DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§4. Calculus of Variations and Surfaces of Constant Mean Curvature 103Now we compute: Recalling that d dt ‖f(t)‖ = f(t) · f ′ (t)and setting x = x ∗ + εξ, wehave‖f(t)‖d1 (dε∣ ‖x u × x v ‖ =(ξuε=0‖x ∗ u × x ∗ × x ∗ v + x ∗ u × ξ v ) · (x ∗ u × x ∗ v) )v‖Next we observe that=(ξ u × x ∗ v + x ∗ u × ξ v ) · n.(ξ u × x ∗ v) · n = ( (ξ × x ∗ v) · n ) u − (ξ × x∗ uv) · n − (ξ × x ∗ v) · n u(x ∗ u × ξ v ) · n = ( (x ∗ u × ξ) · n ) v − (x∗ uv × ξ) · n − (x ∗ u × ξ) · n v ,and so, adding these equations, we obtain(ξ u × x ∗ v + x ∗ u × ξ v ) · n = ( (ξ × x ∗ v) · n ) u + ( (x ∗ u × ξ) · n ) v − ( (ξ × x ∗ v) · n u +(x ∗ u × ξ) · n v)= ( (ξ × x ∗ v) · n ) u − ( (ξ × x ∗ u) · n ) v − ( (ξ × x ∗ v) · n u +(x ∗ u × ξ) · n v)= ( (ξ × x ∗ v) · n ) u − ( (ξ × x ∗ u) · n ) v − ξ · (x ∗ v × n u + n v × x ∗ u).At the last step, we’ve used the identity (U × V) · W =(W × U) · V =(V × W) · U. Theappropriate way to integrate by parts in the two-dimensional setting is to apply Green’s Theorem,Theorem 2.6 of the Appendix, and so we let P =(ξ × x ∗ u) · n and Q =(ξ × x ∗ v) · n and obtain∫∫(ξ u × x ∗ v + x ∗ u × ξ v ) · ndudvD∫∫ ( ((ξ= × x∗v ) · n ) − ( (ξ × x ∗ uu) · n ) ) ∫∫dudv − ξ · (x ∗vv × n u + n v × x ∗ u)dudvD } {{ } } {{ }DQ uP v∫=∂D(ξ × x ∗ u) · n du +(ξ × x ∗} {{ }v) · n} {{ }PQ∫∫dv − ξ · (x ∗ v × n u + n v × x ∗ u)dudv.DSince ξ = 0 on ∂D, the line integral vanishes. Using the equations (††) onp.56, we find thatx ∗ v × n u = a(x ∗ u × x ∗ v) and n v × x ∗ u = d(x ∗ u × x ∗ v), so, at long last, we obtain∫∫∫∫ddε∣ ‖x u × x v ‖dudv = (ξ u × x ∗ v + x ∗ u × ξ v ) · ndudvε=0 DD∫∫= − ξ · (x ∗ v × n u + n v × x ∗ u)dudvD∫∫∫∫= − (a + d)ξ · (x ∗ u × x ∗ v)dudv = − 2Hξ · ndA,DDsince H = 1 2 trS P .We conclude from this, using a two-dimensional analogue of Exercise 2, the followingTheorem 4.2. Among all (parametrized) surfaces with a given boundary curve, the one ofleast area is minimal, i.e., has H =0.This result, indeed, is the origin of the terminology.Next, suppose we wish to characterize those closed surfaces of least area containing a givenvolume V .Tomake a parametrized surface closed, we require that x(u, v) =x 0 for all (u, v) ∈ ∂D.But how do we express the volume constraint in terms of x? The answer comes from the Divergence