A minimal surface has zero mean curvature H = 0. This means the surface is a critical point of the area functional. Does this mean minimal surfaces have the least area among all surfaces with the same boundary?
AYes — zero mean curvature implies global area minimization
BNot necessarily — they are critical points (like saddle points can be critical points of a function), so they locally minimize area but may not globally minimize
CNo — minimal surfaces always maximize area
DOnly for compact surfaces without boundary
Minimal surfaces are critical points of the area functional, meaning the first variation of area vanishes. Like saddle points of functions, they need not be global (or even local) minima. The catenoid, for instance, is minimal but can be deformed to a surface with less area (a pair of disks). Area-minimizing surfaces are always minimal (H = 0), but the converse is false. The terminology 'minimal' is historical and somewhat misleading.
Question 2 Short Answer
The second fundamental form of a surface S ⊂ ℝ³ measures how S curves within the ambient space. The mean curvature H is the trace of the second fundamental form. If the principal curvatures at a point are κ₁ and κ₂, then H = (κ₁ + κ₂)/2 = 0 on a minimal surface. What does this imply about the principal curvatures?
Think about your answer, then reveal below.
Model answer: At every point of a minimal surface, κ₁ = -κ₂. The surface curves equally in opposite directions — it is saddle-shaped at every point (except where both curvatures are zero, which is an umbilic flat point). The Gaussian curvature K = κ₁κ₂ = -κ₁² ≤ 0 at every point. Minimal surfaces therefore have non-positive Gaussian curvature everywhere.
The saddle shape is visible in examples: the catenoid, helicoid, and Enneper's surface all have saddle-shaped geometry at every point. The condition κ₁ = -κ₂ means the surface looks like a saddle in every direction — if it curves up in one principal direction, it curves down equally in the perpendicular direction.
Question 3 True / False
The Plateau problem asks: given a closed curve Γ in ℝ³, does there exist a minimal surface (area-minimizing surface) with boundary Γ?
TTrue
FFalse
Answer: True
The Plateau problem was solved by Jesse Douglas and Tibor Radó in 1930 (Douglas received a Fields Medal for this). For any rectifiable Jordan curve Γ in ℝ³, there exists a disk-type surface of least area spanning Γ. The solution may not be unique, smooth, or embedded — regularity and uniqueness require additional conditions on Γ. The Plateau problem is the founding problem of the calculus of variations in geometry and has been generalized in many directions (higher dimensions, different boundary conditions, singular surfaces).
Question 4 True / False
The only complete minimal surfaces in ℝ³ that are also planes are... planes. More precisely, a complete minimal surface in ℝ³ with finite total curvature ∫|K|dA < ∞ is conformally equivalent to a compact Riemann surface with finitely many punctures.
TTrue
FFalse
Answer: True
This is a deep theorem combining minimal surface theory with complex analysis. The Weierstrass representation expresses minimal surfaces in ℝ³ using holomorphic data (a meromorphic function and a holomorphic 1-form on a Riemann surface). Finite total curvature forces the Riemann surface to have finite topology — it is a compact surface with punctures (the 'ends' of the minimal surface). Classical examples: the plane (genus 0, no punctures), the catenoid (genus 0, two punctures), the Costa surface (genus 1, three punctures).