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RESEARCH STATEMENT
MARIE A. SNIPES
- Introduction
My research interests lie in the areas of geometric measure theory and analysis in Banach spaces. Geometric measure theory (GMT) uses measure theory to analyze geometric prop- erties of sets. This allows one to extend certain aspects of differential geometry and analysis to nonsmooth settings. GMT has close ties to partial differential equations, differential geometry, image processing, and calculus of variations.
The theory originated as a method for solving the Plateau Problem, which seeks the minimal volume surface with a given boundary. For example, if a wire is dipped in a soap solution, the bubble that forms minimizes surface area among all surfaces with the same boundary. In solving the Plateau Problem, one needs a sufficiently general notion of a surface to guarantee that a minimizer actually exists.
The class of surface-like objects that we would like to consider are those over which a differential form can be integrated. If S is such an object, then
S ξ^ exists for all forms^ ξ of suitable degree, so we can associate to S the linear functional whose action is ξ 7 →
S ξ. This leads to the definition of a generalized surface, or current, as a linear functional on differential forms. One particularly important class of currents is the class of so-called flat chains, which is the closure of the space of normal currents (currents of finite mass with finite mass boundary) under the flat norm (see Federer [8]). The space of flat chains includes all rectifiable currents, which are the currents used to solve the Plateau Problem.
Whitney, in [20], defines flat chains from a different point of view. He first considers the space Pk of polyhedral chains (finite sums of oriented k-dimensional polyhedra) with a volume (mass norm) given by Lebesgue measure. The space of flat k-chains is then the completion of Pk under the flat norm, which for A ∈ Pk is given by
|A|[ := inf B∈Pk+ {mass(B) + mass(A − ∂B)}.
Examples of flat chains include fractals (the snowflake curve is a flat 1-chain) and measures (Lebesgue measure on the unit interval is a flat 0-chain).
An important theorem of J. Wolfe identifies the space of linear functionals on flat chains with the space of “flat forms,” L∞-differential forms with L∞-exterior derivatives. In fact, with the flat norm on differential forms given by ‖ω‖[ := max{‖ω‖∞, ‖dω‖∞}, Wolfe’s the- orem states that the space of flat forms is isometrically the dual to the space of flat chains. This point of view yields a general theory of integration where flat forms are integrated over flat chains by duality. 1
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Classically, GMT was developed only in Euclidean space; in recent years there has been active research extending the theory to more general spaces. In 2000, Ambrosio and Kirch- heim [3] developed a theory of currents in metric spaces as linear functionals on “tuples” of Lipschitz functions, and proved an isoperimetric inequality. Following this, Lang [15] developed a local version of this theory, and Wenger [18], [19] generalized the isoperimetric inequality found in [3] and studied convergence properties of metric currents.
Generalizations of flat chains have also been studied. In [1], Adams defined the notion of a flat k-chain in a Banach space by generalizing the definition of mass, or “volume,” of a k-dimensional polyhedral chain. In addition to Adams’s paper on chains in Banach spaces, De Pauw and Hardt have defined and studied chains (via objects called “scans”) in metric spaces (see [6]).
- Results
My research is part of the recent efforts to extend GMT beyond its classical setting. In my thesis I define Banach space versions of flat differential forms, called flat partial forms (Definition 1), and prove the analog to Wolfe’s theorem, that the space of flat partial forms corresponds isometrically to the dual space of Adams’s flat chains (Theorem 1).
The basic motivation for our Banach space definition is the classical definition (see for example [8, p. 17]), where a differential form ω is a map ω : Rn^ → Hom(ΛkRn, R). (Here, ΛkRn^ denotes the k-th exterior power of Rn.) Two forms are equivalent if they agree almost everywhere in Rn. Thus, if ω is a differential form in Rn, ω(p) need only be defined for almost every p ∈ Rn. For such a p, ω(p) is a linear functional on ΛkRn, so ω(p)(ν) is defined for every k-vector ν in ΛkRn.
In a Banach space, one can use the same definition of a differential form if one does not need to consider equivalence classes of forms. For example, the smooth forms in [5] are well-defined in a Banach space. However, a flat form will not have a well-defined value at every point, so, without the use of an ambient measure, we need to describe when two forms are equivalent. Furthermore, if a version of Wolfe’s theorem is to hold in a Banach space, we must be able associate to every k-cochain, or linear functional on flat k-chains, a differential form ω. We wish to use a limiting approach similar to Lebesgue differentiation to define the form associated with a k-cochain, but this limit only exists almost everywhere on every affine k-plane in the direction of that k-plane. In finite dimensions Fubini’s theorem implies that for almost every p in V , the limit ω(p)(ν) is defined in every k-direction (and hence corresponds to a classical differential form), but in an infinite-dimensional space, there is no reason a priori that for even a single p ∈ V , ω(p)(ν) must be defined for every k-vector ν.
Since a classical differential form is a map that produces at each point in V a linear map in Hom(ΛkV, R), we define a partial form to be a certain type of function on a subset of the product space V × ΛkV :
Definition 1. Let V be a Banach space, and ω : E → R a function on a subset E of the product space V × ΛkV. For a point p in V , let Ep := E ∩ ({p} × ΛkV ), and for a k-vector ν, let Eν := E ∩ (V × {ν}).
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Theorem 3. In a Banach space, flat partial k-forms pull back under Lipschitz maps to flat partial k-forms.
To prove Theorem 3, we show that polyhedral chains push forward to flat chains under Lipshitz maps.
A long-term goal of this project is to apply it to the question of the bi-Lipschitz embed- dability of a general metric space in Rk^ (see Section 3.1); Theorem 3 indicates that flat forms are stable under Lipschitz mappings and hence are natural objects to consider in this study.
- Current and Future Work
My future work in this area has two main objectives: first, to use partial forms to address open problems in metric and Banach space analysis, and second, to explore the relationship between (flat) partial forms and various classes of chains and currents in Banach spaces.
3.1. Cartan-Whitney presentations. The generalization of Wolfe’s theorem to Banach spaces has applications to metric space analysis. In particular, Heinonen and Sullivan [14] have identified sufficient conditions under which a metric space can be locally bi-Lipschitzly parametrized by Rn^ for some n. One of their conditions is the existence of a so-called Cartan-Whitney presentation on the metric space. Roughly speaking, a Cartan-Whitney presentation consists of n flat 1-forms, which, when wedged together, produce a volume form that is comparable to Euclidean volume. Since the theory of Cartan-Whitney presentations has only been developed in Euclidean space, the authors in [14] and in the related papers [12] and [13] are restricted to considering only metric spaces which can be embedded in some large-dimensional, ambient Euclidean space. In the context of my research it is natural to ask whether one can define Cartan-Whitney presentations in Banach spaces.
Question 4. Can flat partial 1-forms be used to develop a parallel theory of Cartan-Whitney presentations in Banach spaces?
The first step toward addressing this question is translating the volume requirement of a Cartan-Whitney presentation into a condition that makes sense in our setting. In the Euclidean setting, at almost every point, a 1-form produces a 1-covector, which is naturally associated with a vector in Rn^ (via the Hodge star operator). The precise volume requirement is that the mass of the wedge product of these vectors is bounded uniformly away from zero. I hope to use the Gromov mass and mass* norms (see [11]) on the space of k-vectors to define a volume condition that does not fundamentally depend on the existence of an inner product.
Because any metric space can be embedded in a Banach space, extending Cartan-Whitney theory to Banach spaces would open the door to eliminating the requirement that metric spaces under consideration must be embedded in Euclidean space in the works mentioned above.
3.2. Lp-Theory. In [9] and [10], Gol′dshte˘ın, Kuz′minov, and Shvedov investigate “differ- ential forms of class W (^) p,q∗ ”. These are differential forms in Rn^ with p-integrable coefficient
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functions whose differentials have q-integrable coefficient functions. With this notation, flat forms in Rn^ are of class W (^) ∞∗,∞.
In [10], the authors prove a version of Wolfe’s theorem for differential forms in W (^) p,q∗ ; specifically, they define the notion of a k-dimensional p-cochain and prove that the space of (p, q)-cochains (p cochains c where dc is a q-cochain) is in bijective correspondence with the space W (^) p,q∗. Given a differential form ω in W (^) p,q∗ , one can construct a (p, q)-cochain by an averaging process similar to convolution. However, the proof that each (p, q) cochain can be associated with a form ω is non-constructive.
This contrasts with Whitney’s theory, where one can use a “Lebesgue differentiation” process to construct the form associated to a flat cochain.
Question 5. In Euclidean space, can one explicitly construct the differential form in W (^) p,q∗ that arises from a p-cochain?
I would also like to investigate whether one can extend the theory of W (^) p,q∗ forms to a Banach space.
Question 6. Can we identify the dual space to (p, q)-chains in a Banach space with the space W (^) p,q∗?
One can use Adams’s definition of mass on polyhedral chains in a Banach space to define the space of (p, q)-chains, and hence its dual space, (p, q)-cochains. The next step would be to define a p-integrable form, perhaps as a partial form which is p-integrable on k-dimensional affine spaces. With this definition, the class of partial forms W (^) p,q∗ is then the set of p- integrable forms whose exterior derivatives are q-integrable. The final step is to prove a duality result; one obvious difficulty with this is that the convolution process used in [10] for defining a (p, q)-cochain from a differential form is not natural in a Banach space, where there is no canonical ambient measure.
3.3. k-charges in Banach spaces. De Pauw and Pfeffer define k-charges, which generalize flat differential forms in Euclidean space (see [17]). A k-charge is a linear functional Λ on the space of normal k-currents in Rn^ which satisfies the following property. For all ε > 0 and for all r > 0, there exists a θ > 0 so that for all polyhedral k-chains P supported in the ball of radius r,
(1) Λ(P ) ≤ θ|P |[ + ε(mass(P ) + mass(∂P )).
Since 〈ω, P 〉 ≤ ‖ω‖[ · |P |[ for any flat k-form ω, flat forms are k-charges. Using approx- imation arguments one can easily show that in fact any continuous differential k-form is a k-charge. Laurent Moonens (see [16] or [7]) proves that the converse is almost true.
Theorem 7. (Moonens) Every k-charge Λ can be associated with a pair (ω, ξ) of continuous forms so that Λ(T ) = 〈ω, T 〉 + 〈ξ, ∂T 〉.
Dr. Moonens and I are collaborating on a project investigating whether these results can be extended to Banach spaces. First, we let Nk(V ) denote the space of flat k-chains in a Banach space V that have bounded support, have finite mass, and whose boundaries also have finite mass. A charge in a Banach space is then a linear functional on Nk(V ) that
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References [1] Adams, T. Flat chains in Banach spaces. J. Geom. Anal. 18, 1 (2008), 1–28. [2] Alvarez Paiva, J. C., and Thompson, A. C.´ Volumes on normed and Finsler spaces. In A sampler of Riemann-Finsler geometry, vol. 50 of Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2004, pp. 1–48. [3] Ambrosio, L., and Kirchheim, B. Currents in metric spaces. Acta Math. 185, 1 (2000), 1–80. [4] Benyamini, Y., and Lindenstrauss, J. Geometric Nonlinear Functional Analysis, Volume I, vol. 48 of Colloquium Publications. Amer. Math. Soc., 2000. [5] Cartan, H. Differential forms. Translated from the French. Houghton Mifflin Co., Boston, Mass, 1970. [6] De Pauw, T., and Hardt, R. Rectifiable and flat G-chains in metric spaces. Work in preparation. [7] De Pauw, T., Moonens, L., and Pfeffer, W. F. Charges in middle dimensions. Work in prepa- ration. [8] Federer, H. Geometric Measure Theory, vol. 153 of Die Grundlehren der mathematischen Wis- senschaften. Springer-Verlag, New York, 1969. [9] Gol′dshte˘ın, V. M., Kuz′minov, V. I., and Shvedov, I. A. The integration of differential forms of classes W (^) p,q∗. Sibirsk. Mat. Zh. 23, 5 (1982), 63–79, 223. [10] Gol′dshte˘ın, V. M., Kuz′minov, V. I., and Shvedov, I. A. The Wolf theorem for differential forms of classes W (^) p,q∗. Sibirsk. Mat. Zh. 24, 5 (1983), 31–42. [11] Gromov, M. Filling Riemannian manifolds. J. Differential Geom. 18, 1 (1983), 1–147. [12] Heinonen, J., and Keith, S. Flat forms, bi-Lipschitz parametrizations and smoothability of mani- folds. Work in preparation. [13] Heinonen, J., Pankka, P., and Rajala, K. Quasiconformal frames. Preprint 359, Department of Mathematics and Statistics, Jyv¨askyl¨a, 2007. [14] Heinonen, J., and Sullivan, D. On the locally branched Euclidean metric gauge. Duke Math. J. 114 , 1 (2002), 15–41. [15] Lang, U. Local currents in metric spaces. Lecture notes from the 17th Jyv¨askyl¨a Summer School. [16] Moonens, L. From Kurzweil-Henstock integration to charges in Euclidean space. Dissertation, UCL, Louvain-la-Neuve, 2008. [17] Moonens, L., and Pfeffer, W. F. The multidimensional Luzin theorem. J. Math. Anal. Appl. 339, 1 (2008), 746–752. [18] Wenger, S. Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15, 2 (2005), 534–554. [19] Wenger, S. Flat convergence for integral currents in metric spaces. Calc. Var. Partial Differential Equations 28, 2 (2007), 139–160. [20] Whitney, H. Geometric integration theory. Princeton University Press, Princeton, N. J., 1957.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 E-mail address: [email protected]
