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We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

In this paper, motivated by the result of Osserman and Ru [12], we give a Gauss curvature estimate on an open Riemann surface M whose metric is induced from a nonconstant holomorphic map $$G: M\rightarrow {{\mathbb {P}}}^n({{\mathbb {C}}})$$ .

In this paper, we derive that every n-dimensional ( $$2\le n\le 7$$ ) special Lagrangian cones in $${\mathbb {R}}^{n}\times {\mathbb {R}}^{n}$$ with flat normal bundle must be flat. And we also derive a rigidity result under a certain restriction on Gauss map.

The magneto-micropolar equations are important models in fluid mechanics and material sciences. This paper focuses on the global regularity problem on the 2D magneto-micropolar equations with fractional dissipation. We establish the global regularity for three important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ various technics including the regularization of generalized heat operators on the Fourier frequency localized functions, logarithmic Sobolev interpolation inequalities and the maximal regularity property of the heat operator.

In this paper we investigate surfaces in $$\mathbb {C}P^2$$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh–Vilms type theorems. We also discuss the liftability of an immersion from a surface to $$\mathbb {C}P^2$$ into $$S^5$$ in Appendix A.

In this paper, we prove the sharp weighted bound for certain singular integrals which have non-smooth kernels and do not belong to the class of standard Calderon–Zygmund operators. Our assumptions are weaker than those known in literature, since in particular we do not assume the Cotlar type inequality condition. Applications include sharp weighted estimates for the Riesz transforms associated to the Dirichlet Laplacians on open connected domains, the Riesz transforms associated to the Schrodinger operators with real potentials on the Euclidean spaces, the Riesz transforms associated to the degenerate Schrodinger operators and the Riesz transforms associated to the Schrodinger operators with inverse square potentials.

In this paper, we first give several comparison theorems and their applications in Finsler geometry. Moreover, by means of the Hessian comparison theorems of a real Finsler manifold, Cartan connection, radial flag curvature and tangent curvature, we give Wu’s theorem on a strongly convex weakly Kahler Finsler manifold with a pole. Finally, by using the complex Rund connection and definition of radial flag curvature on a strongly pseudoconvex complex Finsler manifold, we further discuss Wu’s theorem on a strongly convex Kahler Finsler manifold.

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $${\mathfrak {g}}$$ , we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of $${\mathscr {S}}({\mathfrak {g}})$$ on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in $${\mathfrak {g}}\simeq {\mathfrak {g}}^{*}$$ .

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits n-th root for all $$n\ge 1$$. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes Boge’s theorem concerning infinitely divisible measures (commutative case) and Parthasarathy’s result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given.

In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric to $$(S^4, g_0)$$ or $$({\mathbb {C}}P^2, g_{FS})$$ , and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to $$(S^4, g_0)$$ , $$({\mathbb {C}}P^2, g_{FS})$$ , or $$(S^2\times S^2, g_0\oplus g_0)$$ , up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds with $$\mathrm {Ric}=g$$ and nonnegative sectional curvature is bounded above by $$\frac{4}{3}+4^{\frac{1}{3}}$$ . The basic idea of the proofs is to construct an “integrated subharmonic function”, and the main ingredients of the proofs are curvature decompositions (in particular Berger decomposition), the Weitzenbock formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenbock formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdzinski’s argument.

Parreau compactified the Hitchin component of a closed surface $S$ of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of $\pi_1(S)$ on an $\mathbb R$-Euclidean building. In this paper, we use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building., Comment: 26 pages, 4 figures, Minor corrections (typos, etc.)

Let the circle act on a compact almost complex manifold M. In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. By the fixed point data we mean a collection of the multisets of the weights at the fixed points. First, if $$\dim M=2$$, then M is a disjoint union of rotations on two 2-spheres. Second, if $$\dim M=4$$, we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if $$\dim M=6$$, we prove that six types occur for the fixed point data; $$\mathbb {CP}^3$$ type, complex quadric in $$\mathbb {CP}^4$$ type, Fano threefold type, $$S^6 \cup S^6$$ type, blow up of a fixed point of a rotation on $$S^6$$ type, and unknown type that might possibly be realized as a blow up of $$S^2$$ inside a manifold like $$S^6$$. When $$\dim M=6$$, we recover the result by Ahara (J Fac Sci Univ Tokyo Sect IA Math 38(1):47–72, 1991) in which the fixed point data is determined if furthermore $$\mathrm {Todd}(M)=1$$ and $$c_1^3(M)[M] \ne 0$$, and the result by Tolman (Trans Am Math Soc 362(8):3963–3996, 2010) in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $\varphi$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_\alpha$. We investigate the optimal $\beta$ such that the composition operator $\mathcal{C}_\varphi$ maps $\mathcal{D}_\alpha$ boundedly into $\mathcal{D}_\beta$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and use it to consider composition operators generated by polynomial symbols on $\mathcal{H}^p$, finding the first non-trivial results of this type. The embedding also yields a new result for the functional associated to the multiplicative Hilbert matrix., Comment: This paper has been accepted for publication in Mathematische Zeitschrift

Suppose $$L=-\Delta +V$$ is a Schrodinger operator on $$\mathbb {R}^n$$ with a potential V belonging to certain reverse Holder class $$RH_\sigma $$ with $$\sigma \ge n/2$$ . The aim of this paper is to study the $$A_p$$ weights associated to L, denoted by $$A_p^L$$ , which is a larger class than the classical Muckenhoupt $$A_p$$ weights. We first prove the quantitative $$A_p^L$$ bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative $$A_{p,q}^L$$ bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical $$A_{p,q}$$ constant. However, since $$A_{p,q}\subset A_{p,q}^L$$ , the $$A_{p,q}^L$$ constants are smaller than $$A_{p,q}$$ constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the “exp–log” link between $$A_p^L$$ and $$BMO_L$$ (the BMO space associated with L), and show that for $$w\in A_p^L$$ , $$\log w$$ is in $$BMO_L$$ , and that the reverse is not true in general.

In this paper we study sharp generalizations of $$\dot{F}_p^{0,q}$$ multiplier theorem of Mikhlin–Hormander type. The class of multipliers that we consider involves Herz spaces $$K_u^{s,t}$$ . Plancherel’s theorem proves $$\widehat{L_s^2}=K_2^{s,2}$$ and we study the optimal triple (u, t, s) for which $$\sup _{k\in \mathbb {Z}}{\Vert ( m(2^k\cdot )\varphi )^{\vee }\Vert _{K_u^{s,t}}}

Let G be a connected, simply connected, compact semisimple Lie group of dimension n. It has been shown by Clerc (Ann Inst Fourier Grenoble 24(1):149–172, 1974) that, for any $$f\in L^1(G)$$ , the Bochner–Riesz mean $$S_R^\delta (f)$$ converges almost everywhere to f, provided $$\delta >(n-1)/2$$ . In this paper, we show that, at the critical index $$\delta =(n-1)/2$$ , there exists an $$f\in L^1(G)$$ such that $$\begin{aligned} \limsup _{R\rightarrow \infty } \big |S_{R}^{(n-1)/2}(f)(x)\big |=\infty ,\quad \ a.e.\ x\in G. \end{aligned}$$ This is an analogue of a well-known result of Kolmogoroff (Fund Math 4(1):324–328, 1923) for Fourier series on the circle, and a result of Stein (Ann Math 2(74):140–170, 1961) for Bochner–Riesz means on the tori $$\mathbb {T}^{n}, n\ge 2$$ . We also study localization properties of the Bochner–Riesz mean $$S_{R}^{(n-1)/2}(f)$$ for $$f\in L^1(G)$$ .

In this paper we establish some algebraic properties of special L-values attached to Siegel modular forms of half-integral weight, often called metaplectic modular forms. These results are motivated by some “exercises” left by Shimura to the reader in his marvellous book “Arithmeticity in the Theory of Automorphic Forms”.

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern–Weil form can be represented by a given form? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.

Let \(S^1\) be the boundary of the open unit disk \(\mathbb {D}\) and let h be a quasisymmetric homeomorphism from the unit circle \(S^1\) onto itself. Let H be the quasiconformal harmonic extension of h to \(\mathbb {D}\) with respect to the Poincare metric. In this paper, it is shown that, if \(h'(\zeta )=\alpha \ne 0\) at \(\zeta \) in \(S^1\), then when \(z\rightarrow \zeta \) in \(\mathbb {D}\) non-tangentially, $$\begin{aligned} \lim _{z\rightarrow \zeta }\frac{H(z)-H(\zeta )}{z-\zeta }=\alpha \end{aligned}$$ and the complex derivatives \(H_z(z)\) and \(H_{\bar{z}}(z)\) approach \(\alpha \) and 0 respectively, i.e., H has an angular derivative \(\alpha \) at \(\zeta \); conversely, if H has a non-tangential derivative \(\alpha \ne 0\) at \(\zeta \), then \(h'(\zeta )=\alpha \) and hence H has an angular derivative \(\alpha \) at \(\zeta \).

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelof hypothesis for the Riemann zeta-function is equivalent to the Lindelof hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki–Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, which are not special Legendrian deformations, given by harmonic 1-forms.

This paper consists of two parts. In the first part, we use symplectic homology to distinguish the contact structures on the Brieskorn manifolds $\Sigma(2l,2,2,2)$, which contact homology cannot distinguish. This answers a question from [22]. In the second part, we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on $S^7$, distinguished by the mean Euler characteristic of $S^1$-equivariant symplectic homology. Apart from various connected sum constructions, these contact structures can be taken from the Brieskorn manifolds $\Sigma(78k+1,13,6,3,3)$. We end with some considerations about extending this result to higher dimensions., Comment: 33 pages; several small corrections. To appear in Mathematische Zeitschrift. The final publication is available at http://link.springer.com/article/10.1007/s00209-015-1596-3

In this paper we use Kuperberg’s \(\mathfrak {sl}_3\)-webs and Khovanov’s \(\mathfrak {sl}_3\)-foams to define a new algebra \(K^S\), which we call the \(\mathfrak {sl}_3\)-web algebra. It is the \(\mathfrak {sl}_3\) analogue of Khovanov’s arc algebra. We prove that \(K^S\) is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of \(q\)-skew Howe duality, which allows us to prove that \(K^S\) is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group \(K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}\), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that \(K^S\) is a graded cellular algebra.

The main result of this paper is a lower bound for the essential spectrum of Schrodinger operators −Δ+V on Riemannian manifolds. In particular, we obtain conditions on V which imply the discreteness of the spectrum, or equivalently, the compactness of the resolvent.