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Free keywords:
Mathematics, Differential Geometry, math.DG,Mathematics, Analysis of PDEs, math.AP,
Abstract:
This paper concerns integral varifolds of arbitrary dimension in an open
subset of Euclidean space satisfying integrability conditions on their first
variation. Firstly, the study of pointwise power decay rates of the quadratic
tilt-excess is completed by establishing the precise decay rate for
two-dimensional integral varifolds of locally bounded first variation.
Secondly, counter-examples to pointwise power decay rates of the
super-quadratic tilt-excess are obtained. These examples are optimal in terms
of the dimension of the varifold and the exponent of the integrability
condition in most cases, for example if the varifold is not two-dimensional.
Thirdly, these counter-examples demonstrate that within the scale of Lebesgue
spaces no local higher integrability of the second fundamental form, of an at
least two-dimensional curvature varifold, may be deduced from boundedness of
its generalised mean curvature vector.
Amongst the tools are Cartesian products of curvature varifolds.
