A Bernstein theorem for special Lagrangian graphs by Jost J., Xin Y. L.

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  • February 26, 2017
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By Jost J., Xin Y. L.

We receive a Bernstein theorem for distinct Lagrangian graphs in for arbitrary in simple terms assuming bounded slope yet no quantitative limit.

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If O = O ∗ , the orbital O is self-paired; trivially, a diagonal orbital is self-paired. Sims defined the orbital graph G(O) associated with an orbital O as an undirected graph if O is self-paired and a directed graph otherwise, as follows: the vertex-set is , and the edges are all pairs vw for which (v, w) ∈ O. If O is diagonal, then G(O) just has a loop at every vertex in the corresponding orbit; and if O is not self-paired, then G(O ∗ ) is the converse digraph of G(O). Graphs admitting a given group Sims’ construction of orbital graphs can in a sense be reversed.

Dn , then r A2 = deg v ; i ii r the number of edges of G is given by n |E(G)| = deg vi = 12 tr (A2 ); 1 2 i=1 r the number of triangles in G is 1 tr (A3 ). 6 From this we see that the number of edges and the number of triangles in a graph are determined by its eigenvalues. This argument does not extend to k-cycles, with k > 3: the first example of cospectral graphs in Section 2 shows that the number of 4-cycles cannot be determined by the eigenvalues alone. However, the degree sequence adds enough extra information to determine the 4-cycles.

For each i = 1, 2, . . , n, we can write T (ei ) = a1i f1 + a2i f2 + · · · + ami fm , for some scalars a1i , a2i , . . , ami . The rectangular array of scalars   a11 a12 · · · a1n a a22 · · · · a2n  21  A=  · · · ·  am1 am2 · · · amn Introduction 15 is the matrix of T associated with the given bases. The scalar ai j is the ij-entry of A, and we abbreviate the above array by writing A = (ai j ). Since the matrix A has m rows and n columns, we call A an m × n matrix; a matrix for which m = n is a square matrix of order n.

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