# Minimal Surfaces by Tobias H. Colding By Tobias H. Colding

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Is elliptic if there is some λ > 0 such that, for each x ∈ M , v → [φ(x, v/|v|) − λ]|v| is a convex function of v ∈ Tx M . Curvature estimates continue to hold for Φ-stable surfaces. 58) sup |A|2 ≤ C σ −2 . BrΣ 0 −σ As for the minimal surface equation, standard elliptic theory then implies higher derivative estimates (see [Si2]; cf. chapter 16 of [GiTr]). We note that many of the standard tools for minimal surfaces do not hold for general Φ. 1) no longer hold. This introduces significant complications.

Observe also that if a varifold arises from a smooth submanifold, then the first variation formula implies that it is stationary if and only if it has zero mean curvature. 5 If Tj is a sequence of stationary varifolds which converges weakly to a varifold T , then T is also stationary. P ROOF : Given a C 1 vector field X with compact support, we can define a compactly supported continuous function f by f (x, ω) = divω X . 15) Rn ×G(k,n) divω X dTj (x, ω) = 0 . Rn ×G(k,n) Therefore by varifold convergence and by the compact support of f on Rn × G(k, n), we get divω X dT (x, ω) = Rn ×G(k,n) f dT (x, ω) Rn ×G(k,n) f dTj (x, ω) = 0 .

F. J. Almgren, Jr. showed that in dimension three such a cone was also a hyperplane. Finally, J. Simons proved the same theorem for n ≤ 7. 19 (F. J. Almgren, Jr. [Am1] for n = 3 and J. Simons [Sim] for n ≤ 7) The hyperplanes are the only stable minimal hypercones in Rn for n ≤ 7. However, in 1969 Bombieri, De Giorgi, and Giusti [BDGG] gave an example of an area-minimizing singular cone in R8 . 93) Cm = {(x1 , . . , x2m ) | x21 + · · · + x2m = x2m+1 + · · · + x22m } ⊂ R2m are area-minimizing. 3 Weak Bernstein-Type Theorems In this chapter, we will prove a generalization of the classical Bernstein theorem for minimal surfaces discussed in the previous chapters.