arXiv subjects

Jenny Harrison

Publications and source records attributed to Jenny Harrison.

17 recordsLinked to original sources

arXiv2017

General Methods of Elliptic Minimization

We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of competing "surfaces," which span a given bounding set in some ambient space, one with minimal anisotropically weighted area. In particular, rectifiability of a candidate minimizer is proved without the assumption of quasiminimality. Our ambient spaces are a class of Lipschitz neighborhood retracts which includes manifolds with boundary and manifolds with certain singularities. Our competing surfaces are rectifiable sets which satisfy any combination of general homological, cohomological or homotopical spanning conditions. An axiomatic spanning criterion is also provided. Our boundaries are permitted to be arbitrary closed subsets of the ambient space, providing a good setting for surfaces with sliding boundaries.

math.AP↗

arXiv2016

Plateau's Problem: What's Next

Plateau's problem is not a single conjecture or theorem, but rather an abstract framework, encompassing a number of different problems in several related areas of mathematics. In its most general form, Plateau's problem is to find an element of a given collection $\cal{C} $ of "surfaces" specified by some boundary constraint, which minimizes, or is a critical point of, a given "area" function $F:\cal{C}\to \R $. In addition, one should also show that any such element satisfies some sort of regularity, that it be a sufficiently smooth manifold away from a well-behaved singular set. The choices apparent in making this question precise lead to a great many different versions of the problem. Plateau's problem has generated a large number of papers, inspired new fields of mathematics, and given rise to techniques which have proved useful in applications further afield. In this review we discuss a few highlights from the past hundred years, with special attention to papers of Federer, Fleming, Reifenberg and Almgren from the 1960's, and works by several groups, including ourselves, who have made significant progress on different aspects of the problem in recent years. A number of open problems are presented.

math.AP↗

arXiv2015

Operator Calculus of Differential Chains and Differential Forms

Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in this setting, as does Cartesian wedge product. Subspaces of finitely supported Dirac chains and polyhedral chains are both dense, leading to a unification of the discrete with the smooth continuum. We conclude with an application generalizing a simple version of the Reynolds' Transport Theorem to rough domains.

math.DG↗

arXiv2015

Existence and Soap Film Regularity of Solutions to Plateau's Problem

Plateau's soap film problem is to find a surface of least area spanning a given boundary. We begin with a compact orientable $(n-2)$-dimensional submanifold $M$ of $\R^n$. If $M$ is connected, we say a compact set $X$ "spans" $M$ if $X$ intersects every Jordan curve whose linking number with $M$ is 1. Picture a soap film that spans a loop of wire. Using $(n-1)$-dimensional Hausdorff spherical measure as the measure of the size of a compact set $X$ in $\R^n$, we prove there exists a smallest compact set $X_0$ that spans $M$. We also show that $X_0$ is almost everywhere a real analytic $(n-1)$-dimensional minimal submanifold and if $n = 3$, then $X_0$ has the structure of a soap film as predicted by Plateau. We provide more details about the minimizer $X_0$. Primarily, $X_0$ is the support of a current $S_0$ and $M$ is the support of the algebraic boundary of $S_0$. We also discuss the more general case where $M$ has codimension $> 2$.

math.DG↗

arXiv2014

Spanning via Cech Cohomology

Plateau's problem is to find a surface with minimal area spanning a given boundary. In 1960, Reifenberg and Adams developed a definition for "span" using Čech homology, and variants of this definition have been used ever sense. However, limitations of Čech homology resulted in the lack of a natural definition for a boundary consisting of more than one component. The authors avoided this problem in an earlier paper for codimension one surfaces using linking numbers to define spanning sets. In this paper, we show how to use Čech cohomology to provide a similar definition for all dimensions and codimensions.

math.DG↗

arXiv2012

Soap Film Solutions to Plateau's Problem

Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator to solve Plateau's problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Moebius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem because of a lack of a boundary operator. We use the chain complex of differential chains with its continuous boundary operator to solve a general version of Plateau's problem. We find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, both orientable and nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimensions in (\R^n).

math.DG↗

arXiv2012

Operator calculus - the exterior differential complex

We describe a topological predual to differential forms constructed as an inductive limit of a sequence of Banach spaces. This subspace of currents has nice properties, in that Dirac chains and polyhedral chains are dense, and its operator algebra contains operators predual to exterior derivative, Hodge star, Lie derivative, and interior product. Using these operators, we establish higher order divergence theorems for net flux of k-vector fields across nonsmooth boundaries, Stokes' theorem for domains in open sets which are not necessarily regular, and a new fundamental theorem for nonsmooth domains and their boundaries moving in a smooth flow. We close with broad generalizations of the Leibniz integral rule and Reynold's transport theorem.

math.FA↗

arXiv2011

Generalizations of the Cauchy Integral Theorems

We extend the Cauchy residue theorem to a large class of domains including differential chains that represent, via canonical embedding into a space of currents, divergence free vector fields and non-Lipschitz curves. That is, while the classical Cauchy theorems involve integrals over piecewise smooth parameterized curves, these classical theorems actually hold for far more general notions of "curve." We also extend the definition of winding number to these domains and show that it behaves as expected.

math.CV↗

arXiv2011

Geometric Poincaré Lemma

A geometric version of the Poincaré Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications include generalizations of the Intermediate Value Theorem and Rolle's Theorem.

math.AT↗

arXiv2011

Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains 'B(U) defined on an open subset U of a Riemannian manifold M. We show that 'B(U) is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space 'B(U), and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of fundamental operators of the Cartan calculus on differential forms. The space has good properties some of which are not exhibited by currents B'(U) or D'(U). For example, chains supported in finitely many points are dense in 'B(U) for all open U in M, but not generally in the strong dual topology of B'(U).

math.FA↗

arXiv2006

Differential complexes and exterior calculus

In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of ``pointed chains,'' culminating in the chainlet complex. Through this complex, we discover a broad theory of coordinate free, multivector analysis in smooth manifolds for which both the classical Newtonian calculus and the Cartan exterior calculus become special cases. The chainlet operators, products and integrals apply to both symmetric and antisymmetric tensor cochains. As corollaries, we obtain the full calculus on Euclidean space, cell complexes, bilayer structures (e.g., soap films) and nonsmooth domains, with equal ease. The power comes from the recently discovered prederivative and preintegral that are antecedent to the Newtonian theory. These lead to new models for the continuum of space and time, and permit analysis of domains that may not be locally Euclidean, or locally connected, or with locally finite mass.

math-ph↗

arXiv2005

Lectures on chainlet geometry - new topological methods in geometric measure theory

These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. They begin with a new theory of exterior calculus at a single point. This infinitesimal theory extends, by linearity, to a discrete exterior theory, based at finitely many points. A general theory of calculus culminates by taking limits in Banach spaces, and is valid for domains called ``chainlets'' which are defined to be elements of the Banach spaces. Chainlets include manifolds, rough domains (e.g., fractals), soap films, foliations, and Euclidean space. Most of the work is at the level of the infinitesimal calculus, at a single point. The number of limits needed to get to the full theory is minimal. Tangent spaces are not used in these notes, although they can be defined within the theory. This new approach is made possible by giving the Grassmann algebra more geometric structure. As a result, much of geometric measure theory is simplified. Geometry is restored and significant results from the classical theory are expanded. Applications include existence of solutions to a problem of Plateau, an optimal Gauss-Green theorem and new models for Maxwell's equations.

math-ph↗

arXiv2005

Ravello lecture notes on geometric calculus -- Part I

In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key points of the development. For example, $k$-dimensional tangent spaces are replaced by representations of simple $k$-vectors supported in single points as limits of simplicial $k$-chains in a Banach space (much like Dirac monopoles). This subtle difference has powerful advantages that will be explored. Through these ``infinitesimals'', we obtain a coordinate free theory on manifolds that builds upon the Cartan exterior calculus. An infinite array of approximating theories to the calculus of Newton and Lebiniz becomes available and one can now revisit old philosophical questions such as which models are most natural for the continuum or for physics. Applications include three extensions of calculus: calculus on fractals, bilayer calculus (soap bubbles) and discrete calculus. This paper is a draft of the first half of the lectures.

math-ph↗

arXiv2004

Geometric Hodge Star Operator with Applications to the Theorems of Gauss and Green

The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space $$\int_{\partial A} F \cdot n = \int_A div F$$ requires that a normal vector field $n(p)$ be defined a.e. $p \in \partial A$. In this paper we give a new proof and extension of this theorem by replacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedral chains taken with respect to a norm. The operator $\star$ is a geometric dual to the Hodge star operator and is defined on a large class of $k$-dimensional domains of integration $A$ in $n$-space the author calls {\em chainlets}. Chainlets include a broad range of domains, from smooth manifolds to soap bubbles and fractals. We prove as our main result the Star theorem $$\int_{\star A} ω= (-1)^{k(n-k)}\int_A \star ω.$$ When combined with the general Stokes' theorem for chainlet domains $$\int_{\partial A} ω= \int_A d ω$$ this result yields optimal and concise forms of Gauss' divergence theorem $$\int_{\star \partial A}ω= (-1)^{(k-1)(n-k+1)} \int_A d\star ω$$ and Green's curl theorem $$\int_{\partial A} ω= \int_{\star A} \star dω.$$

math-ph↗

arXiv2004

On Plateau's Problem for Soap Films with a Bound on Energy

We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched set which is a union of Lipschitz Jordan curves of finite total length.

math.DG↗

arXiv2004

Cartan's Magic Formula and Soap Film Structures

A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place without canceling each other out and to model triple junctions without introducing extra boundary components. We use chainlet geometry to create dipole cells and mass cells which accomplish these goals and model faithfully all observable soap films and bubbles. We introduce a new norm on chains of these cells and prove lower semicontinuity of area. A geometric version of Cartan's magic formula provides the necessary boundary coherence.

math.CA↗

arXiv1993

Stokes' theorem for nonsmooth chains

Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [{\it Geometric integration theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because we extend the class of integrable {\it domains}. Let $ω$ be an $n$-form defined on $\Bbb R^m$. We show that if $ω$ is sufficiently smooth, it may be integrated over sufficiently controlled, but nonsmooth, domains $γ$. The smoother is $ ω$, the rougher may be $γ$. Allowable domains include a large class of nonsmooth chains and topological $n$-manifolds immersed in $\Bbb R^m$. We show that our integral extends the Lebesgue integral and satisfies a generalized Stokes' theorem.

math.DG↗