The angle criterion

Abstract

Summary.We complete the proof of the angle conjecture on when a pair of oriented m-planes is area-minimizing. The nonzero sum (oriented union) P~ + P2 is area-minimizing if and only if the characterizing angles between them satisfy the inequality

Embedded Special Lagrangian Submanifolds in Calabi-Yau Manifolds

“…They are exactly the characterizing angles between P 1 and P 2 as defined in [10]. Note that one has 0 ≤ n j=1…”

“…We both choose a Lawlor neck (see [10] or section 1) as a local model, connect it to L outside a small ball to construct approximate submanifolds which are Lagrangian, and then apply Hamiltonian deformation to perturb these approximate submanifolds to become special Lagrangian. The main difference between these two works is: In Butscher's situation, the set L \ {p} has two connected components, and thus the first eigenvalues of the approximate submanifolds will tend to zero as the neck size tends to zero.…”

“…We refer to their paper for a detailed discussion on this subject. The followings are some basic definitions: G. Lawlor [10] modified an example of R. Harvey and H. B. Lawson [7] and defined the following submanifolds, which will be called Lawlor necks in this paper:…”

“…If we change the orientation on P 2 , which is denoted by −P 2 , then the characterizing angles between P 1 and −P 2 satisfy n j=1β j = π in the case n = 2 or 3. Change the coordinates such that P 1 = P θ and −P 2 = P −θ , where θ = (β The geometric obstruction for finding a Lawlor neck in n ≥ 4 comes from the following: There is an angle criterion which says that the nonzero sum (oriented union) P 1 + P 2 is area minimizing if and only if the characterizing angles between them satisfy the inequality [10], [16].) Suppose that P 1 and P 2 are two special Lagrangian planes.…”

Calibrations on 𝑅⁸

“…Morgan's conjecture ( [13]; see also [1, problem 5.8]) gives such a condition, and in Chapter 6 we prove that for a pair of 4-planes the condition is necessary for the planes to be simultaneously calibrated (see [1, §6]). Very recently G. Lawlor [18] and D. Nance [16] proved Morgan's conjecture [added in proof].…”