Compactness theorems for problems with unknown boundary

The compactness theorem is proved for sequences of functions that have estimates of the higher derivatives in each subdomain of the domain of definition, divided into parts by a sequence of some curves of class W_2^1. At the same time, in the entire domain of determining summable higher derivatives, these sequences do not have. These results allow us to make limit transitions using approximate solutions in problems with an unknown boundary that describe the processes of phase transitions.

On convergence and compactness in variation with shift of discrete probability laws

We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.

On a topology property for the moduli space of Kapustin–Witten equations

In this article, we study the Kapustin–Witten equations on a closed, simply connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use Taubes’ compactness theorem [C. H. Taubes, Compactness theorems for {\mathrm{SL}(2;\mathbb{C})} generalizations of the 4-dimensional anti-self dual equations, preprint 2014, https://arxiv.org/abs/1307.6447v4] to prove that if {(A,\phi)} is a smooth solution to the Kapustin–Witten equations and the connection A is closed to a generic ASD connection {A_{\infty}}, then {(A,\phi)} must be a trivial solution. We also prove that the moduli space of the solutions to the Kapustin–Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all generic. At last, we extend the results for the Kapustin–Witten equations to other equations on gauge theory such as the Hitchin–Simpson equations and the Vafa–Witten on a compact Kähler surface.

Convergence of manifolds and metric spaces with boundary

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov–Hausdorff (GH) and Sormani–Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably [Formula: see text] rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require non-negative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.