Potential Theory on Minimal Hypersurfaces I: Singularities as Martin Boundaries
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Potential Theory on Minimal Hypersurfaces I: Singularities as Martin Boundaries Joachim Lohkamp1 Received: 26 May 2018 / Accepted: 2 December 2019 / © Springer Nature B.V. 2019
Abstract Area minimizing hypersurfaces and, more generally, almost minimizing hypersurfaces frequently occur in geometry, dynamics and physics. A central problem is that a general (almost) minimizing hypersurface H contains a complicated singular set . Regardless of this we can develop a detailed potential theory on H \ applicable to large classes of linear elliptic second order operators. We even get a fine control over their analysis near . The present paper is the foundational Part 1 of this two parts work. Keywords Potential theory: Boundary harnack inequalities · Minimal hypersurfaces · Singularities · Uniform spaces · Gromov hyperbolicity Mathematics Subject Classification (2010) 30L10 · 31C12 · 49Q15 · 51M10 · 53A10 · 53A30
1 Introduction Area minimizing hypersurfaces and much wider classes of almost minimizing hypersurfaces occur in geometry, dynamics and physics. Examples are horizons of black holes, level sets of geometric flows, and minimizing hypersurfaces used in scalar curvature geometry. A central problem is that a general (almost) minimizing hypersurface H contains a complicated singularity set ⊂ H and H \ degenerates towards in a rather delicate way. Moreover many of the elliptic operators, that one typically studies on H , also degenerate in their own way while we approach . In view of these entanglements the program of this paper (and its second part [28]) may appear surprising: we develop a detailed potential theory on H \ applicable to a large class of linear elliptic second order operators. We even get a fine control over their analysis near . To this end, we first derive boundary Harnack inequalities where we regard as a boundary of the open manifold H \ . We use these central inequalities to deduce a variety of further results. For instance, we observe that is homeomorphic to the Martin boundary. Joachim Lohkamp
[email protected] 1
Mathematisches Institut, Universit¨at M¨unster, Einsteinstrasse 62, M¨unster, Germany
J. Lohkamp
Moreover, each boundary point is minimal. For operators naturally associated with such a hypersurface H , for instance the Jacobi field operator or the conformal Laplacian, we can go even further than Martin theory on H \ . We get stable boundary Harnack inequalities which apply, with the same Harnack constants, to all blow-up hypersurfaces we get from infinite scalings around singular points. This considerably refines the asymptotic analysis towards by dimensional reductions we get from tangent cone approximations.
S -structures This unexpected degree of control is due to a geometric property of H \ , namely its S -uniformity [27]. For the curved space H \ this S -uniformity is the counterpart to uniformity for Euclidean domains. The S -uniformity implies the existence of a canonical conformal hyperbolic unfolding of H \ into some complete Gromov hype
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