Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial order $\leq$) and the topology of certain subspaces of the outgrowth $\beta X\setminus X$. The cases when $\mathfrak{P}$ is either pseudocompactness or realcompactness are studied in more detail.

We study a collection of stability conditions (in the sense of Schmitt) for complexes of sheaves over a smooth complex projective variety indexed by a positive rational parameter. We show that the Harder-Narasimhan filtration of a complex for small values of this parameter encodes the Harder-Narasimhan filtrations of the cohomology sheaves of this complex. Finally we relate a stratification into locally closed subschemes of a parameter space for complexes associated to these stability parameters with the stratification by Harder-Narasimhan types.

We study harmonic functions with polynomial growth on asymptotic cones of a nonnegatively Ricci curved manifold with Euclidean volume growth. Especially, we will give the classification of such harmonic functions.

This paper studies the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces endowed with their Poincaré metrics. This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of any non-linear isoperimetric inequality.

Assume that $M$ is a compact $n$-dimensional manifold and that $N$ is obtained by surgery along a $k$-dimensional sphere, $k\leq n-3$. The smooth Yamabe invariants $\sigma(M)$ and $\sigma(N)$ satisfy $\sigma(N)\geq \min (\sigma(M),\Lambda)$ for a constant $\Lambda$ > 0 depending only on $n$ and $k$. We derive explicit positive lower bounds for $\Lambda$ in dimensions where previous methods failed, namely for $(n,k) \in \{(4,1),(5,1),(5,2), (6,3), (9,1),(10,1)\}$. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.

Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate of the Stokes operator admits a bounded ${\cal H}^\infty$-calculus in ${\mathbb L}_\sigma^p(\Omega)$ for $p\in(\min\{q,q'\},\max\{q,q'\})$. For the proof we use a recent maximal regularity result for the Stokes operator on such domains ([GHHS12]) and an abstract result for the ${\cal H}^\infty$-calculus in complemented subspaces ([KKW06], [KW13]).

Let $G$ be a finite Oliver group. In this paper, we discuss the relation between tangential $G$-representations of smooth one-fixed-point actions on spheres and the Smith equivalence of real $G$-representations.

Let κ be a regular uncountable cardinal, and λ be a cardinal greater than κ. Our main result asserts that if (λ^{< κ})^{<(λ< κ)} = λ^{< κ}, then (p_{κ, λ}(NIn_{κ, λ^{< κ}}))⁺ $\longrightarrow$ ((NS_{κ, λ}^{[λ]< κ})⁺, NS_{κ, λ}s⁺)³ and (p_{κ, λ}(NAIn_{κ, λ^{< κ}}))⁺ $\longrightarrow$ (NS_{κ, λ}s⁺)³, where NS_{κ, λ}s (respectively, NS_{κ, λ}^{[λ]< κ}) denotes the smallest seminormal (respectively, strongly normal) ideal on P_κ (λ), NIn_{κ, λ^{< κ}} (respectively, NAIn_{κ, λ^{< κ}}) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of P_κ(λ^{< κ}), and p_{κ, λ}: P_κ(λ^{< κ}) → P_κ (λ) is defined by p_{κ, λ}(x) = x ∩ λ.

With the aid of the technique of variation of domains developed in Memoirs of Amer. Math. Soc., Vol.209, No.984 (2011), we characterize the pseudoconvex domains with smooth boundary in Hopf surfaces which are not Stein.

We minutely describe the intersection of two real forms in a non-irreducible Hermitian symmetric space $M$ of compact type. In the case where $M$ is irreducible we have already done it in our previous paper. In this paper we reduce the description of the intersection of two real forms to that in some special cases. This reduction is based on the information of the group of all isometries obtained by Takeuchi. We can describe the intersection in the special cases and in all cases. In particular we obtain the intersection number of two real forms in a Hermitian symmetric space of compact type.

We establish the robustness of exponential dichotomies for evolution families of linear operators in a Banach space, in the sense that the existence of an exponential dichotomy persists under sufficiently small linear perturbations. We note that the evolution families may come from nonautonomous differential equations involving unbounded operators. We also consider the general case of a noninvertible dynamics, thus including several classes of functional equations and partial differential equations. Moreover, we consider the general cases of nonuniform exponential dichotomies and of dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ for some function $\rho(t)$.

We show that the resulting manifold by $r$-surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope $r$ satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates $L$-spaces and the left-orderability of their fundamental groups.

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. By making use of the Morse and Ljusternik–Schnirelman theories of critical points, we prove existence theorems of non-trivial solutions of our problem. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of semilinear elliptic boundary value problems with degenerate boundary conditions. The results here extend earlier theorems due to Ambrosetti–Lupo and Struwe to the degenerate case.

We prove asymptotic formulas for mean square values of the Euler double zeta-function $\zeta_2(s_0,s)$, with respect to $\Im s$. Those formulas enable us to propose a double analogue of the Lindelöf hypothesis.

We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots $6_{14}$ and $6_{15}$ are not equivalent. We also show that certain genus two handlebody-knots with a knotted handle decomposing sphere can be determined by their exteriors. As an application, we show that the exteriors of $6_{14}$ and $6_{15}$ are not homeomorphic.

This article studies the asymptotic linking pairing $lk$ on the space of exact 2-forms $B^2(S^3)$ on the 3-sphere $S^3$ through the geometry of Hopf fibrations. Mitsumatsu [7] tried to apply this pairing to 3-dimensional contact topology. He considered a positive definite subspace $P(\xi)$ in $(B^2(M),lk)$ associated with a contact structure $\xi$ on a closed 3-manifold $M$. Further he introduced an invariant of $\xi$, called the analytic torsion. We investigate the case of the standard contact structure on $S^3$ and construct a positive definite subspace of arbitrary large dimension in the $lk$-orthogonal complement of $P(\xi)$. This shows that the analytic torsion is infinite. Also we show that it is infinite even for any closed contact 3-manifold.