This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed...

We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result...

Given a locally integrable structure \({\mathcal {V}}\) over a smooth manifold \(\varOmega \) and given \(p\in \varOmega \) we define the Borel map of\({\mathcal {V}}\) atp as the map which assigns to the germ of a smooth solution of \({\mathcal {V}}\) at p its formal Taylor power series at p. In this work we continue the study initiated in Barostichi et al. (Math. Nachr. 286(14...

We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for \(\Lambda \)-rings. Moreover, Scholze’s \((q-1)\)-adic completion of q-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in...

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective...

We prove several results about the best constants in the Hausdorff–Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.

We solve the isoperimetric problem in the lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are either geodesic spheres or quotients of Clifford tori. We also show that the isoperimetric problem in the lens spaces L(3, 1) and L(3, 2) follows from the proof of the Willmore conjecture by Marques and Neves.

Let K be a knot in the 3-sphere. A slope p / q is said to be characterising for K if whenever p / q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p / q surgery on another knot \(K'\) in the 3-sphere, then K and \(K'\) are isotopic. It was an old conjecture of Gordon, proved by Kronheimer, Mrowka, Ozsváth and Szabó, that every slope is...

Let \(\{u_n\}\) be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold \(K\subset N\) satisfying $$\begin{aligned} \sup _n \ \left( \Vert \nabla u_n\Vert _{L^2(M)}+\Vert \tau (u_n)\Vert _{L^2(M)}\right) \le \Lambda , \end{aligned}$$where \(\tau (u_n)\) is the tension field...

We formulate and prove finite dimensional analogs for the classical Balian–Low theorem, and for a quantitative Balian–Low type theorem that, in the case of the real line, we obtained in a previous work. Moreover, we show that these results imply their counter-parts on the real line.

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over \({\mathbb {Q}}\) that contains a conic defined over \(\mathbb {Q}\).

Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon–Douglis–Nirenberg are investigated. In the whole- and half-space case we construct an explicit formula for the solution and establish coercive \(L^{p}\) estimates. The estimates generalize a famous result of Agmon, Douglis and Nirenberg...

The moduli space of nodal Enriques surfaces is irreducible of dimension 9. A nodal Enriques surface is shown to be the quotient of a double cover of the plane by a lift of the Cremona involution. We also show that this gives a straightforward proof of the known description of the automorphism group for the generic such surface.

We show that every countable group embeds in a group of type \(FP_2\).

Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of \(\overline{\partial }\)-equation on X and prove a Dolbeault–Grothendieck lemma. We obtain fine sheaves \(\mathcal {A}_X^q\) of (0, q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf \(\mathscr {O}_X\). Our construction is based on intrinsic...

We study two closely related problems stemming from the random wave conjecture for Maaß forms. The first problem is bounding the \(L^4\)-norm of a Maaß form in the large eigenvalue limit; we complete the work of Spinu to show that the \(L^4\)-norm of an Eisenstein series \(E(z,1/2+it_g)\) restricted to compact sets is bounded by \(\sqrt{\log t_g}\). The second problem is quantum...

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the...

Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition for the volume of the iterates of the critical set. This generalizes to higher dimensions the well-known equivalence between stability and normality of the critical orbits in dimension one. We also introduce a notion of holomorphic motion of...

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering...

We initiate the systematic study of \(G_2\)-instantons with SU(2)\(^2\)-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on \(\mathbb {R}^4\times S^3\) with its two explicitly known distinct holonomy \(G_2\) metrics, which have different volume growths at infinity...

We construct, using geometric invariant theory, a quasi-projective Deligne–Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded \(A_\infty \)-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a...

We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.