Minimal surface

R^{3}

, this says that the scalar mean curvature is zero, equivalently that its two principal curvatures sum to zero.

Outer area-minimizing surface

An outer area-minimizing surface is a closed surface in a Riemannian initial-data slice whose area is no greater than that of any enclosing competitor. This is a variational property, not a consequence of containment alone. It is important when choosing the horizon-area quantity in the Riemannian Penrose inequality and when examining an apparent-horizon area comparison.

(u, v)

give the immersion

(u - u^{3} /3 + u v^{2}, v - v^{3} /3 + u^{2} v, u^{2} - v^{2})

. Its coordinate functions are harmonic and its metric is

(1 + u^{2} + v^{2})^{2} (d u^{2} + d v^{2})

, so it is a complete immersed minimal surface with self-intersections.

With

A (X, Y) = (\overline{\nabla}_{X} Y)^{⊥}

and

H = tr A

V

satisfies

\frac{d}{d t} Area (M_{t})_{t = 0} = - \int_{M} ⟨ H, V ⟩, d μ + \int_{\partial M} ⟨ V, η ⟩, d σ .

Thus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.

M \subseteq R^{n + m}

and

f

is smooth in the ambient space, then

Δ_{M} (f ∣_{M}) = tr_{TM} (\overline{Hess} f) + ⟨ \overline{\nabla} f, H ⟩ .

For the squared distance

r^{2} (x) = ∣ x ∣^{2}

, a minimal

n

-submanifold satisfies

Δ_{M} r^{2} = 2 n

. On a compact boundaryless manifold,

r^{2}

attains a maximum where the Laplacian is nonpositive, a contradiction for

n > 0

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A minimal surface is a surface that locally minimizes its area for a given boundary. More formally, a minimal surface is defined as a surface with a mean curvature of zero at every point. This means that, at each point on the surface, the surface is as flat as possible and does not bend upwards or downwards. Minimal surfaces can often be described using parametric equations or as graphs of functions.