Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature
نویسندگان
چکیده
Let (M, g̃) be an N -dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation ε∆g̃u + (1− u)u = 0 in M, where ε is a small parameter. Let K ⊂M be an (N − 1)-dimensional smooth minimal submanifold that separatesM into two disjoint components. Assume that K is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that |AK| + Ricg̃(νK, νK) is positive along K. Then for each integer m ≥ 2, we establish the existence of a sequence ε = εj → 0, and solutions uε with m-transition layers near K, with mutual distance O(ε| ln ε|).
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