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- W1966776913 abstract "In this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension. We consider a vector-valued function u : Cl C Rn - > Rm that satisfies the minimal surface system, see equation (1.1) in §1. The graph of u is then a minimal submanifold of Rn+m. We prove an a priori gradient bound under the assumption that the Jacobian of du : Rn - ► Rm on any two dimensional subspace of Rn is less than or equal to one. This assumption is automatically satisfied when du is of rank one and thus the estimate covers the case when m = 1, i.e., the original minimal surface equation. This is applied to Bernstein type theorems for minimal submanifolds of higher codimension. 1. Introduction. The interior gradient bound for solutions to the minimal surface equation was discovered, in the case of two variables, by Finn (3) and in the general case by Bombieri, Di Giorgi and Miranda (1). The a priori bound is a key step in the existence and regularity of minimal surface theory. The estimate has been generalized to other curvature equations and proved by different methods. We refer to the note at the end of chapter 16 of (5) for literature in these directions. Recall a function m : Q C R -> E is a solution to the minimal surface equation if" @default.
- W1966776913 created "2016-06-24" @default.
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- W1966776913 date "2004-01-01" @default.
- W1966776913 modified "2023-10-17" @default.
- W1966776913 title "Interior gradient bounds for solutions to the minimal surface system" @default.
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- W1966776913 doi "https://doi.org/10.1353/ajm.2004.0033" @default.
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