In this article we study the low-temperature limit of a Landau-de Gennes theory. Within all S 2 -valued R-axially symmetric maps (see Definition 1.1), the limiting energy functional has at least two distinct energy minimizers. One minimizer has biaxial torus structure, while another minimizer has split-core segment structure on the z-axis.(1). u is isotropic at x if x is a point singularity of u (see Definition 1.2);(2). u is uniaxial at x if two of the three eigenvalues determined by u are identical and different from zero at x;(3). u is biaxial at x if all the three eigenvalues determined by u are different at x.Moreover a vector field is called director field determined by u if its values are normalized eigenvectors corresponding to the largest eigenvalue in λ 1 , λ 2 , λ 3 .With the above definitions, we can define the so-called biaxial torus and split-core line segment.Definition 1.4. Suppose that T is an open torus in B 1 and u is a given 3-vector field on T . T is called biaxial torus with uniaxial core determined by u if u is uniaxial on a closed simple loop in T . Meanwhile u is biaxial in T except the points on the closed simple loop. In this case the torus T is also called a biaxial torus of the augmented map L rus. Given a segment, we have a straight line, denoted by l, passing across the segment. The segment is called split-core segment associated with a 3-vector field u if there is a neighborhood, denoted by O, containing the segment so that u is uniaxial on O X l except the two end-points of the segment. Meanwhile u is isotropic at the two end-points of the segment and biaxial on O " l. Here and in what follows, we use " to denote the set minus. In this case L rus has split-core segment structure on l.
We wish to correct some notations and results contained in the original article. The main results (Theorems 1.2-1.3) remain unchanged but Lemma 2.1 should be slightly corrected. 1 Let us start from the corrections for some notations and typos.(I). On Page 3 of original article, we defined the cylindersThe center of these cylinders are at (x 0 , t 0 ). In fact, we need (x 0 , t 0 ) to be located on the top of these cylinders. Therefore the definitions of Q r (x 0 , t 0 ), Q * r (x 0 , t 0 ), cQ r (x 0 , t 0 ) and cQ * r (x 0 , t 0 ) on page 3 of original article should be replaced by what is shown below:Moreover, the time interval I r in Lemma 2.4 should be I r = [−r 2s , 0]. (II). In part (3) of Definition 1
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