ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53,58 Okayasu, Takashi, Toyama University, Japan A Bernstein result for minimal graphs of higher codimension Suppose that u : R^{n} → R^{p} satisfies the system of minimal surface equation. Put g = det(δ_{ij} + ∑_{k=1}^{p} \frac{∂u^{k}}{∂x^{i}} \frac{∂u^{k}}{∂x^{j}}). When p = 1, Ecker and Huisken (1990) showed that if |Du(x)| = o(\sqrt{|x|^{2} + |u(x)|^{2}}), then u is linear. In this talk we give an extension of their result to higher codimension. Our main theorem is as follows: Suppose that u satisfies the system of minimal surface equation and its graph graph(u) has flat normal connection. If \sqrt{g(x)} = o((|x|^{2} + |u(x)|^{2})^{r}) for some 0 < r < 1/2, then u is linear.