Consider a codimension 1 submanifold $N^n\subset M^{n+1}$, where $M^{n+1}\subset \mathbb{R}^{n+2}$ is a hypersurface. The envelope of tangent spaces of $M$ along $N$ generalizes the concept of tangent developable surface of a surface along a curve. In this paper, we study the singularities of these envelopes.

There are some important examples of submanifolds that admit a vector field tangent to $M$ and transversal to $N$ whose derivative in any direction of $N$ is contained in $N$. When this is the case, one can construct transversal plane bundles and affine metrics on $N$ with the desirable properties of being equiaffine and apolar. Moreover, this transversal bundle coincides with the classical notion of Transon plane. But we also give an explicit example of a submanifold that does not admit a vector field with the above property.

Consider the nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.

We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish $\mathrm{L}^{p}$ estimates. For the whole space and the half space case we use a reduction of the Stokes equations to $(n-1)$ heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of $\mathcal{R}$-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.

We obtain the analytic continuation of multiple Hurwitz zeta functions by using a simple and elementary translation formula. We also locate the polar hyperplanes for these functions and express the residues, along these hyperplanes, as coefficients of certain infinite matrices.

As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group $GSp(1,1)$ or the definite symplectic group $GSp^*(2)$ over $\mathbb{Q}$ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor $L$-functions as those of paramodular new forms of some specified level on the symplectic group $GSp(2)$ (or $GSp(4)$). This can be viewed as a generalization of the Jacquet–Langlands–Shimizu correspondence to the case of $GSp(2)$ and its inner forms $GSp(1,1)$ and $GSp^*(2)$.

In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from $GL(2)\times B^{\times}$ to $GSp(1,1)$ or $GSp^*(2)$ and a theta lift from $GL(2)\times GL(2)$ (or $GO(2,2)$) to $GSp(2)$. Here $B$ denotes a definite quaternion algebra over $\mathbb{Q}$. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet–Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of $GSp(2)$, which is studied in the appendix by Ralf Schmidt.

For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex body induced by it is of constant width ${\pi}/{2}$.

A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert–Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert–Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac–Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$.

Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.

We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in $\mathbb R^3$, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids in ${\mathbb R}^3$.

We consider weighted Hardy spaces over bidisk ${\mathbb D}^2$ which generalize the weighted Bergman spaces $A_\alpha^2({\mathbb D}^2)$. Let $z,w$ be coordinate functions and $M_{z^Nw^N}$ the multiplication by $z^Nw^N$ for a natural number $N$. In this paper, we study the reducing subspaces of $M_{z^Nw^N}$. In particular, we obtain the minimal reducing subspaces of $M_{zw}$.

We reduce Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension to a special case of the generalized abundance conjecture by establishing an Iitaka type inequality for Nakayama's numerical Kodaira dimension. Our proof heavily depends on Nakayama's theory of $\omega$-sheaves and $\widehat{\omega}$-sheaves. As an application, we prove the subadditivity of the logarithmic Kodaira dimension for affine varieties by using the minimal model program for projective klt pairs with big boundary divisor.

In this work we introduce a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial [3], the affine index polynomial [19] and the zero polynomial [14]. Several applications of this new invariant are discussed.

The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\mathbb{C}$-Fuchsian curves in complex hyperbolic spaces.

It is shown that for each Hecke pair of ergodic discrete measured equivalence relations, there exists a Hecke pair of groups determined by an index cocycle associated with the given pair. We clarify that the construction of these groups can be viewed as a generalization of a notion of Schlichting completion for a Hecke pair of groups, and show that the index cocycle cited above arises from “adjusted” choice functions for the equivalence relations. We prove also that there exists a special kind of choice functions, preferable choice functions, having the property that the restriction of the corresponding index cocycle to the ergodic subrelation is minimal in the sense of Zimmer. It is then proved that the Hecke von Neumann algebra associated with the Hecke pair of groups obtained above is $\ast$-isomorphic to the Hecke von Neumann algebra associated with the Hecke pair of equivalence relations with which we start.

This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electro-magnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.

A new symbol theory for pseudodifferential operators in the complex analytic category is given. Here the pseudodifferential operators mean integral operators with real holomorphic microfunction kernels. The notion of real holomorphic microfunctions had been introduced by Sato, Kawai and Kashiwara by using sheaf cohomology theory. Symbol theory for those operators was partly developed by Kataoka and by the first author and it has been effectively used in the analysis of operators of infinite order. However, there was a missing part that links the symbol theory and the cohomological definition of operators, that is, the consistency of the Leibniz–Hörmander rule and the cohomological definition of composition for operators. This link has not been established completely in the existing symbol theory. This paper supplies the link and provides a cohomological foundation of the symbolic calculus of pseudodifferential operators.