77 research outputs found
Second order rectifiability of integral varifolds of locally bounded first variation
In this work it is shown that every integral varifold in an open subset of
Euclidian space of locally bounded first variation can be covered by a
countable collection of submanifolds of class C^2. Moreover, the mean curvature
of each member of the collection agrees with the mean curvature of the varifold
almost everywhere with respect to the varifold.Comment: v1: 34 pages, no figures; v2: revised presentation, material of
0909.3253 and 0808.3665 reorganised, parts moved to 0909.3253v2, parts from
0909.3253v1 included, this version now depends on 0909.3253v2, comparison to
Hutchinson's second fundamental form added, 40 pages, no figures; v3:
introduction and 4.5 revised, 3.8 and one reference added, one reference
updated, 40 pages, no figure
Pointwise differentiability of higher order for sets
The present paper develops two concepts of pointwise differentiability of
higher order for arbitrary subsets of Euclidean space defined by comparing
their distance functions to those of smooth submanifolds. Results include that
differentials are Borel functions, higher order rectifiability of the set of
differentiability points, and a Rademacher result. One concept is characterised
by a limit procedure involving inhomogeneously dilated sets.
The original motivation to formulate the concepts stems from studying the
support of stationary integral varifolds. In particular, strong pointwise
differentiability of every positive integer order is shown at almost all points
of the intersection of the support with a given plane.Comment: Description of subsequent work added to the introduction, references
and affiliations updated, typographical corrections made; 34 page
Sobolev functions on varifolds
This paper introduces first order Sobolev spaces on certain rectifiable
varifolds. These complete locally convex spaces are contained in the generally
nonlinear class of generalised weakly differentiable functions and share key
functional analytic properties with their Euclidean counterparts.
Assuming the varifold to satisfy a uniform lower density bound and a
dimensionally critical summability condition on its mean curvature, the
following statements hold. Firstly, continuous and compact embeddings of
Sobolev spaces into Lebesgue spaces and spaces of continuous functions are
available. Secondly, the geodesic distance associated to the varifold is a
continuous, not necessarily H\"older continuous Sobolev function with bounded
derivative. Thirdly, if the varifold additionally has bounded mean curvature
and finite measure, the present Sobolev spaces are isomorphic to those
previously available for finite Radon measures yielding many new results for
those classes as well.
Suitable versions of the embedding results obtained for Sobolev functions
hold in the larger class of generalised weakly differentiable functions.Comment: Version initially accepted by Proc. Lond. Math. Soc. (3). The final
printed version will be different. 55 pages, no figure
A novel type of Sobolev-Poincar\'e inequality for submanifolds of Euclidean space
For functions on generalised connected surfaces (of any dimensions) with
boundary and mean curvature, we establish an oscillation estimate in which the
mean curvature enters in a novel way. As application we prove an a priori
estimate of the geodesic diameter of compact connected smooth immersions in
terms of their boundary data and mean curvature. These results are developed in
the framework of varifolds. For this purpose, we establish that the notion of
indecomposability is the appropriate substitute for connectedness and that it
has a strong regularising effect; we thus obtain a new natural class of
varifolds to study. Finally, our development leads to a variety of questions
that are of substance both in the smooth and the nonsmooth setting.Comment: 35 pages, no figure
An isoperimetric inequality for diffused surfaces
For general varifolds in Euclidean space, we prove an isoperimetric
inequality, adapt the basic theory of generalised weakly differentiable
functions, and obtain several Sobolev type inequalities. We thereby intend to
facilitate the use of varifold theory in the study of diffused surfaces.Comment: Awaiting publication in Kodai Math. J. The final printed version will
be different. 14 pages, no figure
Some applications of the isoperimetric inequality for integral varifolds
In this work the isoperimetric inequality for integral varifolds of locally bounded first variation is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifold
Weakly differentiable functions on varifolds
The present paper is intended to provide the basis for the study of weakly
differentiable functions on rectifiable varifolds with locally bounded first
variation. The concept proposed here is defined by means of integration by
parts identities for certain compositions with smooth functions. In this class
the idea of zero boundary values is realised using the relative perimeter of
superlevel sets. Results include a variety of Sobolev Poincar\'e type
embeddings, embeddings into spaces of continuous and sometimes H\"older
continuous functions, pointwise differentiability results both of approximate
and integral type as well as coarea formulae.
As prerequisite for this study decomposition properties of such varifolds and
a relative isoperimetric inequality are established. Both involve a concept of
distributional boundary of a set introduced for this purpose.
As applications the finiteness of the geodesic distance associated to
varifolds with suitable summability of the mean curvature and a
characterisation of curvature varifolds are obtained.Comment: 84 pages, no figure
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