 0 \rightarrow$ equation: $d^2 x^k/ds^2 = 0 \rightarrow x^k = a^k s + b^k$ (straight lines!).

On the sphere $(\theta, \varphi)$: Christoffel symbols are nonzero. The geodesic equation $\rightarrow$ great circles. The Moscow–New York route on a globe does not go “west,” but “through north”—this is a geodesic on the sphere.

Geometric Meaning: Parallel Transport

Covariant acceleration: $\nabla_{\gamma'} \gamma' = \frac{d^2 x^k}{ds^2} + \Gamma^k_{ij} \dot{x}^i \dot{x}^j$.

Geodesic equation: $\nabla_{\gamma'} \gamma' = 0$—the tangent vector is parallel transported along itself.

Intuition: on a curved surface, a “straight line” is a curve along which you “walk straight” (not turning). If you draw a straight line on a sheet of paper and roll the paper into a cylinder, the drawn straight line becomes a geodesic (helical curve or straight generator)—it is still “straight” from the viewpoint of an inhabitant of the cylinder.

Conjugate Points and Global Minimality

Second variation of length: $\delta^2 J = \int [|J'|^2 - K(J, \gamma', J, \gamma')] ds$, where $K$ is Gaussian curvature, $J$ is the Jacobi field.

Jacobi field along a geodesic $\gamma$: $J'' + K \cdot J = 0$ (simplified for surfaces). Zeros—conjugate points.

Jacobi–Bonnet theorem: a geodesic is the shortest curve up to the first conjugate point.

Example: on a sphere of radius $R$, geodesics are great circles. For the point $A = $“north pole,” the conjugate point is the “south pole” (at distance $\pi R$). The arc of a great circle from the north pole to the south is the shortest. But if you continue further (arc

gt; \pi R$), then the other arc through “west” is shorter!

Minimal Surfaces

A minimal surface is an extremal of the area functional $\text{Area}[S] = \iint_\Omega \sqrt{EG - F^2}\ du\ dv$.

The Euler–Lagrange equation $\rightarrow$ condition of zero mean curvature: $H = (\kappa_1 + \kappa_2)/2 = 0$.

Physical meaning: a soap film assumes a minimal surface shape—when there is zero pressure difference, surface tension forces balance at every point. This is indeed $H = 0$.

Classic minimal surfaces:

• Plane: $H = 0$ trivially
• Catenoid: surface of revolution $y = a\cosh(x/a)$. The only minimal surface of revolution (apart from the plane)
• Helicoid: $x = r\cos(az)$, $y = r\sin(az)$, $z = z$. “Twisted” catenoid

Plateau's problem: does a minimal surface exist with a given boundary curve $\Gamma$?

Rado–Douglas theorem (1931): for $\Gamma$ a simple Jordan curve—yes. Jesse Douglas received the first Fields Medal for this in 1936.

Thorough Analysis: Catenoid

Problem: find a surface of revolution of minimal area connecting two circles $x^2 + z^2 = r_0^2$ at $y = \pm h$.

Functional: $\text{Area} = 2\pi \int_{-h}^h r(y)\sqrt{1 + r'(y)^2}, dy$.

Euler–Lagrange equation: $\frac{d}{dy}\left[\frac{r, r'}{\sqrt{1 + r'^2}}\right] - \sqrt{1 + r'^2} = 0$.

Beltrami integral ($F$ independent of $y$): $F - r' F_{r'} = C \rightarrow \frac{r}{\sqrt{1 + r'^2}} = a$.

Solution: $dr/dy = \sqrt{r^2/a^2 - 1} \rightarrow r = a \cosh((y+b)/a)$.

From boundary conditions $r(\pm h) = r_0$: $r_0 = a \cosh(h/a)$. This is a transcendental equation for $a$ (for small $h/r_0$ a solution exists).

Physical application: a soap film between two rings assumes the shape of a catenoid. When separating the rings: if the distance exceeds a critical value, the film “breaks” into two disks. This is a bifurcation of minimal surfaces!

Applications

In architecture, minimal surfaces define optimal shapes of vaults and shells (minimum material for given boundary conditions). In materials science, block copolymers self-organize into structures with minimal surface (gyroid, etc.). In biology, the shape of erythrocytes and cell membranes is determined by variational principles with volume constraints.

Geodesics in General Theory of Relativity

Geodesics are paths minimizing the interval in curved spacetime. In general relativity, the motion of a free body occurs along a geodesic of the Schwarzschild, Kerr, or Friedmann metric. Planetary orbits, deflection of light by the Sun, black holes—all are geodesics in curved geometry. The geodesic equation: $d^2 x^\mu/ds^2 + \Gamma^\mu_{\alpha\beta} (dx^\alpha/ds)(dx^\beta/ds) = 0$, where $\Gamma$ are Christoffel symbols of the metric.

Minimal Surfaces and Bubbles

Minimal surfaces are surfaces with zero mean curvature $H = (k_1 + k_2)/2 = 0$. They minimize area for a given boundary. Classic examples: plane, catenoid (surface of revolution of a catenary), helicoid (Scherk’s twisted surface), Enneper’s surface. In nature, minimal surfaces arise in soap films: surface tension minimizes area.

Plateau's Problem

Plateau's problem: given a closed curve $\Gamma \subset \mathbb{R}^3$. Find a surface of minimal area spanning $\Gamma$. Existence of a solution was proven by Douglas (1931) and Rado (1933)—for this, Douglas received the first Fields Medal in 1936. Modern generalizations: Plateau's problem on manifolds, in higher dimensions, with obstacles—an active area of geometric analysis.

Numerical Methods

• Mean curvature flow: evolution of the surface in the normal direction at speed $H$—the natural gradient flow for the area functional
• Finite element method on surfaces: discretization of the surface with a triangular mesh and numerical solution of PDE
• Level-set methods: representing the surface as the zero level of a function—robust to topological changes

Applications

• Architecture: tent structures (stadiums, exhibition complexes—the roof of Munich Olympic Stadium by Frei Otto) are calculated as minimal surfaces
• Materials science: grain structure in metals, phase boundaries
• Biophysics: shape of biological membranes, vesicles, cell walls
• Computer graphics: creation of smooth surfaces with given contours for 3D modeling
• Tunnel engineering: optimization of tunnel shapes to minimize stress in rock