Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature
By Manuel del Pino, M. Kowalczyk, Jun-cheng Wei, Jun Yang. 2010
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Let $${(\mathcal {M},\tilde{g})}$$ be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation$$\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},$$where $${\varepsilon}$$ is a small parameter. Let $${{\mathcal {K} \subset \mathcal {M}}}$$ be an (N - 1)-dimensional smooth minimal submanifold that separates $${\mathcal{M}}$$ into two disjoint components. Assume that $${\mathcal{K}}$$ is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that $${{|A_\mathcal {K} |^2 + {\rm {Ric}}_{\tilde g} (v _{\mathcal K}, v_{\mathcal K})}}$$ is positive along $${\mathcal{K}}$$. Then for each integer m ≥ 2, we establish the existence of a sequence $${\varepsilon = \varepsilon{_j} \rightarrow 0}$$, and solutions $${u_\varepsilon}$$ with m-transition layers near $${\mathcal{K}}$$, with mutual distance $${O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}$$.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/392031c06544ae287681f1a94f179a89bfa2a854
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In Applied Mathematics
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Citations in the corpus
- Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, background): “es [Gas17]. Upper semicontinuity hinges strongly on the multiplicity one assumption, as the following example suggests: Example 5.2. Let (ui,εi), i= 1,2,..., limi εi = 0, be a sequence constructed by [dPKWY10] to converge, with multiplicity ≥ 2, to a two-sided minimal surface Σ in a closed Riemannia”