By M. M. Schipper

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3 Surfaces of Revolution of Minimum Area 23 In general, we have that cosh p ∂ 2x t cosh2 p (t; p) = cosh p + cosh p = x(t; p) 2 ∂t x0 x0 x20 and t ∂ 2x (t; p) = cosh p + cosh p ∂ p∂ t x0 1+ t sinh p . x0 Using Eq. 22) to eliminate x0 + t sinh p∗ at the critical point p∗ , we obtain cosh2 p∗ + xt0 cosh p∗ sinh p ∂ 2x ∗ (t; p∗ ) = t ∂ p∂ t cosh p ∗ sinh p∗ + x0 cosh p∗ = 1 x(t; p∗ )2 cosh p∗ sinh p∗ . ˙ p∗ ) x20 x(t; Putting all this together, and using Eq. 23), we therefore have that d 2γ cosh2 p∗ (t) = x(t; p ) − ∗ dt 2 x20 ⎡ = x(t; p∗ ) 2 1 x(t;p∗ ) ˙ x20 x(t;p ∗) x(t; p∗ ) =− x20 x(t;p∗ ) 3 − x(t;p ˙ ∗) x(t;p ˙ ∗) x20 2 cosh p∗ ⎢ ⎣1 − x20 x(t;p∗ ) x(t;p ˙ ∗) x(t;p∗ ) x(t;p ˙ ∗) x(0;p∗ ) x(0;p ˙ ∗) 3 − 2 cosh p∗ sinh p∗ x(t;p∗ ) x(t;p ˙ ∗) 3 − x(0;p∗ ) 3 x(0;p ˙ ∗) sinh2 p∗ ⎤ 3 ⎥ x(0;p∗ ) x(0;p ˙ ∗) 3⎦ 3 x(0;p∗ ) x(0;p ˙ ∗) 3 cosh2 p∗ .

Let x∗ be an extremal along which the strengthened Legendre condition is satisfied. Then (c, (x∗ (c)) is a conjugate point to (a, x∗ (a)) if and only if (c, (x∗ (c)) is the limit point of points of intersection between the graph of x∗ and graphs of neighboring extremals drawn from the same initial point (a, x∗ (a)). Proof. 1 and denote the difference with the reference extremal x∗ by Δ (t; p) = x(t; p) − x(t; 0) = x(t; p) − x∗(t). For p = 0, we have Δ (t; 0) ≡ 0, and thus it is possible to factor p from this equation and rewrite Δ in the form Δ (t; p) = pΩ (t; p).

Proof. 4 The Legendre and Jacobi Conditions R(Q + w) ˙ = R Q+ =R 31 − dtd (yR)y ˙ + yR ˙ y˙ y2 Qy2 − Qy2 + Ry˙2 y2 Ry˙ y = 2 = w2 . 27) that exists on the full interval [a, b], let y be a nontrivial solution to the linear differential equation Ry˙ + wy = 0. Then Ry˙ is continuously differentiable and we have that w2 d d wy (Ry) ˙ = (−wy) = −wy˙ − wy ˙ = −w − − Q y = Qy. − dt dt R R Furthermore, as a nontrivial solution, y does not vanish. Note that if y is a solution to Eq. 27) that vanishes at some time c, then so is yα (t) = α y(t) for any α ∈ R, α = 0.

### The Campbell-Hausdorff theorem by M. M. Schipper

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