We define a universal Teichmüller space for locally quasiconformal mappings whose dilatation grows not faster than a certain rate. We prove results of existence and uniqueness for extremal mappings in the generalized Teichmüller class. Further, we analyze the circle maps that arise.

We prove that if S is a smooth reflexive surface in \(\mathbb {P}^3\) defined over a finite field \(\mathbb {F}_q\), then there exists an \(\mathbb {F}_q\)-line meeting S transversely provided that \(q\ge c\deg (S)\), where \(c=\frac{3+\sqrt{17}}{4}\approx 1.7808\). Without the reflexivity hypothesis, we prove the existence of a transverse \(\mathbb {F}_q\)-line for \(q\ge \deg (S)^2\).

We prove the existence of metrics with prescribed Q-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function

We show that if \(f:X\rightarrow B\) is a Lagrangian fibration from a compact connected hyperkähler (and thus Kähler) manifold X onto a projective normal variety B, then f is locally projective. This answers a question raised by L. Kamenova and strengthens a former result (see Campana in Math Z 252:147–156, 2005, Proposition 2.1), according to which the smooth fibres of f are projective.

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at \(\infty \) for the cases \({\mathbb {P}}(1,3)\) and \({\mathbb {P}}(2,2)\) by purely topological methods. We compare them to the

Assume X is a variety over \({\mathbb {C}}\), \(A \subseteq {\mathbb {C}}\) is a finitely generated \({\mathbb {Z}}\)-algebra and \(X_A\) a model of X (i.e. \(X_A \times _A {\mathbb {C}} \cong X\)). Assuming the weak ordinarity conjecture we show that there is a dense set \(S \subseteq {{\,\mathrm{Spec}\,}}A\) such that for every closed point s of S the reduction of the maximal non-lc ideal filtration

In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation $$\begin{aligned} -(\nabla +\mathrm{i} A(x))^2 u+(\lambda V(x)+Z(x))u=f(\vert u\vert ^{2})u\quad \text {in}\, \,{\mathbb {R}}^{2}, \end{aligned}$$ where \(\lambda >0\), f(t) is a continuous function with exponential critical growth,

In this article we use the variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéire 3:77–109, 1986) to prove the existence of a positive solution for the following logarithmic Schrödinger equation $$\begin{aligned} \left\{ \begin{array}{ll} -\,{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{}\quad \text{ in } \,\, {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), &{} \\ \end{array}

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying \(\mathrm {Ric}_{\infty } \ge K>0\). Assuming that equality holds, we show that the 1-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng–Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the

We characterize sandwiched singularities in terms of their link in two different settings. We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links. We describe this self-similarity both in terms of Berkovich analytic geometry and of the combinatorics of weighted dual graphs. We then show that a complex surface singularity is sandwiched

We study the Riemann–Hilbert problem attached to an uncoupled BPS structure proposed by Bridgeland in (“Riemann–Hilbert problems from Donaldson–Thomas theory I”). We show that it has “essentially” unique meromorphic solutions given by a product of Gamma functions. We reconstruct the corresponding connection.

We present a formula to compute the Brasselet number of \(f:(Y,0)\rightarrow (\mathbb {C}, 0)\) where \(Y\subset X\) is a non-degenerate complete intersection in a toric variety X. As applications we establish several results concerning invariance of the Brasselet number for families of non-degenerate complete intersections. Moreover, when \((X,0) = (\mathbb {C}^n,0)\) we derive sufficient conditions

We develop a global Calderón–Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution u and its spatial gradient Du in a nonsmooth domain. The nonlinearity behaves as the parabolic p-Laplacian in Du, its discontinuity with respect to the independent variables is measured in terms of small-BMO,

In this paper, we deal with marginally outer trapped surfaces (MOTS) immersed in the de Sitter space \(\mathbb {S}_1^{n+2}\). In this setting we are able to obtain a Simons formula for the null second fundamental form and under some appropriate constraints on the MOTS, we apply a weak maximum principle in order to guarantee that it must be either a totally geodesic submanifold or isometric to an open

We prove that the Hilbert schemes of 2-points on rational surfaces are numerically rationally connected. The main idea is to show that certain 3-point genus-0 Gromov–Witten invariant of the Hilbert scheme of two points on the complex projective plane is positive and can be calculated enumeratively.

In this article, we study the instanton equation on the cylinder over a closed manifold X which admits non-zero smooth 3-form P and 4-form Q. Our results are (1) if X is a good manifold, i.e., P, Q satisfying \(d*_{X}P=d*_{X}Q=0\), then the instanton with integrable curvature decays exponentially at the ends, and, (2) if X is a real Killing spinor manifold, i.e., P, Q satisfying \(dP=4Q\) and \(d*_{X}Q=(n-3)*_{X}P\)

Let X be a pointed CW-complex. The generalized conjecture on spherical classes states that, the Hurewicz homomorphism\(H: \pi _{*}(Q_0 X) \rightarrow H_{*}(Q_0 X)\)vanishes on classes of\(\pi _* (Q_0 X)\)of Adams filtration greater than 2. Let \( \varphi _s: {\rm Ext} _{\mathcal {A}}^{s}(\widetilde{H}^*(X), \mathbb {F}_2) \rightarrow {(\mathbb {F}_2 \otimes _{{\mathcal {A}}}R_s\widetilde{H}^*(X))}^*

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in \(\mathbb {R}^{n+1}\), giving some conditions under which a translating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in \(R^{n+k}\), namely, if the \(L^n\) norm of the second fundamental form of the soliton is small enough, then

We exhibit a generating function of spin Hurwitz numbers analogous to (disconnected) double Hurwitz numbers that is a tau function of the two-component BKP (2-BKP) hierarchy and is a square root of a tau function of the two-component KP (2-KP) hierarchy defined by related Hurwitz numbers.

Let \({\widetilde{k}}\) be a fixed cubic field, F a quadratic field and \(L=\widetilde{k}\cdot F\). In this paper and its companion paper, we determine the density of more or less the ratio of the residues of the Dedekind zeta functions of L, F where F runs through quadratic fields.