On the analytic reflection of a minimal surface
Jaigyoung Choe · Pacific Journal of Mathematics · 1993
For a long time it has been known that in a Euclidean space one can reflect a minimal surface across a part of its boundary if the boundary contains a line segment, or if the minimal surface meets a plane orthogonally along the boundary. The proof of this fact makes use of H. A. Schwarz's reflection principle for harmonic functions. In this paper we show that a minimal surface, as a conformal and harmonic map from a Riemann surface into R 3, can also be reflected analytically if it meets a plane at a constant angle. THEOREM 1. Let Σ c R3 be a minimal surface with nonempty boundary dΣ and let Π be a plane. Suppose that L c Σ n Π is a C 1 curve, Σ is C 1 along L, and at all points of L the tangent plane to Σ makes a fixed angle 0 C satisfies g(p) g(P*)= (iv) p * G Σ * is a branch point (geometric) if and only if p eΣ is. (v) The map * is a single-valued immersion if Σ is simply con-nected and L is connected, or Σ is doubly connected and L is closed. (vi) If * is single-valued, then Σ * has finite total curvature if and only if Σ does. (vii) If dΣ = L, then Σ is complete. Proof. Let x, y, z be coordinates of R3 such that Π = {(x, y, z): z = 0}. Since x, y, z are harmonic functions on the minimal sur-face Σ, one can find conjugate harmonic (possibly multiple-valued)