Erhard Heinz

On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary

Geometrically speaking these conditions express the fact that x=r(w) represents a surface in R 3 of constant mean curvature H, having finite area A(t) and an assigned boundary F*. Under various conditions on F* and H existence proofs for such surfaces have been given in the literature ([1], [5], [6], [8], [9], and [10]; see also [7] for a complete discussion of the non-parametric case). If one considers the totality of all boundary curves F* lying in the unit ball I~[ < 1, the sharpest result hitherto obtained is due to HILDEBRANDT [6], who proved that in this case the class ~ (F* , H) is nonempty, provided that [HI < 1. On the other hand, from geometric considerations it is very plausible that for a given curve F* the surfaces x =z(w) should cease to exist, if [HI exceeds a certain critical number c(F*). The purpose of this note is to substantiate this fact by estimating the quantity c(F*) for a class of boundary curves F*. Introduce the numbers

On surfaces of constant mean curvature with polygonal boundaries

In Chapter VI of his book [1] on Dirichlets Principle, R. COURANT introduces a real-valued, continuously differentiable function of N variables whose critical points are in one-to-one correspondence with the minimal surfaces ~* spanned in a simple closed polygon F of N + 3 vertices. This result, which admits of important applications to the theory of unstable minimal surfaces (see [1], in particular pp. 233-243), is proved in [1] by first embedding ~* in a certain N-parametric family ~ of minimal vectors {~,} (~0=(~ol, ..., q~N)) and then showing that ~* coincides with the totality of vectors {~,},,c, where C is the set of critical points of the function O(q0=D(~) . The minimal vectors are constructed by solving a modified variational problem for the Dirichlet functional D(x) involving N parameters q~l . . . . . q~NAn essential point then is to show that the solution x =x , is uniquely determined and that it depends continuously on q~. In the present paper we are concerned with extending Courants construction to surfaces of constant mean curvature H spanned in polygons F. Here ~* is the totality of vector functions ~=z(w) (w=u+iv) of class C2(B)c~C~ (B = {[w[ < 1 )) satisfying in B the partial differential equations

Über eine Verallgemeinerung des Plateauschen Problems

AbstractLet T be the domain in ℝN defined by the inequalities O < τ1 < ... < τN < +Π. Put τN+k = Π/2(1+k) (k=1,2,3), τN+4=τ1+2Π, and denote byF(τ) the set of functions x=x(u,v)=(x1(u,v),...,xp(u,v)), (p≥2) of class

Zum Plateauschen Problem für Polygone

C^2 (B) \cap \cap C^0 (\bar B)

Some remarks on minimal surfaces in riemannian manifolds

, where B is the unit disk u2+v2<1, which maps the circular arcs γk={w=eiϕ:τk<ϕ<τK+1} (k=1,..., N+3) into the straight lines containing the edges ak, ak+1 (aN+4=a1) of a polygon Γ⊂IRp. Then we show that the function Θ(τ)=infx∈F(τ) D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

Zu einem Satz von F. E. Browder über nichtlineare Wellengleichungen

Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt − s2=E in the punctured disk 0<(x−x0)2+(y−y0)2<ρ2. Assume thatq is continuous at (x0, y0). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x0,y0) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344).

Zu einem Satz von Hildebrandt über das Randverhalten von Minimalflächen

Das Plateausche Problem ist in der Literatur vielfach behandelt worden (vgl. dazu das Buch [11] von J. C. C. Hitsche). In seiner einfachsten Form verlangt es die Bestimmung einer Minimalflache, die von einer vorgegebenen Raumkurve berandet wird. Zu den bis heute noch weitgehend ungelosten Problemen in diesem Zusammenhang gehort die Frage nach der Losungsanzahl bzw. der Struktur der Losungsmenge. Im folgenden mochte ich uber einige Resultate aus diesem Themenkreis berichten, die ich im Anschlug an fruhere Untersuchungen von I. Marx und M. Shiffman [10] gewonnen habe. In § 1 handelt es sich im wesentlichen um die Charakterisierung der Losungsmenge des verallgemeinerten Plateauschen Problems fur ein Polygon Γ mit (N+3) Ecken durch die kritischen Punkte einer analytischen Funktion Θ in N Variabein (Satz 1). In § 2 beschaftigen wir uns mit dem Problem, den Rang der Hesseschen Matrix von Θ an den kritischen Punkten zu bestimmen (Satz 2). Sine ausfuhrliche Darstellung findet sich in den Arbeiten [4] — [8].