Geometric Analysis Of Tangent Measures

Tangent measures have become one of the most important technical tools in contemporary geometric measure theory and have played a major part in the study of the connection between singular integrals and rectifiability. Tangent measures can be used to describe the local behavior of a measure. There are many applications of tangent measures in many diverse areas, including PDE, the calculus of variations, harmonic analysis and fractals.in this thesis, we will study systematically the geometric properties of tangent measures, and consider the properties of the tangent space to a measure. We will give out the corresponding rectifiability and flatness criterions for a tangent measure or a measure. Meanwhile, we also investigate the geometric properties of tangent measure distributions, varifolds and currents.We obtain a series of interesting results including the structure theorem of tangent measures, the compare theorem of the tangent space to a measure. We prove an existence theorem of a micro tangent set by using the "blow-up" technique, and observe the directional tangent classification theorem. We obtain the rectifiability of 2-varifolds by virtue of the density properties of varifolds for cubes in R3, and give out the conformal invariance properties of tangent measure distributions. We derive the rectifiability properties of currents in metric space, and show the existence of current tangents and prove properties of weak Jacobian currents. We also discuss the Plateau problem and an isoperimetric inequality related.As an application of tangent measures, we also show that a Marstrand-type theorem holds, and give out the concentration of tangent measure distributions.Chapter 1 is devoted to introducing the various definitions of the tangent space to a measure and of the directional tangent. We prove the compare theorem of tangent space, In this section, we also give out some interesting examples.In Chapter 2, we study the geometric properties of tangent measures and the flatness properties of a measure, and give out an existence theorem of a micro tangent set of a measurable set by using the "blow-up" technique and the idea of duality theory. As an application of tangent measures, we obtain the Marstrand-type theorem.Chapter 3 investigates the geometric properties of tangent measure distributions. To get more information about the structure of the set of all tangent measures, we introduce an averaging method: the so-called tangent measure distribution method. Tangent measure distributions (TMD) can be regarded as the generalization of tangent measures. We give out some fine properties, for instance, scaling characteristic, shift-invariance property and so on. We will further focus our attention on concentration of TMD and also on conformal invariance of TMD.In Chapter 4, we will study the varifold theory. We first study the geometric properties of curvature varifolds and give out a structure theorem. We prove thecube-density theorem, and then consider the rectifibability properties of 2-varifolds by using the density properties of 2-varifolds for cubes in R3. We also study the varifold tangent and give an isoperimetric inequality for varifolds.In Chapter 5, we will focus our attention on the study of the current theory. We consider this theory in the following three parts. The first part is to study currents in a metric space. In this part, we derive the mean-value theorem of currents, and the rectifiability properties of currents. We also study the decomposition of currents and the properties of induced measures by the rectifiable currents; The second part is to consider the regularity of currents. We study the regularity for a current by using unit density of Brakke motion. The third part is to study weak Jacobian currents, current tangent and the Plateau problem, and then give out the structure theorem for currents.