The maximum principle for minimal varieties of arbitrary codimension

By B. White. 2009

81 citations recorded by the source corpus.

We prove that an m-dimensional minimal variety in a Riemannian manifold cannot touch the boundary at a point where the sum of the smallest m principal curvatures is greater than 0. We also prove an analogous result for varieties with bounded mean curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0094290e9981c557582830d749ed153d848fb3b9

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory (2016, methodology): “rove a strong maximum principle for stationary varifolds with free boundary in the codimension one case (recall Denition 2.1 and 2.2). Our proof is inspired by the ideas in [22, Proposition 2.5] and [31] although we need to take extra care to ensure that the variation vector eld lies in X tan(M). Th”
A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary (2012, background): “port of W, then the support of Weither touches @B r(x)\Mat a point yin the interior of Mor on the boundary @M. Either case cannot happen by the maximum principle (see Lemma B.1 of [4] or Theorem 1 of [40]). Note that in the general case, we have to perturb by a projection (see [14]) the variation el”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “[44] Brian White.”
Min-max theory for minimal hypersurfaces with boundary (2016, background): “ition is classical if we were to consider Mas a subset of a larger manifold M~ without boundary and we had a varifold V which were stationary in Mn~ . For a proof we refer the reader to White’s paper [42]. However it is straightforward to check that the proof in [42] works in our setting, since the ”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, background): “[38] B. White.”