In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schrödinger operators \((-\Delta +m^{2})^{s}\) with \(s\in (0,1)\) and mass \(m>0\). As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry

Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \(\omega \) that has standard cusp singularities along a divisor D, we show the \(L^2\) projective embedding of (X, D) defined by \(L^k\) is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial

Following the work of Berman and Witt Nyström we present the proof of existence and uniqueness of solutions to transported Monge–Ampère problem on complex compact manifolds with completely integrable torus action. The proof is based on the real theory of optimal transportation and some convex analysis.

In this paper, we give estimates on partial derivatives and logarithmic partial derivatives for holomorphic functions on polydiscs. Estimates will also be utilized to characterize entire solutions of a class of partial differential equations, which gives a new form of Picard’s theorem and its higher dimensional version.

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold \(M\times \mathbb {R}\), where M is asymptotically flat. If the initial hypersurface \(F_0\subset M\times \mathbb {R}\) is uniformly spacelike and asymptotic to \(M\times \left\{ s\right\} \) for some \(s\in \mathbb {R}\) at infinity, we show that a mean curvature flow starting

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality $$\begin{aligned} |\Delta u - u_t| \le \frac{M}{|x|^2} |u|, \end{aligned}$$ with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.

This paper aims to define and study currents and slices of currents in the Heisenberg group \({\mathbb {H}}^n\). Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties

Suppose \(p\ge 1\), \(w=P[F]\) is a harmonic mapping of the unit disk \({\mathbb D}\) satisfying F is absolutely continuous and \({\dot{F}}\in L^p(0, 2\pi )\), where \({\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})\). In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., \(\Vert w_z\Vert _{L^p}\) and \(\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}\), where

The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces”, Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly

A mixed symmetric Chernoff type inequality for two convex domains can be proved, furthermore its stability properties can also be obtained in the Hausdorff and \(L_2\) metrics, respectively.

Let M be an open Riemann surface and let \(\varLambda \subset M\) be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions \(M\rightarrow {\mathbb {R}}^n\), \(n\ge 3\), with prescribed values on \(\varLambda \) and whose generalized Gauss map \(M\rightarrow \mathbb {CP}^{n-1}\), \(n\ge 3\), avoids n hyperplanes of \(\mathbb {CP}^{n-1}\) located in

The existence of nontrivial solutions for the following kind of Klein–Gordon–Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u-(2\omega +\phi )\phi u=f(x,u),&{}\quad x\in {{\mathbb {R}}}^{3},\\ \Delta \phi =(\omega +\phi )u^{2},&{}\quad x\in {{\mathbb {R}}}^{3}, \end{array}\right. \end{aligned}$$ is investigated, where \(\omega >0\) is a constant, \(V\in C({{\mathbb {R}}}^{3}

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points

We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt–Caffarelli–Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland–Hayman

We characterise the real extreme points of the unit ball of \(m^0_\Psi \), the complex extreme points of the unit ball of \(m_\Psi \) and the real extreme and exposed points of the unit ball of \((m_\Psi ^0)'\). Using these characterisations we show that, depending on the length of the extreme points, the multipliers of \(m^0_\Psi \) are either constant multiple of the identity or diagonal operators

In a domain \(\Omega \subset {\mathbb {R}}^{n}\) whose boundary is smooth except on a set \({\mathcal {C}}\) of codimension (in \(\partial \Omega \)) k, the behavior of a nonparametric prescribed mean curvature hypersurface \(z=f\left( \mathbf{x}\right) \) in the vertical cylinder \(\Omega \times {\mathbb {R}}\) at a point \({\mathcal {O}}\in {\mathcal {C}}\) can be largely unknown when \(n\ge 3,\)

We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean

We show that if an exponential polynomial \(\sum _{i=1}^m P_i(z)e^{Q_i(z)}\), where \(P_i\), \(Q_i\in \mathbb C[z]\), is a dth power, \(d\ge 2\), of an entire function g, then g itself is also an exponential polynomial. We also study when a multivariable polynomial with moving targets of slow growth evaluated at unit arguments can be a dth power of an entire function. Finally, we formulate a boundary

In this note, we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to \(S^1 \times S^{n-1}\) for each \(n \ge 2\) in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely that the profile curve is convex, we remove the entropy assumption. Supposing

Assuming the bilinear reverse Hölder’s condition, we character weighted inequalities for the bilinear maximal operator on filtered measure spaces. We also obtain Hytönen–Pérez type weighted estimates for the bilinear maximal operator. Our approaches are mainly based on the new construction of bilinear versions of principal sets and the new Carleson embedding theorem on filtered measure spaces. In particular

In this paper, as an application of Zalcman’s lemma in \(C^n,\) we give a sufficient condition for normality of a family of holomorphic functions of several complex variables, which generalizes previous known one-dimensional results of H.L. Royden and W. Schwick.

Let \(\Omega \) be a two-dimensional exterior domain with smooth boundary \(\partial \Omega \) and \(1< r < \infty \). Then \(L^r(\Omega )^2\) allows a Helmholtz–Weyl decomposition, i.e., for every \(\mathbf{u}\in L^r(\Omega )^2\) there exist \(\mathbf{h} \in X^r_{\tiny {\text{ har }}}(\Omega )\), \(w \in {\dot{H}}^{1,r}(\Omega )\) and \(p \in {\dot{H}}^{1,r}(\Omega )\) such that $$\begin{aligned}

In this paper, we study the so-called Hankel measures on the open unit disk. We obtain several new characterizations for such measures and answer a question raised by J. Xiao in 2000.

We show there exists a nontrivial \(H^1_0\) solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. In fact, the solution we construct satisfies both the no-slip boundary condition and vanishing at infinity. This means that the solution is in some sense actually a paradoxical solution since the fluid

We prove regularity of solutions of the \(\bar{\partial }\)-problem in the Hölder–Zygmund spaces of bounded, strongly \(\mathbf{C}\)-linearly convex domains of class \(C^{1,1}\). The proofs rely on a new analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the \(\bar{\partial }\)-problem

Extremal length is a classical tool in 1-dimensional complex analysis for building conformal invariants. We propose a higher-dimensional generalization for complex manifolds and provide some ideas on how to estimate and calculate it. We also show how to formulate certain natural geometric inequalities concerning moduli spaces in terms of a complex analogue of the classical Riemannian notion of systole

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non-compact. We construct families of (non-integrable) almost complex structures on \({{\mathbb {R}}}^4\), such that the space of closed J-anti-invariant forms is infinite dimensional, and also 0- or 1-dimensional. In the compact case, we construct 6-dimensional almost complex manifolds with arbitrary large anti-invariant

We derive a very general condition for a sense-preserving harmonic mapping with dilatation a square to be injective in the unit disk \({{\mathbb {D}}}\) and to admit a quasiconformal extension to the extended complex plane. The analysis depends on geometric properties of an extension of the Weierstrass–Enneper lift to the extended plane that glues the parametrized minimal surface to a complementary

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the boundedness of a space-time integral norm of the scalar curvature.

Cartan (J Math Pures Appl 96:1–114, 1931) firstly introduced bounded anti-circular domains centered at the origin in \({\mathbb {C}}^2\). In this paper, we give the classification and realization of a class of bounded anti-circular domains centered at the origin with the Thullen condition. Their isotropic subgroups and holomorphic automorphism groups are also determined.

In this paper, we consider the rigidity for an \(n(\ge 4)\)-dimensional submanifold \(M^n\) with parallel mean curvature in the space form \({\mathbb {M}}^{n+p}_{c}\) when the integral Ricci curvature of M has some bound. We prove that, if \(c+H^2>0\) and \(\Vert {\mathrm{Ric}_{-}^{\lambda }}\Vert _{n/2}< \epsilon (n,c, \lambda , H)\) for \(\lambda \) satisfying \( \tfrac{n-2}{n-1} (c+H^2) < \lambda

The \({\hat{B}}_n^{(1)}\)-hierarchy is constructed from the standard splitting of the affine Kac–Moody algebra \({\hat{B}}_n^{(1)}\), the Drinfeld–Sokolov \({\hat{B}}_n^{(1)}\)–KdV hierarchy is obtained by pushing down the \({\hat{B}}_n^{(1)}\)-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) use loop group factorization to construct Darboux transforms (DTs)

We use the Suita conjecture (now a theorem) to prove that for any domain \(\Omega \subset \mathbb {C}\) its Bergman kernel \(K(\cdot , \cdot )\) satisfies \(K(z_0, z_0) = \hbox {Volume}(\Omega )^{-1}\) for some \(z_0 \in \Omega \) if and only if \(\Omega \) is either a disk minus a (possibly empty) closed polar set or \(\mathbb {C}\) minus a (possibly empty) closed polar set. When \(\Omega \) is bounded

We study the cohomology with high tensor powers of Nakano q-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the order of estimates are optimal. Besides, estimates for the modified Dirac operator on

For two constants \(K\ge 1\) and \(K'\ge 0\), suppose that f is a \((K,K')\)-quasiconformal self-mapping of the unit disk \({\mathbb {D}}\), which satisfies the following: (1) the inhomogeneous polyharmonic equation \(\Delta ^{n}f=\Delta (\Delta ^{n-1} f)=\varphi _{n}\) in \({\mathbb {D}}\)\((\varphi _{n}\in {\mathcal {C}}(\overline{{\mathbb {D}}}))\), (2) the boundary conditions \(\Delta ^{n-1}f=\varphi

We prove a Carathéodory-type extension of branched quasisymmetric (BQS) homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carathéodory-type extension of geometric quasiconformal mappings between two such domains provided the two domains are both Ahlfors Q-regular and support a Q-Poincare inequality when

In this paper, we show that if (X, g) is an oriented four-dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in X of appropriate spin enjoy the Calabi–Yau property, meaning that every immersed surface of this type from a bordered Riemann surface can be uniformly approximated by complete superminimal surfaces with Jordan boundaries. The proof uses the theory

We prove that a 4-dimensional \(C^2\) conformally compact Einstein manifold with Hölder continuous scalar curvature and with \(C^{m,\alpha }\) boundary metric has a \(C^{m,\alpha }\) compactification. We also study the regularity of the new structure and the new defining function. This is a supplementary proof of Anderson’s work and an improvement of Helliwell’s result in dimension 4.

In this paper, we introduce a horizontal F-energy functional for maps from a Riemannian manifold to a Carnot group. The critical maps of this functional are referred to as CC-F-harmonic maps. Under suitable conditions on the Hessian of the distance function and the degree of F(t), we obtain several Liouville-type theorems for CC-F-harmonic maps from some complete Riemannian manifolds to Carnot groups

In 2006 Carbery raised a question about an improvement on the naïve norm inequality \(\Vert f+g\Vert _p^p \le 2^{p-1}(\Vert f\Vert _p^p + \Vert g\Vert _p^p)\) for two functions f and g in \(L^p\) of any measure space. When \(f=g\) this is an equality, but when the supports of f and g are disjoint the factor \(2^{p-1}\) is not needed. Carbery’s question concerns a proposed interpolation between the

In this paper we study spectral stability of the \(\bar{\partial }\)-Neumann Laplacian under the Kohn–Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the \(\bar{\partial }\)-Neumann Laplacian when either the operator or the underlying domain is perturbed.

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton \(\Sigma \subset {\mathbb {R}}^{n+1}\) for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in \({\mathbb {R}}^{n+1},\, n\ge 3 \) is the axisymmetric “bowl soliton.”

We propose to study positive harmonic functions satisfying a nonlinear Neuman condition on a compact Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary. A precise conjecture is formulated. We discuss its implications and present some partial results. Related questions are discussed for compact Riemannian manifolds with positive Ricci curvature and convex boundary.

We study the density of polynomials in \(H^2(E,\varphi )\), the space of square integrable functions with respect to \(\mathrm{e}^{-\varphi }\mathrm{d}m\) and holomorphic on the interior of E in \({\mathbb {C}}\), where \(\varphi \) is a subharmonic function and dm is a measure on E. We give a result where E is the union of a Lipschitz graph and a Carathéodory domain, which we state as a weighted \(L^2\)-version

The Hilbert metric on convex subsets of \({\mathbb {R}}^n\) has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subsets of complex projective spaces (see also L. Dubois in J Lond Math Soc (2) 79(3):719–737, 2009) and give examples of geometric applications. Basic examples include the hyperbolic metric on complex hyperbolic spaces, the n-punctured

In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like $$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t})

For \(\alpha \in {\mathbb {R}}\), let \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) be the corresponding power weight function on \({\mathbb {R}}\) (respectively, the corresponding periodic power weight function, also defined on \({\mathbb {R}} \)). It is known that \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) belongs to \(A_{1}\left( {\mathbb {R}}\right) \) (respectively, \(A_{1}\left( {\mathbb

In this paper, we consider the martingale Hardy spaces defined with the help of the mixed \(L_{\overrightarrow{p}}\)-norm. Five mixed martingale Hardy spaces will be investigated: \(H_{\overrightarrow{p}}^{s}\), \(H_{\overrightarrow{p}}^S\), \(H_{\overrightarrow{p}}^M\), \(\mathcal {P}_{\overrightarrow{p}}\), and \(\mathcal {Q}_{\overrightarrow{p}}\). Several results are proved for these spaces, like

Our primary objective in this paper is to study a class of k-Hessian equations with zero-order and first-order terms in unbounded domains. Among others, we derive Phragmén–Lindelöf and Liouville type results. The zero-order terms involved are required to satisfy Keller–Osserman type conditions.

We develop a new semiclassical calculus in analytic regularity, and apply these techniques to the study of Berezin–Toeplitz quantization in real-analytic regularity. We provide asymptotic formulas for the Bergman projector and Berezin–Toeplitz operators on a compact Kähler manifold. These objects depend on an integer N and we study, in the limit \(N\rightarrow +\infty \), situations in which one can

Abstract We prove local Lipschitz regularity and non-degeneracy estimates for viscosity solutions to one-phase free boundary problems governed by non-homogeneous fully non-linear equations with unbounded ingredients and quadratic growth in the gradient. We use the approach in De Silva (Interfaces Free Bound 13:223–238, 2011) to show that flat or Lipschitz free boundaries are graphs \(C^{1, \alpha }\)

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity

In this paper, we establish some estimates of level sets of holomorphic functions. Relying on obtained estimates we compute some of the weighted log canonical thresholds of plurisubharmonic functions. Finally, we prove the analyticity of the sublevel sets of weighted log canonical thresholds of plurisubharmonic functions.

If X is a projective variety and G is an algebraic group acting algebraically on X, we provide a counter-example to the existence of a G-equivariant extension on the formal semi-universal deformation of X.

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces \(\varSigma \subset {\mathbb {R}}^{n+1}. \) We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume, and exponential volume growth for

This paper may be viewed as a companion paper to Greenblatt (arXiv:1910.04547). In that paper, \(L^2\) Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove \(L^p\) to \(L^q_s\) estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here \(q \ge

This paper is mainly concerned with the existence of extremals for the fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser inequalities. We first establish the existence of extremals of the fractional Caffarelli–Kohn–Nirenberg inequalities in Theorem 1.1. It is also proved that the extremals of this inequality are the ground-state solutions of some fractional p-Laplacian equation. Then, we use

In this paper, we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat