arXiv subjects

Joachim Lohkamp

Publications and source records attributed to Joachim Lohkamp.

15 recordsLinked to original sources

arXiv2022

Potential Theory on Gromov Hyperbolic Spaces

Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Here we extend Ancona's potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.

math.DG↗

arXiv2022

The Secret Hyperbolic Life of Positive Scalar Curvature

This survey introduces to the hyperbolic unfolding correspondence that links the geometric analysis of minimal hypersurfaces with that of Gromov hyperbolic spaces. Problems caused from hypersurface singularities oftentimes become solvable on associated Gromov hyperbolic spaces. Applied to scalar curvature geometry this yields smoothing schemes that eliminate such singularities.

math.DG↗

arXiv2022

Scalar Curvature Splittings II: Removal of Singularities

In Part I of this paper we have seen that any singular compact area minimizer in a positive scalar curvature manifold admits a conformal deformation to some minimal factor geometry that shares many properties with the minimizer, like the Ahlfors regularity, the validity of Poincare inequalities and the presence of tangent cones with positive scalar curvature. In this Part II we show that these geometries admit surgery style arguments eliminating the singular sets.

math.DG↗

arXiv2022

Scalar Curvature Splittings I: Minimal Factors

Scalar curvature constraints can be studied by means of splitting procedures. The success of this strategy depends on the control we can get on its splitting factors. We introduce canonical so-called minimal splitting factors. They have positive scalar curvature while other properties strongly resemble those of area minimizing hypersurfaces. This includes the presence of Poincare, Sobolev and isoperimetric inequalities and the fact that singular points admit tangent cones but now with positive scalar curvature.

math.DG↗

arXiv2018

Contracting Maps and Scalar Curvature

We explain how to derive largeness constraints in scalar curvature geometry using some basic splitting results and the potential theory on singular area minimizing hypersurfaces. This includes a variety of results like the non-existence of positive scalar curvature metrics on enlargeable manifolds or simplified proofs for positive mass theorems.

math.DG↗

arXiv2018

Potential Theory on Minimal Hypersurfaces I: Singularities as Martin Boundaries

This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.

math.DG↗

arXiv2018

Potential Theory on Minimal Hypersurfaces II: Hardy Structures and Schrödinger Operators

We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal growth of positive solutions of Lw = 0 towards the singular set is a stable property. It persists under perturbations or blow-ups of the underlying spaces. This is the key result to develop a dimensional induction scheme for the asymptotic analysis of these operators near the singular set.

math.DG↗

arXiv2018

Minimal Smoothings of Area Minimizing Cones

In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in scalar curvature geometry saying that any (n-1)-dim. homology class of a compact manifold M^n with positive scalar curvature can be represented by a smooth (!) compact hypersurface that admits a positive scalar curvature metric.

math.DG↗

arXiv2018

Potential Theory on Gromov Hyperbolic Manifolds of Bounded Geometry

In the 1980s Alano Ancona developed a profound potential theory on Gromov hyperbolic manifolds of bounded geometry. Since then, such hyperbolic spaces have become basic in geometry, topology and group theory. In this paper we make Ancona's original work, addressed to a rather advanced audience, approachable for a wider readership without firm background in potential theory. We streamline the arguments to make the results depend only on coarse geometro-analytic invariants and we derive useful details not found in the original sources. We explain remarkable relations of these results to recent developments in the potential theory and quasi-conformal geometry of Euclidean domains.

math.DG↗

arXiv2018

Hyperbolic Unfoldings of Minimal Hypersurfaces

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the hypersurface and its singularity set. In this paper, this is used to prove the existence of hyperbolic unfoldings: canonical conformal deformations of the regular part of these hypersurfaces into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to the singular set.

math.DG↗

arXiv2015

Skin Structures on Minimal Hypersurfaces

This is the first in a series of papers where we develop new structural elements on singular area minimizing hypersurfaces, the skin structures. They disclose previously unapproachable and largely unexpected geometric and analytic properties of such hypersurfaces.

math.DG↗

arXiv2015

Hyperbolic Geometry and Potential Theory on Minimal Hypersurfaces

This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Martin boundary for many classical elliptic operators.

math.DG↗

arXiv2015

Skin Structures in Scalar Curvature Geometry

This is the third in a series of papers on the geometry and analysis of singular area minimizing hypersurfaces. We show how to derive obstruction and structure theories for scalar curvature constraints without imposing dimensional or topological restrictions on the underlying manifold. To this end, we use skin structures to merge minimal hypersurface methods with surgery arguments.

math.DG↗

arXiv2008

Inductive Analysis on Singular Minimal Hypersurfaces

The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how such minimal hypersurfaces inherit positive scalar curvature from their ambience resp. how to smooth singular minimal hypersurfaces to regular hypersurfaces with positive mean curvature.

math.DG↗

arXiv1992

Negatively Ricci curved manifolds

In this paper we announce the following result: ``Every manifold of dimension $\ge3$ admits a complete negatively Ricci curved metric.'' Furthermore we describe some sharper results and sketch proofs.

math.DG↗